At the atomic scale, solids can be modeled by molecular mechanics or
molecular dynamics, which have become very useful tools in studying
crystal structure, defect dynamics and material properties. However due
to the computational complexity, the application of these models are
usually limited to very small spatial and temporal scales. On the other
hand continuum models, such as elasticity, elastodynamics and their
finite element (or finite volume) formulations, have been widely used to
study processes at much larger scales. But the constitutive relation
involved in these continuum models may be ad hoc, and fails to count for the presence of microstructure in the material. /vskip .1in

\noindent In this talk I will present a multiscale model, which couples the
atomistic and continuum models concurrently. The macroscale model
evolves the system at continuum scale, and the atomistic model, which
only involves a small number of atoms, estimate the constitutive data
and defect structure. I will show the estimate of the modeling error as
well as various applications of this new model.

In this talk, I will survey some well known local regularity results for various classes of minimal submanifolds, and discuss some recent developments in the theory for the class of immersed stable minimal hypersurfaces with branch point singularities. I will also present an existence result for this latter class, and discuss implications of the theory to the global structure of immersed, stable minimal hypersurfaces of low dimension.

Using a method of Howe we obtain explicit bounds on the decay of
matrix coefficients of unitary representations of simply connected groups
of type $A-D-E$. Moreover, we show that the minimal representation has matrix coefficients of slowest decay, and is isolated in the unitary dual of $E_7$ and $E_8$. This is a joint work with Hung Yean Loke.

Advances in the fabrication of small devices have stimulated interest in low-temperature kinetic processes on crystal surfaces. In most experimental situations, nanoscale solid structures decay with time having a lifetime that typically is a large power of the feature size and increases with decreasing temperature. Strategies for skirting the lifetime limitations involve processing at ever-lower temperatures for ever-smaller structure sizes. At temperatures below the roughening transition crystal surfaces evolve via the motion of interacting steps at the nanoscale, and may develop macroscopically flat parts known as ``facets''. The mathematical description of surface evolution at such temperatures is an area of intensive research. \vskip .1in

\noindent The subject of this talk is a continuum description of the morphological relaxation of crystal surfaces in $(2+1)$ dimensions below the roughening temperature by use of $PDEs$. For processes limited by the diffusion of point defects (``adatoms'') on terraces between steps and the attachment and detachment of atoms to and from steps, the surface height profile outside facets is described via a nonlinear, fourth-order $PDE$ that accounts for step line-tension energy $g1$ and step-step repulsive interaction energy $g3$. The $PDE$ is derived from the difference-differential equations for the motion of individual steps, and, alternatively, via a continuum surface free energy. Particular solutions to the $PDE$ are shown to plausibly unify experimental observations of decaying biperiodic surface profiles. To further test the $PDE$, the facet evolution of axisymmetric profiles is treated analytically as a free-boundary problem. For long times, axisymmetric shapes and $g3/g1< 1$, singular perturbation theory is applied for self-similar shapes close to the facet. Scaling laws with $g3/g1$ are derived for the boundary-layer width, maximum slope and facet radius; and a universal $ODE$ for the slope profile is derived and solved uniquely via applying effective boundary conditions at the facet edge. The scaling results compare favorably with numerical solutions of the difference-differential equations for the step positions.

In this talk, we first present an interpolation error estimate in $L^p$ norm ($1\leq p\leq \infty$) for finite element simplicial meshes in any spatial dimensions and then discuss its applications to computational geometry and numerical solution of PDEs. We show that an asymptotically optimal error estimate can be obtained under near optimal meshes. A sufficient condition for a mesh to be nearly optimal is that it is quasi-uniform under a new metric defined by a modified Hessian matrix of the function to be interpolated. We further show such estimates are in fact asymptotically sharp for strictly convex functions. \vskip .1in

\noindent To illustrate the useful of our results, we present an efficient polygonal curve simplification algorithm which improve the computational cost to be optimal. We also briefly discuss some interesting and related problems in the computational geometry, such as sphere covering and optimal polytope approximation of convex bodies. \vskip .1in

\noindent The above interpolation error estimate is useful for approximating functions with anisotropic singularity. Thus it can be applied to convection diffusion problem with small diffusion parameter $\varepsilon$, of which solutions usually present boundary layers or interior layers. For a type of 1-D problems, we have carefully designed a special streamline diffusion finite element method whose discretization error is shown to be uniformly governed by the interpolation error in maximum norm. For problems in multidimensions, we shall discuss some practical issues in the algorithms especially the homotopy with respect to the parameter $\varepsilon$.

Tsen's theorem says roughly that polynomials of low degree in many
variables over the field of meromorphic functions on a compact Riemann
surface always have solutions. I will discuss work with Harris, Starr,
and Mazur which suggests that this theorem is best understood in
connection with the geometry of rational curves.

We survey some recent development of a priori estimates for fully nonlinear elliptic equations without convexity. Such equations as
special Lagrangian equations, Isaacs equations arise from differential
geometry, stochastic control theory. Note that the theory of a priori
estimates for fully nonlinear elliptic equations with convexity condition
is well developed.

In a letter to Andre\' Weil in 1967, Langlands suggested the existence of a relation between two seemingly unrelated mathematical objects: Galois representations and automorphic representations. \vskip .1in

\noindent Since then, the work of many mathematicians has focused on isolating and constructing algebraic varieties whose geometry is supposed to explain the existence of such correspondences. \vskip .1in

\noindent For correspondences defined over a number field, this role is played by Shimura varieties. \vskip .1in

\noindent In my talk I will discuss some aspects of the geometry of the Shimura varieties and how they reflect Langlands\' conjectures.

A noncommutative analogue of a cyclotomic field extension in the realm of central simple algebras is a Schur algebra, namely a finite dimensional central simple algebra $A$ over a field $K$ which is generated over $K$ by a finite subgroup of the multiplicative group of $A$. Its projective version, a projective Schur algebra, is the noncommutative analogue of a radical field extension, namely a finite dimensional central simple algebra $A$ over a field $K$ which is generated over $K$ by a subgroup $G$ of the multiplicative group of $A$ such that $G/K*$ is finite. The talk will focus on projective Schur algebras and the subgroup of the Brauer group that they determine.

I will begin by introducing the Birch and Swinnerton-Dyer conjecture in the context of abelian varieties attached to modular forms, and discuss some of the main results about it. I will then introduce Mazur's notion of visibility of Shafarevich-Tate groups and explain some of the basic facts and theorems. Cremona, Mazur, Agashe, and myself carried out large computations about visibility for modular abelian varieties of level $N$ in $J_0(N)$. These computations addressed the following question: If $A$ is a modular abelian variety of level $N$, how much of the Shafarevich-Tate group of $A$ is modular of level $N$, i.e., visible in $J_0(N)$. The results of these computations suggest that often much of the Shafarevich-Tate group is NOT modular of level $N$. This suggests asking if every element is modular of level $N*m$, for some auxiliary integer $m$, and if so, what can one say about the set of such $m$? I will finish the talk with some new data and thoughts about this last question, which is still very much open.

In this talk, we consider a vector valued nonlinear wave equation of the unknown $u(t, x) \in R^n$. Suppose the energy density of the equation contains a nonlinear potential $V(u)/\epsilon^2$ which achieves its minimal value $0$ on a submanifold $M$ in $R^n$. As $\epsilon$ approaches $0$, i.e. as this potential approaches infinity, we are interested in the convergence of finite energy solutions. Through a multi-scale formal asymptotic expansion involving rapid oscillations, J. Keller and K. Rubinstein (1991) found that the singular limits of those solutions satisfy a hyperboic PDE system. We rigorously justified this convergence procedure and the local well-posedness of this system. In particular, when the initial data is well prepared, the limit system reduces to the wave map equation targeted on $M$. The comparison between the structures of the wave equation and the limit system and a more general picture of Hamiltonian PDEs with strong potentials, will also be briefly discussed.

Random Bernoulli matrices (the entries are i.i.d Bernoulli random variables, taking value 1 and -1 with probability half) are of interest in several areas: combinatorics, mathematical physics, theoretical computer science, to mention a few. On the other hand, very little has been proved about this model. In this talk, I am going to discuss several basic and rather notorious questions and mention few recent results, obtained with Terry Tao. \vskip .1in

\noindent Some detailed proofs will be presented in a subsequent seminar.

The study of various moments of automorphic $L$--functions plays a crucial role in analytic number theory. In this talk, we will formulate precise conjectures for the asymptotics of some classical moment problems and discuss their importance. Then, our main objective will be to outline a proof of the fourth moment conjecture of quadratic Dirichlet $L$--functions over rational function fields.

There exists a polynomial $f(x)$ of degree $n$ with integer coefficients which is irreducible over the rationals but reducible modulo $p$ for all primes $p$ if and only if $n$ is not a prime number. The same result holds with "reducible mod $p$" replaced by "reducible over $Q_p$", and generalizes to arbitrary global fields. (Joint work with Bob Guralnick and Murray Schacher)

Let $p$ be an odd prime number, and consider the twisted affine Fermat curve $$x^p + y^p = \delta$$ with a rational $\delta.$ A well-known theorem of Faltings implies that, for $p\ge 5$, the twisted affine Fermat curve has finitely many rational points. When $\delta = 1$, it has just two (trivial) rational points, thanks to Wiles' proof of Fermat's Last Theorem. In this talk, we will introduce a different idea to study twisted affine Fermat curves. It is based on the connection between the central value of the Hasse-Weil $L$--function associated to the twisted affine Fermat curve and the rank of its Jacobian, as predicted by the Birch and Swinnerton-Dyer conjecture. We will give a sufficient (effective) condition for the twisted affine Fermat curves to have no rational points in terms of the non-vanishing at the central point of certain $L$--functions. Then, using analytic methods, we will conclude that our sufficient condition is
satisfied infinitely often, for any prime $p$. (This is joint work with Y. Tian).

I will talk about a joint work with Sunhi Choi and David Jerison in which we prove that for small positive time, the solution to the one-phase Hele-Shaw flow, starting from an initial surface with small Lipschitz constant, is smooth. Along the way we obtain upper and lower bounds on the speed of the free boundary in terms of the initial data.

For the very large nonlinear systems that arise in meteorology and oceanography, the available observations are not sufficient to initiate a numerical forecasting model. Data assimilation is a technique for combining the measured observations with the model predictions in order to generate accurate estimates of the expected system states - both current and future. Four-dimensional variational assimilation techniques (4D-Var) are attractive because they deliver the best statistically linear unbiased estimate of the model solution given the available observations and their error covariances. The optimal estimates minimize an objective function that measures the mismatch between the model predictions and the observed system states, weighted by the inverse of the covariance matrices. The model equations are treated as strong constraints. \vskip .1in

\noindent Gradient methods are used, typically, to solve the large-scale constrained optimization problem. Currently popular is the ``incremental 4D-Var" procedure, in which a sequence of linearly constrained, convex, quadratic cost functions are minimized. We show here that this procedure approximates a Gauss-Newton method for treating nonlinear least squares problems. We review the known convergence theory for this method and then investigate the effects of approximations on the convergence of the procedure. Specifically we consider the effects of truncating the inner iterations and of using approximate linear constraints in the inner loop. To illustrate the behaviour of the method, we apply incremental 4D-Var schemes to a discrete numerical model of the one-dimensional nonlinear shallow water equations.

I will talk about a new family of generating functions labelled by affine permutations which are affine analogues of Stanley symmetric functions. These functions are related to the reduced words of affine permutations. I'll give some basic properties of these functions and conjecture other properties. As applications, I will show how these functions are related the k-Schur functions of Lapointe, Lascoux and Morse as well as the cylindric Schur functions of Postnikov. Finally I will discuss conjectural relationships between these functions and the cohomology of the affine flag variety.

We are going to investigate the determinant of $M(n)$, a random $(n by n)$ matrix with i.i.d. Bernoulli entries. In particular, we show that with high probability, the determinant (in absolute value) almost reaches Hadamard's upper bound.

This talk continues the survey talk "On random Bernoulli matrices I", but no background is required. (Joint work with T. Tao).

Denoising is a fundamental procedure in image processing and computer vision. The total variation based image denoising model of Rudin, Osher, and Fatemi (ROF) has become one of the standard techniques in the field, and has been applied to a variety of image restoration tasks. In the first part of the talk, I will consider anisotropic versions of the ROF energy and describe properties of their minimizers (joint work with Stanley Osher). In the second part, I will show that certain variants of the ROF model turn out to be convex formulations of non-convex optimization problems encountered in shape denoising and image segmentation applications, thus allowing us to find global minimizers of these non-convex problems via standard convex minimization techniques (joint work with Tony Chan and Mila Nikolova).

It is known since 1954 that every $3$-manifold bounds a $4$-manifold. Thus, for instance, every $3$-manifold has a surgery diagram. There are many proofs of this fact, including several constructive ones, but they do not bound the complexity of the $4$-manifold. Given a $3$-manifold $M$ of complexity $n$, we show how to construct a $4$-manifold bounded by $M$ of complexity $O(n^2)$, for suitable notions of ``complexity". It is an open question whether this quadratic bound can be replaced by a linear bound. \vskip .1in

\noindent The natural setting for this result is shadow surfaces, a representation of $3$- and $4$-manifolds that generalizes many other representations of these manifolds. One consequence of our results is some intriguing connections between the complexity of a shadow representation and the hyperbolic volume of a $3$-manifold. \vskip .1in

\noindent (Joint work with Francesco Costantino.)

