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#### 2003 Archive

##### Solution of the truncated parabolic moment problem

We provide concrete (computable) necessary and sufficient conditions for the existence of a representing measure, supported in a prescribed parabola p(x,y) =0, for moment data B := B(2n)={B_{i,j}: i, j, >= 0, i+j <=2n}. There exists a positive Borel measure u, supported in p(x,y)=0, such that B_{i,j} is the u-moment for x^i y^j (i+j <= 2n) if and only if the associated moment matrix M(n)(B) is positive semi-definite, recursively generated, has a column dependence relation p(X,Y) = 0, and satisfies rank M(n)(B) <= card V(B), where V(B) is the algebraic variety naturally associated to the data B(2n). (Joint work with R.E. Curto)

##### The Nature of Singularities in Cosmological Solutions of Einstein's Equations

The Hawking- Penrose singularity theorems tell us that cosmological solutions of Einstein's equations generically contain a singularity. But these theorems tell us little about what happens near such a singularity. Do the gravitational fields necessarily grow without bound? Can causality break down? What about the Cosmic Censor? Work done during the past ten years--both analytical and numerical--has gotten us a lot closer to answers to these questions. We survey this work, discussing both the mathematical ideas and the physical implications. We also discuss the likely direction of future studies.

##### Random tree-automorphisms

We study randomness