We study the problem of testing whether two populations have the same law, by comparing kernel estimators of the two density functions. The proposed test statistic is based on a local empirical likelihood approach. We obtain the asymptotic distribution of the test statistic and propose a
bootstrap approximation to calibrate the test. A simulation study is carried
out, in which the proposed method is compared with two competitors, and a procedure to select the bandwidth parameter is studied. The proposed test can be extended to more than two samples and to multivariate distributions. (joint work with Ricardo Cao)

We consider the complexity of properly learning concept classes, i.e. when the learner must output a hypothesis of the same form as the unknown concept. We present the following new upper and lower bounds on well-known concept classes: \vskip .1in

$\bullet$ \hangindent=29pt We show that unless $NP = RP$, there is no polynomial-time learning algorithm for DNF formulae where the hypothesis is an OR-of-thresholds. Note that as special cases, we show that neither DNF nor OR-of-thresholds are properly learnable unless $NP = RP$.

$\bullet$ \hangindent=29pt Assuming that $NP \not \subseteq DTIME(2^{n^\epsilon})$ for a certain constant $\epsilon <1$ we show that it is not possible to learn size $s$ decision trees by size $s^{k}$ decision trees for any $k \geq 0$. Previous hardness results for learning decision trees held for $k \leq 2$.

$\bullet$ \hangindent=29pt We present the first non-trivial upper bounds on properly learning DNF formulae and decision trees. In particular we show how to learn size $s$ DNF by DNF in time $2^{\tilde{O}(\sqrt{n \log s})}$, and how to learn size $s$ decision trees by decision trees in time $n^{O(\log s)}$.

The global existence/blowup of smooth solutions for the 3D incompressible Euler equations has been one of the most outstanding open problems. By exploring a local geometric property of the vorticity field along one vortex filament, we establish a sharp relationship between the geometric properties of the vorticity field and the maximum vortex stretching. This new understanding reveals new subtleties in the 3D Euler flow, and leads to an improved result of the global existence of the 3D Euler equation under assumptions that are consistent with recent numerical observations.

Algebraic quantization is one of several mathematical counterparts to the physical notion of ``quantization''. It's goal is to describe non-commutative algebras using generators satisfying nice commutation relations. \vskip .1in

\noindent Let $\frak g$ be a Lie algebra. We denote by $U(\frak g)$ its enveloping algebra and by $D(\frak g)$ the quotient skew field (to be defined and explained). \vskip .1in

\noindent $G-K$ theorem claims that the skew field $D(\frak g)$ is generated by $2n = N(N+1)$ elements $p_1,\,\dots ,\,p_n,\, q_1,\,\dots,\,q_n$ satisfying the canonical
commutation relations
$$
[p_i,\,p_j] = [q_i,\,q_j] = 0,\quad [p_i,\,q_j] = \delta_{ij}
$$
and by $N$ elements $z_1,\,\dots ,\,z_N$ which are in the center of $U(\frak g)$. \vskip .1in

\noindent Let $\frak p_N$ be a parabolic subalgebra in $\frak g$ isomorphic to $\frak{gl}(N)\times \Bbb{R}^N$ (the stabilizer of a non-zero row vector in the standard realization). The crucial fact is that $D(\frak p_N)$ contains $2N$ elements satisfying canonical commutation relations such that their centralizer is isomorphic to $D(\frak p_{N-1})$.

Schubert varieties are classical objects of study in algebraic geometry; their study often reduces to easy-to-state combinatorial questions.Gorensteinness is a well-known measure of the "pathology" of the singularities of an algebraic variety. Gorensteinness is a condition that is logically weaker than smoothness but stronger than Cohen-Macaulayness. We present a non-recursive, combinatorial characterization of which Schubert varieties in the flag variety are Gorenstein. Our answer is in terms of generalized permutation pattern avoidance conditions. I'll explain the algebraic geometric and representation (Borel-Weil) theoretic applications of this work. I will also describe further combinatorial questions. \vskip .1in

\noindent This is a joint project with Alexander Woo, see math.AG/0409490.

I will survey what is known concerning the basic behavior of random walks on finitely generated groups in the simple case where the probability measure driving the walk is finitely supported, symmetric and non-degenerate. Here, basic behavior refers to the behavior of the probability of return to the starting point. For those finitely generated groups that can be realized as a closed subgroup of a Lie group, the possible behaviors are classified into three types (classical behaviors). However, many other behaviors are possible even in the class of solvable groups.

We are originally motivated by a problem in genome rearrangement, which is to ask what is the rate of evolution induced by certain large-scale mutations called inversions. If we think of a chromosome as being made up of $n$ markers (the genes), a reversal is a mutation that chooses a segment of the chromosome and flips it around to reverse its order. Traditionally, biologists have studied these questions with parsimony methods. However, it was observed numerically by Bourque and Pevzner that this method provides accurate results only if the number of mutations is small enough. \vskip .1in

\noindent Our work provides a theoretical explanation for this fact. We consider the cleaner but similar problem of random transpositions. Let $\sigma_t$ be the composition of $t$ random uniform transpositions. $\sigma_t$ can be viewed as a random walk on some Cayley graph of the symmetric group. If $D_t$ is the distance between $\sigma_t$ and its starting point, we prove that $D_t$ undergoes a phase transition at critical time $n/2$, from a linear to a sublinear behavior. We will then focus on the consequences of this result for the geometry of the graph. Roughly speaking, our results say that the non-smooth behavior of the random walk can be associated with a change in the Gromov hyperbolicity of the space.

Part of this talk is based on joint work with Rick Durrett.

\parindent=0pt
{\bf SCHEDULE

Thursday, January 27, 2005}

Natural Sciences Building, Room 1205

1:30-2:00 p.m.: Registration

2:05-2:15 p.m.: Welcoming remarks

2:15-3:15 p.m.: Susan Holmes, "Even Odds"

3:15-4:00 p.m.: Refreshment Break

4:00-5:00 p.m.: Ron Graham, "Unfinished Business" \vskip .1in

\parindent=0pt
{\bf Friday, January 28, 2005}

Natural Sciences Building, Room 1205

9:30-10:30 a.m.: Eric Rains, "Random Matrix Perturbations"

10:30-11:15 a.m.: Refreshment Break

11:15 a.m.-12:15 p.m.: Laurent Saloff-Coste, "Some Problems Concerning Gaussian Convolution Semigroups on Compact Groups"

12:15-1:45 p.m.: Lunch Break (on your own)

1:45-2:45 p.m.: Donald Ylvisaker, "Lotteries"

2.45-3:30 p.m.: Refreshment Break

3:30-5:00 p.m.: Open session to discuss connections with Diaconis' work

5:30-6:30 p.m.: No host bar, UCSD Faculty Club

6:30 p.m.: Conference Dinner, UCSD Faculty Club (\$25 per person -- prepayment required by

January 12, 2005). \vskip .1in

\parindent=0pt
{\bf Saturday, January 29, 2005}

Center Hall, Room 115 (note change of room from that on Thursday and Friday)

9:30-10:30 a.m.: Robin Pemantle, "Multivariate asymptotics (why probabilists need to know
math)"

10:30-11:00 a.m.: Refreshment Break

11:00 a.m.-12:00 noon: Barry Mazur, "It is a Story"

12:00 noon-2:00 p.m.: Lunch Break (on your own)

2:00-3:00 p.m.: David Aldous, "Constrained Ising Models: From One-Dimensional Theory to
Storage in Ad Hoc Networks"

3:00-3:30 p.m.: Break

3:30-4:30 p.m.: Persi Diaconis, "A Few of My Favorite Things"

In 1999 Khovanov introduced homology groups for classical links
such that coefficients of the Jones polynomial appear as alternating
sums of the ranks of these groups. In the talk, down to earth adaptation of this construction will be discussed. It is generalized to a class of virtual links (or abstract Gauss diagrams). The original chain complex defining the Khovanov homology is huge, but can be replaced by its deformation retract which is essentially smaller than the original one.

We will consider low bounds for the number of lines meeting given 4 disjoint smooth closed curves in the real projective 3-space in a given cyclic order. Similarly, we estimate the number of circles meeting in a given cyclic order given 6 disjoint smooth closed curves in Euclidean 3-space. The estimations are formulated in terms of linking numbers of the curves and based on a study of a surface swept by projective lines meeting 3 given disjoint smooth closed curves and a surface swept by circles meeting 5 given disjoint smooth closed curves. These results admit generalizations to higher dimensions and more complicated patterns of intersections. Lines and circles can be replaced by configurations of curves of other kinds; linking numbers, by Vassiliev invariants.

Optimization plays a significant role in data mining: the process of analyzing data in order to extract useful patterns and relations such as clusters and classes. Clustering, a major branch of unsupervised machine learning, is amenable to a fruitful application of optimization theory. This leads to effective algorithms such as the k-median clustering algorithm and novel methods for the suppression of irrelevant features in clustering. Classification on the other hand, a mainstay of supervised machine learning and data mining, is an extremely rich field of application for optimization theory and its algorithms. Support vector machines ($SVM$s) constitute the core of modern classification theory. $SVM$s have been extensively used in the last decade, even though they were introduced some forty years ago. Through the use of nonlinear kernel functions, SVMs are powerful tools not only in classification theory but also in function approximation as well as nonconvex function optimization. Kernels allow the introduction of complex nonlinear structures into classifiers and nonlinear function approximation by using linear programming only. \vskip .1in

\noindent Topics such as the above will be presented as well as applications to medicine and bioinformatics.

We consider biased random walk on supercritical percolation clusters in $Z^2$. We show that the random walk is transient and that there are two speed regimes: If the bias is large enough, the random walk has speed zero, while if the bias is small enough, the speed of the random walk is positive. \vskip .1in

\noindent This is based on joint work with Nina Gantert and Yuval Peres.

This talk is about a theory of local newforms for irreducible, admissible representations of the group $GSp(4,F)$, where $F$ is a non-archimedean local field. We will present a joint work with Brooks Roberts, where we proved the existence of a local newform theory analogous to Casselman's theory for $GL(2,F)$. A key feature of our theory is to consider fixed vectors under a family of open-compact subgroups called the paramodular groups. Local newforms provide a link between classical modular forms and automorphic representations. Our theory therefore has applications to Siegel modular forms of degree $2$; this will be discussed in a talk by Brooks Roberts.

This talk is about a long-term project to verify the full Birch and
Swinnerton-Dyer conjecture for every elliptic curve in Cremona's book,
except the $18$ optimal curves of rank $2$. The methods involve Kolyvagin's Euler system of Heegner points, computation, and whatever else one can use. I'll begin with a discussion of the Birch and
Swinnerton-Dyer conjecture for elliptic curves, and methods for
computing the true order of Shafarevich-Tate groups, along with
frustrating difficulties that have yet to be overcome.

A direct combinatorial construction of symmetric group representations,
avoiding a priori use of external concepts (such as Young tableaux), is presented. The construction leads to the study of old and new combinatorial structures; e.g., distinguished intervals in Bruhat posets, shuffles and induction, generalized Young forms and convexity in
Cayley graphs. (This is joint work with Ron Adin and Francesco Brenti)

Special Lagrangian submanifolds are submanifolds of a Ricci-flat Kahler manifold that are both minimal and Lagrangian. We will introduce some basic facts about special Lagrangian geometry and then describe a gluing construction for special Lagrangian submanifolds.

This talk is about joint work with Ralf Schmidt, and follows his talk on local newforms for $GSp(4)$ (in the representation theory seminar on Tues, Feb 1st). In this talk we will discuss a conjecture which describes a theory of Siegel modular newforms of degree two with respect to the paramodular group. Using our local theory, we will indicate how this conjecture follows from a corresponding conjecture in the language of automorphic representations. Finally, we will discuss how the corresponding conjecture is implied by the conjectural detailed structure of the discrete spectrum of $GSp(4)$.

Let ``Fred" be a finite partially ordered set, labelled by the numbers 1, 2, 3, up to $n$ so that, whenever an element $p$ is below an element $q$ in the poset, the label of $p$ (a natural number) is less than the label of $q$. (The permutation $123...n$ is a ``linear extension" of the poset Fred.) \vskip .1 in

\noindent For example, consider the zig-zag-shaped poset with four elements $1,2,3,4$, whose partial ordering is given by $1<3>2<4$. \vskip .1 in

\noindent Look at the linear extensions, that is, the permutations in $S_n$ that respect the partial ordering of Fred, by which we mean the following: If the element labelled $i$ in Fred is below the element labelled $j$ in Fred, then the number $i$ must come before the number $j$ in the permutation. In our example, there are 5 linear extensions: $1234$, $2134$, $1243$, $2143$, and $2413$. \vskip .1 in

\noindent Now take your favourite linear extension and count the number of descents, the number of places where a bigger number comes immediately before a smaller number. In our example, the number of descents in the linear extensions is given by 0, 1, 1, 2, and 1, respectively. Let $H_k$ be the set of linear extensions with $k$ descents and let $h_k$ be the number of such extensions. The zig-zag has $h_0=1$, $h_1=3$, and $h_2=1$. \vskip .1 in

\noindent The $h$-vector in this case---(1,3,1)---is symmetric. Around 1971 Stanley proved that the $h$-vector of a ranked poset is always symmetric. At the 1981 Banff Conference on Ordered Sets, Stanley asked for a bijective proof of this fact. To wit, if a naturally-labelled poset of size $n$ is ranked (all maximal chains have the same number of elements $r+1$), Stanley wanted to find a bijection between the set of linear extensions with $k$ descents and the set of linear extensions with $n-1-r-k$ descents. \vskip .1 in

\noindent We establish such a bijection, thus solving Stanley's problem from 1981.

We observe $n$ points in the unit d-dimensional hypercube. We want to know whether these points are uniformly distributed or whether a small fraction of them are actually concentrated near an object, such as a curve or sheet, which is only known to belong to some regularity class. \vskip .1in

\noindent We argue that this hypothesis testing problem is relevant for the task of detecting structures in galaxy distributions. \vskip .1in

\noindent We consider classes of Holder immersions and study the asymptotic power of the Generalized Likelihood Ratio Test (GLRT), or Scan Statistic, in this setting. \vskip .1in

\noindent We also address computational issues. In turns out that some exact calculations are feasible in some situations, via Dynamic Programming. \vskip .1in

\noindent However, in general, exact computations are known to be $NP$-hard. Approximations are nevertheless possible, at least in theory. Via custom-built graphical structures, it is possible to translate this computational task into some variation of ``The Traveling Salesman Problem", famous in Computer Science and Operations Research. \vskip .1in

\noindent We extend this study to higher order contact, which models recent experiments in Perceptual Psychophysics. \vskip .1in

\noindent Collaborators: David Donoho (Stanford), Xiaoming Huo (Georgia Tech) and Craig Tovey (Georgia Tech).

We consider primitive algebras satisfying additional conditions of locality. These conditions come from the theory of conformal algebras and lead to the description of certain subalgebras of the first Weyl algebra.

We are going to discuss a recent result of Reitzner which shows that the volume of a random polytope generalized by a Poisson point process inside a fixed smooth convex body K satisfies the central limit theorem.

Paramaterized Complexity looks at the time requirements for problems as a function of both instance size and other measures of the difficulty of an instance. For example, while finding an independent set of size $K$ in a graph is an $NP$-complete problem, for each fixed $K$, the problem is clearly solvable in polynomial time through exhaustive search. Parameterized complexity answers such questions as: can this problem can be solved in polynomial-time for $K$ a growing function $K(n)$; and, to what extent can the naive $n^{O(K)}$ algorithm be improved? There have been a number of very interesting connections made recently between parameterized complexity, the exponential time hypothesis (that 3SAT requires time $2^\Omega(n)$) and several other classical notions. I will discuss some of these, giving the necessary background material. If time permits I will also talk about applications to online models.

The goal of this work is to give a new unified axiomatic approach to the representation theory of Coxeter groups and their Hecke algebras. Building upon fundamental works by Young and Kazhdan-Lusztig (and motivated by ideas of Vershik and Ram), we propose a direct combinatorial construction. For simply laced Coxeter groups, this assumption yields explicit simple representation matrices, generalizing the Young forms. For the symmetric groups the resulting representations are completely classified and contain the irreducible ones. \vskip .1in

\noindent This is a joint work with Ron Adin and Francesco Brenti.

This talk will present a generalized version of the First Order Stark Conjecture. After presenting some cases where the generalization is true, we give two cases where the abelian condition is not met. We discuss some possible consequences for the Generalized Stark Question.

Many patterns of cell and tissue organization are specified during development by gradients of morphogens, substances that assign different cell fates at different concentrations. One of the central questions in cell and developmental biology is to identify mechanisms by which the morphogen gradient systems might achieve robustness to ensure reproducible embryonic patterns despite genetic or environmental fluctuations. \vskip .1in

\noindent Recently, through computations and analysis of various bio-chemical models and examination of old and new experimental data, we found a set of new mechanisms for enhancing robustness of cell-cell signaling through non-signaling cell surface molecules (e.g., HSPG). In addition, we examined the roles of diffusive ligands (e.g., Sog) on the formation and robustness of BMP (Bone Morphogenetic Protein) gradients in the Drosophila embryo. In this talk, I shall also discuss some mathematical and computational challenges associated with such study, and present a new class of numerical algorithms for reaction-diffusion equations arising from biological models.

Let $K$ be a polytope of volume one in $R^d$. Choose n random points in $K$ with respect to the uniform distribution. We are going to prove that the key functionals of the convex hull of these points satisify the central limit theorem.

The space $M(n)$ of $n\times n$ matrices is a Poisson manifold. Gelfand-Zeitlin theory gives rise a maximal Poisson commutative algebra of functions on $M(n)$. We show that the corresponding Poisson vector fields are globally integrable and give to a new commutative group $A$ of Poisson automorphisms on M(n). The orbits of $A$ are explicitly given and give rise to new decompositions of $M(n)$. \vskip .1in

\noindent The group $A$ leads to a solution of a classical analogue of the Gelfand-Kirillov conjecture

Assume $v_1,...,v_n$ are unit vectors in $d$-dimensional space whose sum is zero. Can you reorder these vectors as $v_{i_1},...,v_{i_n}$ so that each partial sum $s_k=\sum _{j=1}^k v_{i_j}$ is bounded by a constant that depends only on dimension? The answer is yes, you can. This is going to be the topic of the lecture. The proof is based on linear algebra. Further applications of the proof method will also be presented.

In joint work with Jean-Pierre Wintenberger we have proved Serre's conjectures in some cases. Our work also leads to a strategy to prove the general case of Serre's conjecture. To get the strategy to work, one needs to generalize the modularity lifting results of Wiles and Taylor. These generalizations, while not yet carried out fully, seem relatively accessible. In my talk I will report on this work.

Let $A(n)$ be the minimum area of convex lattice $n$-gons. (Here lattice is the usual lattice of integer points in $R^2$.) G. E. Andrews proved in 1963 that $A(n)>cn^3$ for a suitable positive $c$. We show here that $\lim A(n)/n^3$ exists. Our computations suggest that the value of the limit is very close to $0.0185067\ldots$. It turns out further that the convex lattice $n$-gon $P_n$ with area $A(n)$ has elongated shape: After a suitable lattice preserving affine transformation $P_n$ is very close to the ellipsoid whose halfaxis have length $0.00357n^2$ and $1.656n$. This is joint work with Norihide Tokushige.

Yau and Aubin showed that a compact Kahler manifold with ample canonical bundle admits a Kahler-Einstein metric. The question of whether there exist constant scalar curvature metrics in classes away from the canonical class remains open, and is expected to be related to a notion of stability in the sense of geometric invariant theory. This idea comes from a well-known conjecture of Yau. It will be discussed how the J-flow of Donaldson and X. X. Chen is connected to this problem via the functional known as the Mabuchi energy. We find necessary and sufficient conditions for convergence of the J-flow. And, when the J-flow develops singularities, we show that, in some cases, estimates can be derived away from a subvariety. These can be used to prove, in two dimensions, a weak form of a conjectural remark of Donaldson that if the J-flow does not converge then it should blow up over some curves of negative self-intersection. It will be discussed how these results can be applied to prove properness of the Mabuchi energy for some Kahler classes, and, conjecturally, how they relate to notions of stability due to Tian and Ross-Thomas. (Joint work with J. Song)

Invited Speakers: \vskip .1 in

\noindent Jeff Cheeger (New York University)

\noindent Zhiqin Lu (UCI)

\noindent Bill Minicozzi (Johns Hopkins University)

\noindent Ngaiming Mok (University of Hong Kong)

\noindent Mei-Chi Shaw (University of Notre Dame)

\noindent Brian White (Stanford University) \vskip .1 in

\noindent The 12th SCGAS will be held at the Department of Mathematics of University of California at San Diego on Saturday, February 19, 2005 and Sunday, February 20, 2005. \vskip .1 in

\noindent Registration starts at 9 am Saturday morning (on the 4th floor of AP&M.) The first talk will be at 10:00am and the last talk will be right after lunch of Sunday, to allow for travel. \vskip .1 in

\noindent Graduate students, fresh Ph.D.s and minorities are especially welcome to join our annual seminar. Partial financial support is available. \vskip .1 in

\noindent The Seminar is supported by the NSF.

This talk is based on data from a study of the development of mathematical ideas and ways of reasoning of a focus group of students who did mathematics together throughout their public school and early university years. Findings from the study, now in its seventeenth year, indicate how students' early ideas and images are later elaborated and presented in symbolic expressions of generalized mathematical ideas. From interview data, we hear from participants as young adults how they viewed their mathematical activity in structuring their investigations and justifying their solutions. Video episodes from the study will be shown to illustrate student reasoning over the years. From interviews conducted in their senior-high school and early university years, we learn about particular conditions that were important to participants in doing meaningful mathematics. The Private Universe Project in Mathematics (PUP-Math), produced by the Science Media Group of the Harvard Astrophysical Observatory, illustrates with six, one-hour, videotapes the first twelve years of the study. The tapes and accompanying materials are available at their website, http://www.learner.org. Sample problems from the long-term study are available on the website for the Robert B. Davis Institute for Learning (RBDIL), Rutgers University: www.gse.rutgers.edu/rbdil.

This talk will present a new method for computing the Igusa class polynomials of a primitive quartic $CM$ field. For a primitive quartic $CM$ field, $K$, we compute the Igusa class polynomials modulo $p$ for certain small primes $p$ and then use the Chinese remainder theorem and a conjectural bound on the denominators to construct the class polynomials. We also provide an extension to genus $2$ of the algorithm for determining endomorphism rings of elliptic curves. Our algorithm can be used to generate genus $2$ curves over a finite field with a given zeta function. \vskip .1in

\noindent Joint work with Kirsten Eisentraeger, University of Michigan.

In contrast to Marsden-Weinstein reduction we introduce the notion
of symplectic induction that raises dimensions rather than lowering them. We show that the symplectic induction of certain coadjoint orbits of a Lie group $G$ lead to coadjoint orbits of a larger group. In particular, coadjoint orbits of the five exceptional groups arise from certain coadjoint orbits of classical groups. For example, Tits-Koecher machinary associates to certain Jordan algebras coadjoint orbits of a compact Lie compact classical group. The split forms of $E_{6}$. $E_{7}$ and $E_{8}$ arise, respectively, from the symplectic induction of the coadjoint orbits associated to rank 1 Joradan algebras over the
reals, complexes and quaternions.

The well-known Max Cut problem asks for the largest bipartite subgraph of a graph $G$. This problem has been the subject of extensive research, both from the algorithmic perspective in computer science and the extremal perspective in combinatorics. Let $n$ be the number of vertices and $e$ the number edges of $G$, and let $b(G)$ denote the size of the largest bipartite subgraph of $G$. The extremal part of the Max Cut problem asks to estimate $b(G)$ as a function of $n$ and $e$. This question was first raised almost forty years ago and attracted a lot of attention since then. \vskip .1in

\noindent In this talk we survey some old and recent bounds on the size of the largest bipartite subgraphs for various classes of graphs and obtain some new results. In particular we show that every $K_4$-free graph $G$ with $n$ vertices can be made bipartite by deleting at most $n^2/9$ edges. This proves an old conjecture of Erdos.

Let $S=\{0,1,4,9,...\}$ be the set of squares. There is a unique set $R$ of nonnegative integers such that every positive integer $k$ can be written in the form $s+r (s \in S, r \in R)$ in an even number of ways. Are the only even numbers in $R$ those of the form $2 n^2$? Does the set $R$ have positive density? \vskip .1in

\noindent The more general problem is to develop methods for describing $R$ for a wide variety of initial sets $S$. Specifically, I will talk about sets $S$ that are the range of a quadratic polynomial, the Thue-Morse set, and random sets. I will ask more questions than I am able to provide answers for. \vskip .1in

\noindent This is joint work with Dennis Eichhorn and Joshua N. Cooper.

A locally restricted composition of $n$ is a sequence of positive integers summing to $n$, where the allowed values of each part depend on the values of the $k$ preceding parts. For compositions of $n$ with $k=0$ (all compositions) or with $k=1$ and adjacent parts unequal (Carlitz compositions) the number of compositions, behavior of the largest part and other information (e.g., number of parts) have been studied. I'll discuss \vskip .1in

\noindent (a) similar results we have obtained for $k=1$ when the restriction is on the part difference and \vskip .1in

\noindent (b) current research on the general case.\vskip .1in

\noindent This is joint work with Rod Canfield and Bill Helton.

In this talk we will dicuss an extension of the classical $F$. and $M$. Riesz theorem for holomorphic functions to the continuous solutions of real analytic involutive structures

Ramanujan graphs are regular graphs whose nontrivial eigenvalues are small in absolute value. They have good expanding properties, and hence have broad applications in computer science. In particular, they give rise to very efficient communication networks. In this survey talk we shall discuss different explicit constructions of such graphs using group theory and number theory, and show how character sum estimates are used to estimate the eigenvalues. The eigenvalues of Ramanujan graphs have close ties to Fourier coefficients of automorphic forms. We shall give a flavor of this connection.

Recently, we have introduced a Bergman function for any strongly pseudoconvex variety with only isolated singularities. This is a global biholomorphic invariant for strongly pseudoconvex variety. We shall show that our Bergman function is a complete invariant of a family of strongly pseudoconvex varieties.

A volatility derivative is a financial contract whose payoff depends on the realized volatility of some other asset. \vskip .1in

\noindent The market for volatility derivatives is nascent but emerging. Recent theoretical breakthroughs have made it possible to robustly replicate the payoffs on a wide variety of volatility derivatives. We survey these developments and provide empirical evidence on the magnitude of the variance risk premium embedded in the prices of standard options.

Many physical problems are governed by constrained hyperbolic systems including Maxwell's equations of electromagnetism and Einstein's equations of gravitation. Numerically solving these systems accurately requires the development of special methods to guarantee constraint satisfaction. This talk will discuss various ways to augment standard and least-squares finite element methods to aid in this goal.

Picard's Little Theorem (1879) says that an entire function whose range omits two (distinct, finite) values must be constant. We survey some of the developments in this area of the past 125 years, explain how they connect with function theory on finite domains (such as the disc), and wind up with a new approach to some longstanding open problems.

In this talk we shall survey the problem of analytic regularity of solutions for ``sums of squares" operators satisfying Hormander condition. We shall also present some new positive results obtained in collaboration with N. Hanges.

This will be a conclusion of last term's seminar on rational homotopy theory. I will try to explain what A-infinity, L-infinity, and C-infinity algebras (also known as homotopy associative, Lie and commutative algebras) are, and why they are a useful framework for the theorems of rational homotopy theory.

We prove that any Reinhardt domain of two-dimensional complex space with a $C^2$ boundary of constant Levi curvature $K$ must be a sphere of radius $1/K$.

Galois groups with finite ramification over $Q$ are the "fundamental groups" of number theory. Most of what we know about them stems from their action on certain $p$-adic vector spaces. In this talk, I will describe their action on certain trees, which promises to throw a different kind of light on fundamental groups. Let $K$ be a number field and $f(x)$ in $K[x]$ be a polynomial whose critical points are preperiodic under iteration of $f$. Then every $K$-rational specialization of the tower of iterates of $f:P^1 --> P^1$ is finitely ramified. This leads to a number of open problems about the nature of the corresponding "iterated monodromy" representations of the Galois group of $K$. This is joint work with Christian Maire (Toulouse) and Wayne Aitken (CSU San Marcos).

I will discuss a compactness result for various classes of Riemannian metrics in dimension four; in particular the method applies to anti-self-dual metrics, Kaehler metrics with constant scalar curvature, and metrics with harmonic curvature. With certain geometric assumptions, the moduli space can be compactified by adding metrics with orbifold singularities. Similar results were obtained previously for Einstein metrics, but our analysis differs substantially from the Einstein case in that we do not assume any pointwise Ricci curvature bound. This is joint work with Gang Tian.

The number of left to right minima of a permutation is generalized to
Coxeter (and closely related) groups, via an interpretation as the number of ``long factors" in canonical expressions of elements in the group. This statistic is used to determine a covering map, which `lifts' identities on the symmetric group $S_n$ to the alternating group $A_{n+1}$. The covering map is then extended to `lift' known identities on $S_n$ to new identities on $S_{n+q-1}$ for every positive integer $q$, thus yielding $q$-analogues of the known $S_n$ identities. Equi-distribution identities on certain families of pattern avoiding permutations follow. The cardinalities of subsets of permutations avoiding these patterns are given by extended Stirling and Bell numbers. The dual systems (determined by matrix inversion) have combinatorial realizations via statistics on colored permutations.
Joint with Amitai Regev.

We consider the problem of estimating a vector of location parameters restricted to a cone in the presence of an unknown scale parameter. Examples include estimating the location vector when the parameter ordering is known (or partially known), or when it is known that all means are positive. A standard estimator when sampling from a multivariate normal distribution is the MLE. In the unrestricted case, this estimator is the vector of sample means, which is dominated (in three and higher dimensions) by the James-Stein estimator among others. We study improvements of the James-Stein type for the restricted parameter case in the general setting of a spherical symmetry location family. The development is based on a general result that shows how improved estimators of location in the multivariate normal case with known scale can be extended to give improved estimators in the general spherically symmetric case with unknown scale.

We consider permutations which are $1324$ and $\overline{2143}$ avoiding, where $\overline{2143}$ avoiding means that it is $2143$ avoiding with the additional Bruhat restriction $\{2\leftrightarrow3\}$. In particular, for every permutation $\pi$ we will construct a linear map $L_\pi$ and a labeled graph $G_\pi$ and will show that the following three conditions are equivalent: $\pi$ is $1324$ and $\overline{2143}$ avoiding; $L_\pi$ is onto; $G_\pi$ is a forest. If time allows we will give a constructive proof showing that the $n$th Catalan number gives a lower bound for the number of such permutations in $S_n$. \vskip .1in

\noindent This answers a conjecture of Woo and Yong, which shows that such permutations characterize which Schubert varieties are factorial.

We consider the estimation of a large number of Garch models, of the order of several hundreds. Our interest lies in the identification of common structures in the volatility dynamics of the univariate
(or even low-dimensional multivariate) time series. To do so, we classify the series in an unknown number of clusters. Within a cluster, the series share the same model and the same parameters. Each cluster contains therefore similar series. We do not know a priori which series belongs to which cluster. The model is a finite mixture of distributions, where the component weights are unknown parameters and each component distribution has its own conditional mean and variance. Inference is done by the Bayesian approach, using data augmentation techniques. Simulations and an illustration using data on US stocks are provided.

In my book with V. Guillemin ``The Spectral Theory of Toeplitz Operators'', we defined Toeplitz projectors on a compact contact manifold, which are analogues of the Szego projector on a strictly pseudo-convex boundary. If $X$ is a contact manifold, the kernel of a Toeplitz projector, just as the Szego kernel, has a holonomic singularity including a polar and a logarithmic term, and its trace is well defined. \vskip .1in

\noindent We show that the trace of a Toeplitz operator only depends on the contact structure of $X$. If $X$ is the three sphere equipped with any contact form, this invariant vanishes (Y. Eliashberg has shown there are many nonisomorphic such contact forms); this makes it not unlikely that the trace vanishes identically.

Is there any simple way to describe walks, starting from the origin, in the positive quadrant between lattice points, each in a direction N, S, E or W, in terms of pattern avoidance in permutations? Does there exist a pattern with the property that the number of $n$-permutations avoiding it equals the number of $n$-permutations having cycles of length at most $k$?

To answer these questions and to provide other interesting facts, we introduce the concept of partially ordered patterns (POPs), which generalize the generalized patterns introduced by Babson and Steingrimsson in 2000. We also discuss some of the results on POPs in the literature and suggest few problems to solve.

In this talk we will introduce a boundary value problem for Ricci flow and discuss the short-time, long-time existence and long time behavior.

A problem formulated by I.M. Gelfand in the 1950s is to reconstruct
the metric tensor of a compact Riemannian manifold with boundary, from data on the spectrum of its Laplace operator, with the Neumann boundary condition, and the behavior at the boundary of the normalized eigenfunctions. \vskip .1in

\noindent The first ingredient that goes into the resolution of such an ``inverse problem'' is a uniqueness theorem, but further work beyond establishing uniqueness is required. This arises because of the ``ill posedness'' associated with inverse problems. That is, various ``large'' perturbations of the unknown region can yield small perturbations of the observed data. The key to stabilizing an ill-posed inverse problem is to have appropriate a priori knowledge of the unknown domain so that a search for the solution can be confined to a ``compact'' family of possible domains. In this context, the suitable notion is that of Gromov compactness, and one key to stabilizing Gelfand's inverse problem involves establishing such compactness. This is done under fairly weak hypotheses on the geometry of the unknown domain, including bounds on its curvature (to be precise, its Ricci tensor) and on the curvature of its boundary. Estimates for solutions to a naturally occuring elliptic boundary value problem for the metric tensor play a central role. \vskip .1in

\noindent The speaker will discuss some of these matters, which have been treated in joint work with M. Anderson, A. Katsuda, Y. Kurylev, and M. Lassas.

Many physical problems are governed by constrained hyperbolic systems including Maxwell's equations of electromagnetism and Einstein's equations of gravitation. Numerically solving these systems accurately requires the development of special methods to guarantee constraint satisfaction. This talk will discuss various ways to augment standard and least-squares finite element methods to aid in this goal.

The demand for new methodology for the simultaneous testing of many
hypotheses is driven by numerous modern applications in genomics, imaging, astronomy, finance, etc. A key feature of these problems
is their high dimensionality, meaning tests of thousands of hypotheses may be considered simultaneously. The problem of controlling measures of error are particularly important in order to counter the effects of ``data snooping'' (or ``data mining''). The goal of the lectures is develop some classes of techniques that effectively deal with the problem of multiplicity. \vskip .1in

\noindent A classical approach to dealing with multiplicity is to require
decision rules that control the familywise error rate (FWER). We will develop some stepwise procedures that offer either finite sample or asymptotic control. It will be shown that the problem of controlling the FWER for stepdown tests can be reduced to the problem of controlling the Type 1 error of single testing; as such, the ``subset pivotality'' often used in the literature are unnecessary. The use of resampling methods can provide improved ability to detect false hypotheses by implicitly estimating the dependence structure of the test statistics. Some optimality results will be presented, as well as some applications to set estimation problems in underidentified econometric models. \vskip .1 in

\noindent In many applications, particularly if the number of hypotheses is large, one might be willing to tolerate more than one false rejections if
the number of such cases is controlled. Therefore, we will replace control of the FWER by the $k$-FWER, the probability of $k$ or more false rejections. We will also consider the false discovery proportion (FDP) defined as the number of false rejections divided by the total number of rejections. The false discovery rate of Benjamini Hochberg (1995) controls $E(FDP)$. We will also discuss suitable methods to control the tail probability $P \{ FDP > \gamma \}$ for any $\gamma$.
Resampling, subsampling, and permutation methods can yield valid procedures.

The demand for new methodology for the simultaneous testing of many
hypotheses is driven by numerous modern applications in genomics, imaging, astronomy, finance, etc. A key feature of these problems
is their high dimensionality, meaning tests of thousands of hypotheses may be considered simultaneously. The problem of controlling measures of error are particularly important in order to counter the effects of ``data snooping'' (or ``data mining''). The goal of the lectures is develop some classes of techniques that effectively deal with the problem of multiplicity. \vskip .1in

\noindent A classical approach to dealing with multiplicity is to require
decision rules that control the familywise error rate (FWER). We will develop some stepwise procedures that offer either finite sample or asymptotic control. It will be shown that the problem of controlling the FWER for stepdown tests can be reduced to the problem of controlling the Type 1 error of single testing; as such, the ``subset pivotality'' often used in the literature are unnecessary. The use of resampling methods can provide improved ability to detect false hypotheses by implicitly estimating the dependence structure of the test statistics. Some optimality results will be presented, as well as some applications to set estimation problems in underidentified econometric models. \vskip .1 in

\noindent In many applications, particularly if the number of hypotheses is large, one might be willing to tolerate more than one false rejections if
the number of such cases is controlled. Therefore, we will replace control of the FWER by the $k$-FWER, the probability of $k$ or more false rejections. We will also consider the false discovery proportion (FDP) defined as the number of false rejections divided by the total number of rejections. The false discovery rate of Benjamini Hochberg (1995) controls $E(FDP)$. We will also discuss suitable methods to control the tail probability $P \{ FDP > \gamma \}$ for any $\gamma$.
Resampling, subsampling, and permutation methods can yield valid procedures.

DPLL (for Davis, Putnam, Logemann, and Loveland) algorithms
form the largest family of contemporary algorithms for SAT
(the propositional satisfiability problem) and are widely
used in applications. While the behavior of DPLL algorithms
on unsatisfiable formulas has been extensively investigated
(by studying the refutation size in Resolution proof system),
little research was done towards proving lower bounds on
satisfiable instances. We fill this gap by constructing
satisfiable formulas which are provably difficult for
a wide class of DPLL algorithms.

In the second part of the talk I will survey some lower bounds on
other restricted computational models, which include a
generalization of our results for backtrack SAT algorithms
and proving integrality gaps for Lovasz-Schrijver hierarchy
for the linear relaxations of a (hypergraph) vertex cover.

The talk is based on joint work with Edward Hirsch, Dmitry Itsykson,
Allan Borodin, Joshua Buresh-Oppenheim,
Russell Impagliazzo, Avner Magen, Toniann Pitassi,
Sanjeev Arora, and Iannis Tourlakis

To every genus-2 curve C over a finite field $F_q$, one can associate the
characteristic polynomials of the Frobenius endomorphism acting on the
Jacobian of C. This polynomial --- also known as the *Weil polynomial* of C --- is of the form
$$x^4 + a*x^3 + b*x^2 + a*q*x + q^2$$
where a and b are integers.

We will use the Brauer relations, applied to a certain biquadratic number
field, to show that no curve over $F_q$ gives rise to the Weil polynomial
with $a = 0$ and $b = 2 - 2*q$. The same method can be used to show that the Weil polynomials with $a = 0$ and with other values of b (subject to certain elementary restrictions) *do* occur; this was carried out by Daniel Maisner.

These results, combined with earlier work, allow us to easily determine
the Weil polynomials that arise from genus-2 curves with ordinary
Jacobians.

We trace the theory for finite sample signal extraction, moving from stationary signal and noise, to nonstationary signal and stationary noise, to nonstationary signal and noise. These concepts are developed through some key examples, including the airline model. Our final goal is to present exact formulas for finite sample signal extraction estimates and mean squared errors, within a fairly general class of unobserved component models.

The demand for new methodology for the simultaneous testing of many
hypotheses is driven by numerous modern applications in genomics, imaging, astronomy, finance, etc. A key feature of these problems
is their high dimensionality, meaning tests of thousands of hypotheses may be considered simultaneously. The problem of controlling measures of error are particularly important in order to counter the effects of ``data snooping'' (or ``data mining''). The goal of the lectures is develop some classes of techniques that effectively deal with the problem of multiplicity. \vskip .1in

\noindent A classical approach to dealing with multiplicity is to require
decision rules that control the familywise error rate (FWER). We will develop some stepwise procedures that offer either finite sample or asymptotic control. It will be shown that the problem of controlling the FWER for stepdown tests can be reduced to the problem of controlling the Type 1 error of single testing; as such, the ``subset pivotality'' often used in the literature are unnecessary. The use of resampling methods can provide improved ability to detect false hypotheses by implicitly estimating the dependence structure of the test statistics. Some optimality results will be presented, as well as some applications to set estimation problems in underidentified econometric models. \vskip .1 in

\noindent In many applications, particularly if the number of hypotheses is large, one might be willing to tolerate more than one false rejections if
the number of such cases is controlled. Therefore, we will replace control of the FWER by the $k$-FWER, the probability of $k$ or more false rejections. We will also consider the false discovery proportion (FDP) defined as the number of false rejections divided by the total number of rejections. The false discovery rate of Benjamini Hochberg (1995) controls $E(FDP)$. We will also discuss suitable methods to control the tail probability $P \{ FDP > \gamma \}$ for any $\gamma$. Resampling, subsampling, and permutation methods can yield valid procedures.

JOINT SEMINAR WITH UCI HELD at UC Irvine, MSTB 254

THIS SEMINAR IS NOT BEING HELD AT UCSD BUT AT UC IRVINE

One of the most frequently studied problems in the context of information
dissemination in communication networks is the broadcasting problem.
We propose here several time efficient, centralized as well as fully distributed procedures for the broadcasting problem in random radio networks. In particular we show how to perform a centralized broadcast
in a random graph $G_p=(V,E)$ of size $n=|V|$ and expected average degree $d=pn$ in time $O(\ln n/\ln d+\ln d).$ Later we present a randomized
distributed broadcasting algorithm with the running time $O(\ln n).$
In both cases we show that the presented algorithms are asymptotically optimal by deriving lower bounds on the complexity of radio broadcasting in random graphs. In these proofs we determine some structural properties in random graphs which may be of independent interest. We should note here that the results of this paper hold with probability $1-o(1/n)$.

This is a joint work with Leszek Gasieniec from the University of Liverpool.

JOINT SEMINAR WITH UCI HELD at UC Irvine, MSTB 254

THIS SEMINAR IS NOT BEING HELD AT UCSD BUT AT UC IRVINE

Abstract: Given a cuspidal representation of a symplectic group
the theta correspondence method returns an automorphic
representation of an orthogonal group. It is possible to decide
whether the image is cuspidal by considering a tower of theta
correspondences. To determine the first place when the image
of a given representation is nonzero is the fundamental problem
of the theta correspondence method. We give upper and lower
bounds for it in terms of the wave front set of the initial
representation.

This is a joint work with D. Ginzburg

The Dehn function of a finite presentation of a group $G = \langle A \mid R \rangle$ gives the least upper bound for the number of relators that must be applied to a word $w \in G$ that is trivial i.e., $w =_G 1$, in order to reduce $w$ to the trivial word. Up to a natural equivalence on functions, the Dehn function is a quasi-isometry invariant of the group $G$. The study of Dehn functions gained importance after Gromov's seminal theorem: a finitely presented group $G$ has sub-quadratic Dehn function if and only if $G$ has linear Dehn function if and only if the Cayley graph of $G$ is $\delta$-hyperbolic.

In this talk we will outline the various definitions and ideas in the subject. Then we will address the basic question: what possible functions can arise as Dehn functions of finitely presented groups? We will outline the construction of the so-called {\it snowflake groups} which give many new examples of Dehn functions. The results presented are joint work with Noel Brady, Martin Bridson and Max Forester.

The Dehn function of a finite presentation of a group $G = \langle A \mid R \rangle$ gives the least upper bound for the number of relators that must be applied to a word $w \in G$ that is trivial i.e., $w =_G 1$, in order to reduce $w$ to the trivial word. Up to a natural equivalence on functions, the Dehn function is a quasi-isometry invariant of the group $G$. The study of Dehn functions gained importance after Gromov's seminal theorem: a finitely presented group $G$ has sub-quadratic Dehn function if and only if $G$ has linear Dehn function if and only if the Cayley graph of $G$ is $\delta$-hyperbolic. \vskip .1in

\noindent In this talk we will outline the various definitions and ideas in the subject. Then we will address the basic question: what possible functions can arise as Dehn functions of finitely presented groups? We will outline the construction of the so-called {\it snowflake groups} which give many new examples of Dehn functions. The results presented are joint work with Noel Brady, Martin Bridson and Max Forester.

Given a fixed convex body $K$, choose $n$ points randomly on the boundary of $K$ according to a ``uniform" distribution, and call the convex hull of these $n$ points a Random Inscribing Polytope. We will discuss some recent asymptotic results on the volume of these polytopes. Namely, we will prove a lower bound on the variance, some concentration result, and an analogue of the central limit theorem.
This is joint work with Ross Richardson and Van Vu.

A $-1/1$-polytope in $\Bbb{R}^n$ is, by definition, the convex hull of a
subset of the vertices of the unit cube $[-1,1]^n.$ Let $g(n)$ denote the
maximal number of facets such a polytope can have. Fukuda and Ziegler asked how $g(n)$ grows with $n.$ Fleiner, Kaibel and Rote have shown that $g(n)\leq 30 (n-2)!$ for sufficiently large $n,$ and this is the best known upper bound on $g(n).$ In a major advancement, B\'{a}r\'{a}ny and P\'{o}r obtained the lower bound $g(n)\geq (c n/\ln n)^{n/4},$ where $c>0$ is a universal constant, and gave heuristics of why one might expect $g(n)$ to be of the order of $n^{n/2}$ (this is believed to be the right order of magnitude for $g(n)$). We show that $g(n)\geq (cn/\ln n)^{n/2},$ with $c>0$ a universal constant.

The talk will start by explaining certain episodes on a way from the
halfspace depth in multivariate location (``the Tukey depth") through depth in general data-analytic situations (models?) toward the psychedelic experience of a new notion of depth in the location-scale model, Location-Scale Depth, and its most tractable version, the Student
depth. The latter has a couple of entertaining theoretical and computational properties, stemming from the fact that it is nothing
but the bivariate halfspace depth interpreted in the Poincar\'e plane
model of the Lobachevski geometry - in particular, invariance with
respect to the M\"obius group and favorable time complexities of
algorithms. The practical implications involve a new fancy location-scale typical value, the Student median, as well as somewhat extravagant graphical tool for exploring distributional properties of univariate samples, a sort of cousin to the quantile-quantile plot. \vskip .1in

\noindent However, perhaps more than those particular accomplishments it may be worthy to note potential new views on data and questions that the process raises: the role of invariance (if any) in data analyses, whether there can be such a thing as median in sophisticated situations, and, more generally, whether classical rank-based nonparametrics can be elevated beyond their traditional (essentially) univariate setting; and also, whether we may be missing yet some brave new data worlds in the realm of non-Euclidean geometry.

Time course studies of gene expression are essential in biomedical research to understand biological phenomena that evolve in a temporal fashion. Microarray technologies make it possible to study genome-wide temporal differences in gene expression profiles between different experimental conditions. In this paper, we introduce a functional hierarchical model for detecting emporally differentially expressed (TDE) genes between two experimental conditions for cross-sectional designs, where the gene expression profiles are treated as functional data and are modeled by basis function expansions. Monte Carlo EM algorithm is developed for estimating both the gene-specific parameters and the hyperparameters in the second level of the modeling. We use a direct posterior probability approach to bound the rate of false discovery at a pre-specified level. We evaluate the methods by simulations and application to a microarray time course gene expression data on C. elegans developmental processes. Simulation results suggested that the procedure performed better than the two-way ANOVA in identifying TDE genes, resulting in both higher sensitivity and specificity. Genes identified from the C. elegans developmental data set showed clear patterns of changes between the two experimental conditions.

Let $M_n$ be a random matrix whose entries are i.i.d Bernoulli random variables and $Q_n$ be a random symmetric matrix whose upper diagonal entries are i.i.d Bernoulli random variables. We prove: \vskip .1in

\noindent 1. P ($Q_n$ is singular)$= 0$($n^{-1/8+ \epsilon }$) (Costello, Tao and Vu). \vskip .1in

\noindent 2. P($M_n$ is singular)$=0$ (($3/4$)$^n$) (Tao and Vu). \vskip .1in

\noindent The first result answers a question of B. Weiss, posed in the early 1990s. The second improved an earlier bound ($.999^n$) of Kahn, Komlos and Szemeredi from 1995. \vskip .1in

\noindent From 2:00 p.m. to 2:30 p.m., Costello will talk about ($1$). Vu will continue from 3 p.m. to 3:30 p.m. with the beginning of the proof of ($2$). The rest of the proof comes next week.

In seminal paper, Gessel-Reutenauer construct a truly remarkable bijection between words in the free monoid over an infinite alphabet and multisets of primitive necklaces. Using this bijection Gessel-Reutenauer derive a variety of beautiful symmetric function identities with deep implications in permutation enumeration. In this talk we review the Gessel-Reutenauer results, derive some applications
and state some challenging combinatorial problems.

This talk is a report on joint work with S. Molchanov. One set of results involves the behavior of sums of the form $\sum_{i=1}^{N(n)}\exp{\beta(\sum_{i=1}^n V_{ij}}$ where $V_{ij}$ are iid random variables. We identify rates of growth of $N(n)$ which give stable limit laws for properly normalized and centered sums, and other rates which give rise to a Central Limit Theorem holding for these sums. Another aspect of the work, which is related, considers sums of the form $\sum_{x \in Q_{L(n)}} u(n,x)$, where the function $u(t,x)$ is the solution of parabolic Anderson model and $Q_L$ is a box in $Z^d$ of radius L. Again the limit behavior of such sums depends on the rate of growth of $L(n)$. The results for this setting give a relation between intermittency and the so-called quenched-to-annealed transition.

One of fundamental results on local geometry of nondegenerate real hypersurfaces in complex space is the construction of normal forms developed by S. S. Chern and J. Moser. We will discuss a generalization of this construction to Levi degenerate hypersurfaces of finite type in $\Bbb C^2$. As a main consequence, using a convergence result of M. S. Baouendi, P. Ebenfelt and L. P. Rothschild, we obtain an explicit solution to the problem of local biholomorphic equivalence. Another application gives precise information on the dimension of the stability group. We will also mention some open problems for hypersurfaces in higher dimensions.

This is a history talk about the three Lehmers... D. N. Lehmer, D. H. Lehmer, and Emma Lehmer and the computing world they created that was centered at Berkeley. \vskip .1in

\noindent One of their main mathematical interests throughout the period 1900 - 1975 was factoring integers or proving that a prime is a prime. (No, Gertrude Stein did Not say ``A prime is a prime is a prime".) \vskip .1in

\noindent A dramatic moment came in 1970 when factoring a large, composite integer N by exclusion. The method considered best for factoring large integers for at least 150 years and which was the favored method used by the Lehmers on their sieves, was shown to be much less powerful than finding non-trivial $x$ and $y$ in the congruence $x2 \equiv y2$ (mod $N$) on a computer. The latter is the basic method that has been used and improved on ever since to carry out large-scale factoring.

We discuss a refinement of Rubin's integral version of Stark's Conjecture for abelian $L$-functions of arbitrary order of vanishing at $s=0$. This generalizes the $v$-adic refinement of the abelian, order of vanishing $1$ integral Stark Conjecture formulated by Gross in the 1980s, and predicts a link between special values of derivatives of $p$-adic and global $L$--functions. Time permitting, we will also show how our refinement relates to a recent strengthening of Gross's Conjecture due to Tate.

The Bergman kernel in complex analysis has always appeared
to be an abstractly defined thing having to do with infinite sums of orthogonal functions, or with solutions to a PDE, or with extremal problems, or with the Riesz Representation Theorem. I will explain a way of looking at the kernel that connects it to algebra, and I will show how it can be used to convert problems in conformal mapping to problems in algebraic geometry. Would you believe that the Bergman
kernel $K(z,w)$ of a multiply connected domain in the plane is just a rational combination of two explicit and simple analytic functions of one complex variable?

The problem of approximating m data points $(x_i, y_i)$ in $R^n+1$, with a quadratic function $q(x,p)$ with s parameters $s<=m$, is considered. The parameter vector $p$ in $R^s$ is determined so as to satisfy three conditions: (1) $q(x,p)$ must underestimate all m data points, i.e. $q(x_i,p)<=y_i$, $i=1,....m$. (2) The error of approximation is to be minimized in the $L1$ norm. (3) The eigenvalues of $H$ are to satisfy specified lower and upper bounds, where $H$ is the Hessian of $q(x,p)$ with respect to $x$. Approximation of the data by a sum of negative Gaussians is also considered.

In early 2005 D. Felix conjectured that, were it to ever happen, this would be 'the world's most awesomest advancement talk, ever'. This question has already been the focus of much current research, and if true, would have far reaching consequences in the theory of Ph.D. getting. We will discuss the history of the problem, outline some of the known special cases, and describe some recent progress on the general problem. Time permitting, we will also present some interesting potential applications to hyper-awesome theory.

Which linear differential operators preserve harmonic functions? Even on Euclidean space, this is a deceptively simple question. The answer may be expressed in terms of conformal geometry and the AdS/CFT correspondence.

Recent years have seen an increased interest in congestion control for the internet. Typically, congestion control algorithms have been designed based upon fluid models of communication networks. In this talk we propose an alternative random-matrix model of communication networks that use $TCP$-like congestion control algorithms. We show that essential features of such networks may be captured using this model in a simple manner using basic results from positive linear systems. These results suggest several strategies for designing congestion control strategies.

Paul Erdos asked the following question: How many edges can a subgraph of the $n$-hypercube, $Q_n$, have that contains no $4$-cycle? The conjectured bound is $({1 \over 2}+o(1))e(Q_n)$ where $e(Q_n)$ denotes the number of edges of the hypercube. The best known result is due to Chung who also considered the more general question of $2k$ cycles where $k$ is even. More recently Alon, Radoicic, Sudakov, and Vondrak extended Chung's results to get Ramsey type results for $2k$ cycles where $k$ is odd. \vskip .1in
\noindent In the talk we will review the developments of the problem and where it stands now.

Broadcasting algorithms have a various range of application in differentfields of computer science. In this paper we study the communicationcomplexity of simple randomized broadcasting algorithms in random-like networks. We start our analysis on the classical random graph model, i.e., a graph $G_p$ with $n$ nodes is constructed by letting any two arbitrary nodes be connected with probability $p$. First, we state some combinatorial results which are necessary for our main study. \vskip .1in

\noindent Then, we consider a modified version of the random phone call model intorduced by Karp et al., and show that the communication complexity of the corresponding broadcasting algorithm is bounded by an asymptotically optimal value in almost all connected random graphs. More precisely, we show that if $p$ exceeds some threshold, then we are able to broadcast any information $r$ in a random graph $G_p$ of size $n$ within $O(\log n)$ steps by using at most $O(n \max\{ \log \log n, \log n/\log d\})$ transmissions of $r$, where $d=pn$ denotes the expected average degree in $G_p$. This result holds with probability $1- o(1/n^c)$, where $c$ is a constant, even if $n$ and $d$ are unknown to the nodes of the graph. \vskip .1in

\noindent The main result of the paper can be extended to other random graph models as well. A slight modification of our algorithm results in asymptotically optimal communication overhead for certain types of the random power law graphs defined by Chung and Lu. It is worth mentioning that such random power law graphs are often used to model large scale real world networks such as the Internet. \vskip .1in

\noindent The algorithm we present in this paper is simple, scalable, and robust. It can efficiently handle restricted communication failures and certain changes in the size of the network. In addition, our methods and the auxiliary combinatorial results might be useful for further investigation on this field.

In this talk, I will present several classical results and remarkable new developments in Ramsey theory on the integers and reals. A system of linear equations is called partition $k$-regular if for every $k$-coloring of the positive integers, there exists a monochromatic solution to the given system of linear equations. Generalizing classical theorems of Schur and van der Waerden, Richard Rado classified those systems of linear equations that are partition $k$-regular for all positive integers $k$ in his famous 1933 dissertation {\it Studien zur Kombinatorik}. Rado further conjectured in his dissertation that there exists a function $K:N \to N$ such that if a linear equation $a_1x_1+ \cdots +a_nx_n=b$ is partition $K(n)$-regular, then it is partition $k$-regular for all positive integers k. D. Kleitman and I recently settled the first nontrivial case of this conjecture, known as Rado's Boundedness Conjecture. In particular, if $a$, $b$, $c$, and $d$ are integers, and if every $36$-coloring of the positive integers contains a monochromatic solution to $ax+by+cz=d$, then every finite coloring of the positive integers must have a monochromatic solution to $ax+by+cz=d$. The degree of regularity of an equation $a_1x_1+ \cdots +a_nx_n=0$ over $R$ is the largest positive integer $r$ (if it exists) such that every $r$-coloring of $R-{0}$ has a monochromatic solution to $a_1x_1+ \cdots +a_nx_n=0$. In 1943, Rado extended the results of his dissertation by classifying those equations that have finite degree of regularity over $R$. Motivated by recent results of S.\ Shelah and A.\ Soifer, R.\ Radoi\v{c}i\'{c} and I found equations whose degree of regularity over $R$ is dependent on the axioms for set theory. For example, in the Zermelo-Fraenkel-Choice (ZFC) system of axioms, we show there exists a $3$-coloring of the nonzero real numbers without a monochromatic solution to $x+2y=4z$. However, in a consistent system of axioms with limited choice studied by R.\ Solovay in 1970, every $3$-coloring of the nonzero real numbers contains a monochromatic solution to $x+2y=4z$. Time permitting, I will discuss applications to several related problems.

A one-dimensional lattice gas is a system of particles, each attached to one of a finite number of binding sites around a circle. Time advances in discrete steps; at each time step the particles may hop from one site to a neighboring one, and nearby particles may interact with one another. I will discuss a novel lattice gas model in which the interaction of adjacent particles can create new binding sites or remove existing ones. Although the basic interactions are time-reversible, numerical simulations show an irreversible tendency for the number of sites to grow with time, roughly as . I will explain this behavior, exhibit other solutions of the model in which the motion of the particles and the number of sites are periodic in time, and point out remaining open research problems. This is based on joint work with Karin Baur and David Meyer.

Let $G$ be a finite group and $V$ a finite-dimensional representation of $G$. A {\it compression\/} of $V$ is an equivariant morphism $\phi\colon V \to X$ such that $G$ acts faithfully on the image $\phi(V)$. The main question is the following: \vskip .1in

\noindent How far can we compress a given group action, i.e. what is the minimal possible dimension of $X$? \vskip .1in

\noindent This minimal dimension depends only on the group $G$ and is called {\it covariant dimension\/} of $G$. For example, if $G$ is commutative, then its covariant dimension equals its rank. But in general, the answer is not known. For the symmetric group $S_n$ there are upper and lower estimates. They were first proved by J. Buhler and Z. Reichstein via the so-called {\it essential dimension\/} of $G$ which is defined similarly to the covariant dimension, but allowing rational compressions $\phi\colon V \to X$. We will introduce the notion of compression and covariant dimension, give a few basic results and discuss somerecent joint work with G.W. Schwarz.

In this talk we will discuss the shape of embedded disks with bounded mean curvature. In particular, we will prove that an embedded disk with bounded mean curvature and Gaussian curvature large at a point contains a multi-valued graph around that point on the scale of the norm square of the II fundamental form. Roughly speaking, it looks like helicoids. This generalizes Colding and Minicozzi's result for minimal surfaces.

We propose a new hypothesis that supports the theory of a blitzkrieg waged by human hunters against some species of animals. We suggest that some species were doomed when encountering human hunters because of their tendency to live in large flocks. We introduce a mathematical model that, at least in theory, shows that animals living in large flocks during catastrophic times are more susceptible to mass extinctions than animals living in small flocks. We also show that in non catastrophic times aggregation may help survival.

Have you ever seen someone juggle and wondered how they do it? Or, are you able to juggle but have wondered how the process might be described mathematically? In this talk, I will introduce the concept of a juggling sequence (and what jugglers call “siteswap notation”) and how juggling sequences can be used to describe simple juggling patterns. I will also address some of the mathematical questions related to juggling sequences, such as “Which periodic sequences are jugglable?” In addition, I will illustrate some simple juggling patterns by juggling them (when I'm not picking up the balls off the floor) and I'll have a juggling animator program available to illustrate patterns not accessible to my juggling skills. \vskip .1in
\noindent Refreshments will be provided! \vskip 2in

Canonical lifts of elliptic curves have been used both for point counting (determining the zeta function) and for constructing $CM$ invariants. In genus $2$, Mestre introduced an algorithm for determining the zeta function of a curve using canonical lifts. In this talk I will describe an algorithm for determining $CM$ invariants in the moduli space of genus $2$ curves.

Any algebraic curve in projective $3$-space that is not contained in a
plane has degree at least $3$ -- that is, it meets any plane in at least
3 points. Moreover, any curve of degree $3$, can be parametrised (in
suitable coordinates) by \vskip .1in

$t--> (t, t^2, t^3)$ \vskip .1in

\noindent This was known 150 years ago, and since that time many mathematicians have used and generalized the result. I will describe some of the ideas involved, including recent work of mine with Mark Green, Klaus Hulek and Sorin Popescu.

Multibody dynamics and its various applications in vehicle analysis, aerospace engineering, robotics, and biomechanics have experienced a significant development over the last decades. From the numerical analysis point of view, the question of constraints and their discretization is one of the key issues in this field. The first part of the talk concentrates on systems of rigid bodies and the equations of constrained mechanical motion. The properties of these differential-algebraic equations (DAEs) are discussed and an algorithm for real-time applications with hardware-in-the-loop features is introduced. A simulation of a vehicle with trailer illustrates the approach. In the second part of the talk, deformable bodies are included in terms of a dynamic saddle point problem as basic mathematical model. It turns out that the DAE index and the inf-sup condition are closely related here. Moreover, a combination of the HHT and the RATTLE time integrators is applied to the method of lines and its reversed counterpart. The talk closes with simulations of pantograph and catenary dynamics.

Let $Q=(Q_0,Q_1)$ be a directed graph or ``quiver", and $i\mapsto$ $V_i$ a vector space assigned to each vertex in $Q_0$. The closures of the orbits of $\prod_{Q_0} GL(V_i)$ on
$\prod_{Q_1}Hom(V_{ta},V_{ha})$ are called quiver loci.

\vskip .1in

\noindent In the case that $Q$ is an ADE Dynkin diagram, Reineke showed that each quiver loci is the proper image of a homogeneous vector bundle over a flag manifold. Kempf had earlier used this setup to prove geometric statements about such images. \vskip .1in

\noindent I'll explain how we've extended Kempf's results to apply to this case, and a formula for the equivariant cohomology class (or ``multidegree'') of a quiver locus. These multidegrees were previously only known in type $A$. \vskip .1in

\noindent This work is joint with Mark Shimozono.

The Random to top operator $Tn$ is the sum of the cycles ($i,i+1,..n$) for $i=1..n$. This operator has occurred in various contexts that range from algebraic geometry to statistics. It has played a crucial role in the proof of several recent results. In this lecture we completely determine its action in the regular representation of $Sn$.

The Pseudo-Polar FFT (PPFFT), which was developed and presented by Averbuch et. al, computes Discrete Fourier Transform coefficients on a nearly polar grid. The special structural properties of the
Pseudo-Polar grid allow a computational complexity of $O(n^2\log(n))$
for an $n$ by $n$ image (as opposed to $O(n^4)$ in the polar case). Averbuch et. al. also presented a weighted version of the PPFFT that is nearly orthogonal and can be used for the application of an extremely fast iterative inverse solver. The PPFFT is directly related to a highly accurate and fast version of the Discrete Radon Transform that possesses the same desirable computational properties as the PPFFT, the Fast Slant-Stack Transform. In many instances and applications the PPFFT can be a very good substitute for the Polar FT and its superior computational properties can speed up many related algorithms by several orders of magnitude. Classical applications of the Polar FFT include rotational registration of images and reconstruction in medical and biological imaging. The preliminary part of this talk will introduce the basics of 2D DFTs in Cartesian, Polar and PP grids using matrices and vectors notation. Later, a new direct an exact inverse PPFFT will be presented, the algorithm is based on a preprocessing step in which an optimal set of weights is computed for the given image size, these weights perfectly orthogonalize the
columns of the transform's matrix so that the inverse problem can be solved exactly by a single application of the Adjoint PPFFT which can be computed as well in $O(n^2\log(n))$ complexity. The results will be generalized to 3D as well as to Radon Transforms in 2D and 3D.

I will introduce cubic curves in the context of solving equations. I will then motivate and describe aspects of the Birch and Swinnerton-Dyer conjecture, which predicts exactly when a cubic curve has infinitely many solutions over the rational numbers. This talk should be accessible to anybody with an interest in mathematics. \vskip .1in

\noindent Refreshments will be provided!

The formulation of the 13th Problem in Hilbert's address of 1900 to the International Congress of Mathematicians in Paris allows many different interpretations. The most general one was solved by Kolmogorov in 1957. However, the more natural "algebraic" form of the problem is still completely open. \vskip .1in
\noindent We will describe Hilbert's 13th Problem in general and will give a algebraic form of it. Then we explain some very classical results and a modern approach to them. Finally, we shortly introduce a more restrictive form of the 13th Problem for which Abhyankar has recently given a positive solution.

Let $T$ be a collection of functions from $X$ to $Y$, and call them ``tableaux". I'll describe a simplicial complex whose facets correspond exactly to these tableaux, with many good properties. \vskip .1in

\noindent Something particularly nice happens if $X$ is a poset, $Y$ is totally ordered, and the tableaux are weakly order-preserving functions; then the simplicial complex is homeomorphic to a ball. (This is a sign of an interesting complex, and I'll recall why.) \vskip .1in

\noindent This work is joint with Ezra Miller and Alex Yong (both in Minnesota).

We describe some of the recent advances on the interface between the theory of Levy processes and fractal geometry. We will mention how these results help resolve old questions in the area of Levy processes.
A common theme in this talk is that oftent imes the analysis of Levy processes leads us to study a class of random fields that are called ``additive Levy processes." Much of this talk is based on collaborative efforts with Yimin Xiao.

So you have an image, taken perhaps by a digital camera during a family outing or with expensive equipment such as an MRI scanner. What do you want to do with this image? Maybe you want to make it cleaner, sharper, fix defects, or circle important parts. Or maybe you want the computer to do it for you. In this talk, we touch on these topics and reveal some of the mathematics behind automated digital image processing. \vskip .1in

\noindent Refreshments will be provided! \vskip 2in

Let $M$ be a compact manifold of dimension $n \geq 3$. Along the Yamabe flow, a Riemannian metric on $M$ is deformed according to the equation ${{\partial g}\over{\partial t}} = -(R_g - r_g) \, g$, where $R_g$ is the scalar curvature associated with the metric $g$ and $r_g$ denotes the mean value of $R_g$. It is known that the Yamabe flow exists for all time. Moreover, if $3 \leq n \leq 5$ or $M$ is locally conformally flat, then the solution approaches a metric of constant scalar curvature as $t \to \infty$. I will describe how this result can be generalized to dimensions $6$ and higher under a technical condition on the Weyl tensor. The proof requires the construction of a suitable family of test functions.

The MacDonald polynomials have long been a source of interesting combinatorial questions. In a recent paper, Haglund, Haiman and Loeher managed to give an entirely combinatorial description of them.
They showed that their description defines polynomials indexed not just by partitions, but indeed by any diagram of cells in the positive integer
lattice. However, the combinatorial coefficients they describe are
somewhat unwieldly. I will show how these coefficients can be expressed as certain $q,t$ analogs of multinomial coefficients, and give a recursive construction for rectangular shapes.

In this work I discuss the $sigma-$function of a graph. I will discuss the $sigma-$function and the $\vartheta-$function of Lov\'asz, cliques,
coloring, and much more. I will show the results of experiments (pictures!), give some basic structure theory, and mention conjectures and open problems. If time permits I will talk about applications to
graph drawing and clique finding. This talk should be accessible to all. This is joint work with Fan Chung.

I present an introduction into the mechanisms of generation of spikes and bursts in neurons and in Hodgkin-Huxley-type neuronal models. Then I present a deterministic model that reproduces spiking and bursting behavior of known types of cortical neurons. The model combines the biologically plausibility of Hodgkin-Huxley-type dynamics and the computational efficiency of integrate-and-fire neurons. Using this model, one can simulate tens of thousands of spiking cortical neurons in real time (1 ms resolution) using a desktop PC. \vskip .1in

\noindent MATLAB and C++ code, as well as pdf file of the paper are available at http://www.nsi.edu/users/izhikevich/publications/spikes.htm

Given 52 playing cards, how many shuffles does it take to approximately randomize the deck? More generally, how long does it take a finite Markov chain to get close to its stationary distribution? In this talk, I'll introduce the spectral profile as a tool for proving upper and lower bounds on convergence rates. This approach extends the commonly used spectral gap method, and allows us to recover and refine previous conductance-based estimates of mixing time. I will illustrate how the spectral profile technique is applied in several models, including groups with moderate growth, the fractal-like Viscek graphs, and the torus. This work is joint with Ravi Montenegro and Prasad Tetali.

A regression function is a curve that describes the relation between two random variables $X$ and $Y$. For each value of the variable $X$, this curve reflects the average value taken by the variable $Y$. For example, $X$ might represent the height of individuals and $Y$ their weight, and the regression curve would give, for every possible value of the height, the average weight of individuals sharing that same height. In many real life applications, the regression curve that describes the relation between two given variables is unknown, but the value of ($X,Y$) is observed for a certain number of individuals. The unknown regression curve can then be estimated by finding the curve that ``fits well" the observed values of ($X,Y$). The estimated curve can be determined by parametric techniques, where we suppose that the curve is of a well specified type (for example we assume that the curve is a polynomial of order $3$ and we try to find the 3rd order polynomial that best fits the data). In this talk, we introduce nonparametric techniques of estimation, where no assumption is made on the shape of the curve. \vskip .1in

\noindent Refreshments will be provided!

Commutative rings and algebraic geometry are tightly linked, since an algebraic variety is the set of solutions to some set of polynomials in a polynomial ring. Many people have tried to find theories of geometry which have a similar relationship to noncommutative rings. I will give an introduction to one such remarkable theory which is called noncommutative projective geometry, with many examples.

If $R$ is a commutative ring and $I$ an ideal, the associated graded
ring ${\rm gr}\, R := R/I + I/I^2 + I^2/I^3 + \ldots$ is a very useful
replacement for $R$ for many algebraic and geometric purposes.
Frequently, though, it will have nilpotent elements even when $R$ didn't. \vskip .1in

\noindent I'll discuss a replacement for it, essentially due to Samuel, Rees, and Nagata, using the ``homogenization" of the $I$-adic filtration.
This new filtration is better geometrically motivated, and its associated graded ring never has nilpotents. \vskip .1in

\noindent The applications include nilpotent-free versions of intersection theory on smooth varieties, and of Gr\"obner bases.

We shall consider the Dirichlet problem (for the Laplace operator) with rational data on the boundary of a domain in the plane. We shall characterize those simply connected domains for which all solutions are rational in terms of their Riemann maps and Bergman kernels.

Coxeter defined a natural representation of a reflection group, for any graph. This is extended to the associated braid group. This was originally motivated by questions about unitarizibility of certain representations.

Let $f_m(a,b,c,d)$ denote the maximum size family of a family $F$ of subsets of an $m$-element set so that there is no pair of sets $A,B$ in $F$ such that: \vskip .1in

\noindent (i) $A$ and $B$ have at least $a$ points in common \vskip .01in
\noindent (ii) $B$ has at least $b$ points not in $A$ \vskip .01in
\noindent (iii) $A$ has at least $c$ points not in $B$ \vskip .01in
\noindent (iv) There are at least $d$ points not in $A$ or $B$ \vskip .1in

\noindent By symmetry we can assume $a >= d$ and $b >= c$. We show that $f_m$(a,b,c,d) has order of magnitude $m^{a+b-1}$ if either $b>c$ or $a,b >= 1$. We also show $f_m$(0,b,b,0) has order $m^b$ and $f_m(a,0,0,d)$ has order $m^a$. This can be viewed as a result concerning forbidden configurations, and provides further evidence for a conjecture of Anstee and Sali. Our key tool is a strong stability version of the Ahlswede-Khachatrian Complete Intersection Theorem, which is of independent interest. \vskip .1in

\noindent This is joint work with Richard Anstee.

Let $A$ be an $n*n$ random symmetric matrix. In this talk, I'll survey history and recent developments on the asymptotic distribution of eigenvalues of $A$. This will be self-contained and accessible to general audience. Work in progress with Van Vu.

Parameter error equations in adaptive filtering naturally occur as vector difference equations with random coefficients depending on the realization of the regressors. Our aim in this seminar will be to understand the various concepts of and conditions for the exponential stability of the homogeneous parts. Specifically, we shall analyze the scalar case in some detail and then proceed to untangle the significantly weaker results available in the vector case. The role of correlation and dependence will be explored to understand the unusual distinction between the cases. Please bring your mobile phones, in their off mode, since they will be used as very common example of adaptive filtering and I would like to point to them.

We start by discussing some general properties of asymptotically
flat static solutions to the vacuum Einstein equations. Then we prove a
rigidity theorem of Schwarzchild spacetimes among all such solutions with horizon(minimal surface) boundary. In the end, we relate our discussion to Bartnik's quasi-local mass definition.

I will introduce ``spaces", the basic object of study in geometry and topology. I will try to motivate these objects and describe how one may study them. Concepts like vector fields and Euler characteristics will be introduced. We may go into more complicated constructions if time permits. \vskip .1in

\noindent Refreshments will be provided!

Starting with the examples of Dirichlet's class number formula and the Birch--Swinnerton-Dyer conjecture, we give a broad overview of the expected relations between the arithmetic of algebraic objects (such as varieties) and the special values of their $L$-functions. Our formulation will deliberately tend toward lowbrow and down-to-earth, rather than aiming for the highest possible generality. \vskip .1in

\noindent Note: Watkins is not a new faculty member at UCSD, but he is visiting one (Stein) for the week.

I will describe recent computations regarding special values of
symmetric power elliptic curve L-functions. For the symmetric cube,
these were undertaken by Buhler, Schoen, and Top in the mid 1990s.
Recently the calculation of Euler factors at bad primes has been
expedited by the work of Phil Martin; combined with the availability
of mutli-processor clusters, the time seemed ripe for a large-scale
data-gathering project, and I will share some of the preliminary results.

Real quaternions were introduced by Hamilton in 1843 and for a while were the only advanced mathematics taught in some American universities. After briefly discussing some of their classical and modern applications outside number theory, we shall focus on rational quaternion algebras and their many applications to arithmetic. As sample results we have chosen to discuss the problem of representation of numbers by quadratic forms, a theorem of Gross-Zagier, and Ramanujan graphs. Our intention is to point to the scope and diversity of
applications, ranging from classical to current, of quaternion algebras.

All nontrivial elements in higher $K$-groups of toric varieties over a class of regular rings are annihilated by iterations of the natural Frobenius type endomorphisms. This is a higher analog of the triviality of vector bundles on affine toric varieties. In the talk I will assume no $K$-theoretical background and no familiarity with toric varieties. The main $K$-theoretical concepts will be presented in an informal and completely accessible way, without giving a single formal definition.

Satisfiability is a canonical NP-complete problem that requires to find a $0$-$1$ satisfying assignment for a given set of Boolean constraints. I will consider the Random Walk algorithm that heuristically tries to find a solution by the random walk in the Boolean hypercube, that converges to a satisfying assignment. I will prove the linear upper bound for the expected number of steps of the walk for the class of random small density $3$-SAT formulas. This convergence follows the from geometric investigation of the convex hull of n randomly chosen points in the space.

We will consider a hat guessing game. This game is composed of
$n$ players who have one of $k$ different colored hats placed on their
heads they are allowed to see what other players are wearing, but not
their own hat. They then must guess their own hat. No communication is allowed. Before the hats are placed the players are allowed to come up with a public strategy. The goal of the strategy is to maximize the
guaranteed number of correct guesses. We will show that the best possible is $\lfloor n/k \rfloor$. \vskip .1in

\noindent We will give a hyper-hypercube interpretation of the game which will allow us to generate some balanced strategies in the 2-colored version of the game. We will also discuss the limited hats game and give a bound for it by using the hyper-hypercube interpretation.

One of the central questions in Geometric Analysis is the interplay between the curvature of the manifold and the spectrum of an operator. In this talk, we will be considering the Hodge Laplacian on differential forms of any order $k$ in the Banach Space $L^p$. In particular, under sufficient curvature conditions, it will be demonstrated that the $L^p\,$ spectrum is independent of $p$ for $1\!\leq\!p\!\leq\! \infty.$ The underlying space is a $C^{\infty}$-smooth non-compact manifold $M^n$ with a lower bound on its Ricci Curvature and the Weitzenb\"ock Tensor. The further assumption on subexponential growth of the manifold is also necessary. We will see that in the case of Hyperbolic space the $L^p$ spectrum does in fact depend on $p.$ As an application, we will show that the spectrum of the Laplacian on one-forms has no gaps on certain manifolds with a pole and on manifolds that are in a warped product form. This will be done under weaker curvature restrictions than what have been used previously; it will be achieved by finding the $L^1$ spectrum of the Laplacian.

Among quadrilaterals with sides of specified lengths, there is a unique quadrilateral that encloses the maximum possible area. This extremal quadrilateral is cyclic, that is, its vertices lie on a circle. The extremal pentagons are also cyclic. In general, n-sided polygons with sides of specified lengths that enclose the maximum area are cyclic. These facts are used to solve the isoperimetric problem: Among closed plane curves of length $L$, find the curves that enclose the maximum area. Solution: the circles of length $L$. \vskip 3in

I will state the Birch and Swinnerton-Dyer conjecture, and discuss
some work done toward it.

In Iwasawa theory, one attempts to relate arithmetic objects arising from Galois cohomology groups to analytic objects, specifically $p$-adic $L$-functions, usually at the level of a cyclotomic $Z_p$-extension of a number field. For some time, I have been interested in operations in the Galois cohomology of number fields with restricted ramification, particularly cup products. I will explain various applications of these operations, for instance to Iwasawa theory over certain nonabelian $p$-adic Lie extensions of number fields. I will then discuss the broad outline of a conjecture relating an inverse limit of such cup products up the cyclotomic tower to the two-variable $p$-adic $L$-function for Hida families of cusp forms congruent to Eisenstein series modulo $p$.

The speaker is dealing with commutative hypergroups whose base space is a compact subset $K$ of the $k$-dimensional Euclidean space and whose convolution in the set $M(K)$ of bounded measures on $K$ is defined via sequences of $k$-variable polynomials on $K$. Examples of such hypergroups are the unit interval, the closed unit square, the disk, the parabolic biangle, and the simplex. \vskip .1in

\noindent There is an elaborate harmonic analysis available for compact commutative hypergroups based on the generalized translation operation in $M(K)$. Significant results have been established in analogy to but remarkably distinct from the classical framework of a compact Abelian group. We just note that in general there is no dual hypergroup attached to the given hypergroup $K$, that the Plancherel measure is rarely full, and that positive definite functions can be unbounded. Nevertheless, one has a Haar measure for $K$ and an extended Fourier-Stieltjes theory. \vskip .1in

\noindent Applications of commutative hypergroups point in various directions: Hecke algebras, second order differential operators, and limit theorems of probability theory are just three of them. In the talk to be given the speaker restricts himself to cone-embedded polynomial hypergroups $K$ and describes canonical representations of processes with independent increments in $K$.

Any profinite group can be viewed as a probabilistic space. This approach was explored intensively during the last years. On one hand, the people have been interested in properties of this probabilistic space like to be PFG (positively finite generated). A profinite group is called PFG if for some $k$, random $k$ elements of the group generate it with positive probability. On the other hand, the probabilistic aspect of profinite groups is used in the solution of different kind of problems. In the first direction we present new characterizations of PFG profinite groups which permit us to prove that an open subgroup of a PFG profinite group is also PFG. In the second direction we solve a problem
posed by A. Mann, showing that there exist a constant $c$ such
that the number of finite groups of order $n$ which can be defined
by $r$ relations is at most $n^{cr}$.

A $CR$ manifold with conical singularities is a manifold with conical singularities together with a $CR$ structure on its regular part that behaves in a specified manner near the conical points. I will discuss general aspects of such $CR$ manifolds and an embedding theorem analogous to that of Boutet de Monvel in the case of compact strictly pseudoconvex $CR$ manifolds with conical singularities.

We will review recent works on the passage from discrete (atomistic) description of matter to continuum descriptions, which provide a rigorous ground for some hybrid multiscale models.
This is joint work with Xavier Blanc (University Paris 6), and Pierre-Louis Lions (College de France).

We will consider a hat guessing game. This game is composed of
$n$ players who have one of $k$ different colored hats placed on their
heads they are allowed to see what other players are wearing, but not
their own hat. They then must guess their own hat. No communication is allowed. Before the hats are placed the players are allowed to come up with a public strategy. The goal of the strategy is to maximize the
guaranteed number of correct guesses. We will show that the best possible is $\lfloor n/k \rfloor$. \vskip .1in

\noindent We then consider several variations of the game including the limited vision problem, constructing balanced strategies and the hat placement problem.

Polymerase Chain Reaction (PCR) is an in vitro technique aimed at creating multiple copies of a given nucleotide sequence. In principle, after each PCR cycle there should be a duplication of the molecules so that after n cycles there should be $N(0) 2^n$ molecules, where $N(0)$ is the number of initial molecules. In practice there is a probability $p$ of duplication at each cycle, with $p$ less than one. This value measures the efficiency of the duplication process and it may be constant through the cycles or varying in a random or deterministic way. We use the age dependent branching process model to estimate the efficiency and show the consistency and asymptotic Normality of the quasi-likelihood estimator. We also present a simulation study done by Claude Gravel (M.Sc. thesis, UQAM, 2005) on the small sample properties of the estimator. These results are useful for the related important problem of estimating the initial number of molecules (the concentration) if unknown.

The problem of detection arises in a wide variety of settings, such as analyzing galaxy catalogs; object tracking; road tracking; satellite imagery, which includes detecting fires as well as man-made objects on land or sea; environment monitoring, which includes controlling pollution levels of water sources; detection of disease outbreaks and detection/localization of whales in the ocean from acoustic data. \vskip .1in

\noindent The scan statistic, also known as matched filter and deformable template, has been applied in such settings and many others. The scan statistic will be described, some theoretical and computational issues will be presented and some details will be given for specific applications. \vskip .1in

\noindent Refreshments will be provided!

The Full Transformation Semigroup $T_n$ is the semigroup of all maps from a set of size $n$ to itself. The representation theory of $T_n$ is closely tied to that of the Symmetric Group $S_n$, which it contains. However the usual questions are considerably more difficult to answer because the associated semigroup algebra ${\cal C} T_n$ is not semisimple. \vskip .1in

\noindent In this talk we define a deformation of ${\cal C} T_n$, introducing a parameter with the aim of making the algebra generically semisimple. We show that the irreducible modules of the deformed algebra are in fact isomorphic to those of the Rook Monoid, which have a simple combinatorial description due to Cheryl Grood. We derive character formulas, and discuss the specialization to certain "bad" values of the parameter.

We consider $k$ independent Galton-Watson random trees whose offspring distribution is in the domain of attraction of any stable law. We prove that conditionally on the total progeny being equal to $n$, when $n$ and $k$ tend towards infinity, under suitable rescaling, the associated coding random walk and height process converge in law on the Skohorod space respectively towards the ``first passage bridge" of a stable Levy process with no negative jumps and its height process.

Given the three kinds of puzzle pieces pictured on the left, define a puzzle to be a decomposition of a triangle into puzzle pieces (such that the edges match up, like in a jigsaw puzzle). Call a puzzle ``rigid'' if there is no other puzzle with the same outer boundary. A lot can be proven about puzzles (we'll do (1) and (2) in the talk): \vskip .05in
\noindent 1. The number of $0$s on one side equals the number of $0$s on each of the other two sides - see if you can prove this one before the talk! \vskip .05in
\noindent 2. The lines in the puzzle pieces can all be simultaneously straightened (as in the right-hand picture) if and only if the puzzle is rigid. \vskip .05in
\noindent 3. There is an easy 1:1 correspondence between rigid puzzles and inequalities on the eigenvalues of the sum of two Hermitian matrices. \vskip .05in
\noindent 4. The statement ``Given four generic lines in space, there are exactly two others that touch all four,'' and others like it, can be turned into puzzle-counting statements. \vskip .1in
\noindent Refreshments will be provided!

In this talk, I will discuss some structural and rigidity properties of a class of manifolds satisfying some weighted Poincare inequality. These theorems can be viewed as a generalization of a theorem of Witten-Yau on $AdS/CFT$ correspondence. It also generalizes some known results in hyperbolic geometry.

We describe hierarchy of coarse-grained stochastic processes that approximate, on larger scales, microscopic lattice systems simulated by (kinetic) Monte Carlo algorithm. In applications the microscopic model is often coupled with continuum $PDE$ description on larger scales. We present mathematical tools for error estimation of the coarse-graining procedure. A posteriori estimates on the loss of information between the coarse-grained and the microscopic probability measure can guide the coarse-graining procedure at different parts of the computational domain. In specific examples of lattice spin dynamics we demonstrate that coarse-grained $MC$ leads to significant CPU speed up of simulations of metastable phenomena, e.g., estimation of switching times or nucleation of new phases. Some recent results obtained in joint work with M. Katsoulakis and A. Sopasakis will be presented.

This talk considers the problem of constructing optimal designs for regression models when there are several independent variables and some of them are not under the control of the experimenter. A variable that is not under control can have known values before the experiment is realized. The first case is briefly discussed in the literature. The aim of this work is to provide equivalence theorems for the second case and the mixture of both cases. Iterative algorithms for generating approximate optimal designs are given and a real case of lung cancer is discussed. Assuming an optimal design was carried out and then a new patient arrives the marginal design changes. Thus, we want to find the optimal joint design with this new restriction. Different approaches are considered to deal with this typical situation.

A gradient Gibbs measure is the projection to the gradient variables
$\eta_b=\phi_y-\phi_x$ of the Gibbs measure of the form
$$
P(d\phi)=Z^{-1}\exp\Bigl\{-\beta\sum_{\langle
x,y\rangle}V(\phi_y-\phi_x)\Bigr\}d\phi,
$$
where $V$ is a potential, $\beta$ is the inverse temperature and $d\phi$
is the product Lebesgue measure. The simplest example is the (lattice)
Gaussian free field $V(\eta)={1 \over 2}\kappa\eta^2$. A well known result of Funaki and Spohn asserts that, for any uniformly-convex $V$, the possible infinite-volume measures of this type are characterized by the {\it tilt}, which is a vector $u\in{\bf R}^d$ such that $E(\eta_b)=u\cdot b$ for any (oriented) edge $b$. I will discuss a simple example for which this result fails once $V$ is sufficiently non-convex thus showing that the conditions of Funaki-Spohn's theory are generally optimal. The underlying mechanism is an order-disorder phase transition known, e.g., from the context of the $q$-state Potts model with sufficiently large $q$. Based on joint work with Roman Koteck\'y.

Kenneth Arrow's Impossibility Theorem essentially states that in the presence of three or more candidates there is no way to hold a fair election. That this statement is true in practice should not be a surprise to anyone familiar with our current electoral system. It is a little more surprising that it holds even in the abstract world of mathematics. \vskip .1in
\noindent In this talk we shall define social welfare functions, discuss a reasonable set of fairness criteria, and sketch a proof of Arrow's theorem. Along the way we shall touch on topics in combinatorics, geometry, logic, and set theory. To end on a positive note, we shall show that fair voting methods do exist when the number of voters is infinite. \vskip .1in
\noindent Prerequisites: A small amount of basic set theory and linear algebra will be assumed, but all important terms will be defined as we go. \vskip .1in
\noindent Refreshments will be provided!

An important problem in science and engineering is the determination of the effects of uncertainty or variation in parameters and data on the output of a deterministic nonlinear operator. The Monte-Carlo method is a widely used tool for determining such effects. It employs random sampling of the input space in order to produce a pointwise representation of the output. It is a robust and easily implemented tool. Unfortunately, it generally requires sampling the operator very many times. Moreover, standard analysis provides only asymptotic or
distributional information about the error computed from a particular realization. \vskip .1in

\noindent We present an alternative approach for this problem that is based on techniques borrowed from a posteriori error analysis for finite element methods. Our approach allows the efficient computation of the gradient of a quantity of interest with respect to parameters at sample points. This derivative information is used in turn to produce an error estimate for the information, thus providing a basis for both deterministic and probabilistic adaptive sampling algorithms. The deterministic adaptive sampling method can be orders of magnitude faster than Monte-Carlo sampling in case of a moderate number of parameters. The gradient can also be used to compute useful information that cannot be obtained easily from a Monte-Carlo sample. For example, the adaptive algorithm yields a natural dimensional reduction in the parameter space where applicable.

The classical Bohr-Sommerfeld quantization condition gives very precise spectral results for selfadjoint semiclassical differential operators in dimension one, while important difficulties usually appear in higher dimensions. In this talk, we would like to discuss the recent results of a spectral analysis for non-selfadjoint perturbations of selfadjoint operators in dimension two. Assuming that the underlying classical flow of the unperturbed part possesses invariant Lagrangian tori satisfying a Diophantine condition, we obtain a complete asymptotic description of all eigenvalues in suitable regions of the complex spectral plane. This result, joint with Johannes Sjostrand and San V\~u Ngoc, can be viewed as a version of the Bohr-Sommerfeld rule in the non-selfadjoint two-dimensional case.

In joint work with Benson Farb, we show that $K_g$, a certain subgroup of the mapping class group of a genus $g$ surface, is not finitely generated. $K_g$ is the group generated by Dehn twists about null-homologous (separating) curves; it is also known as the kernel of the Johnson homomorphism out of the Torelli group. No background will be assumed.

We begin by surveying some highlights of the theory of increasing and
decreasing subsequences of permutations, including (1) connections with Young tableaux and the $RSK$ algorithm, (2) the expected length is($w$) of the longest increasing subsequence of a random permutation $w$ of $1$,$2$,...,n, and (3) the limiting distribution of is($w$) due to Baik, Deift, and Johansson. We will then discuss how these results carry over to (complete) matchings $M$ on the vertices $1$,$2$,...,$2n$. The analogue of increasing/decreasing subsequences will be shown to be related to crossings and nestings of $M$.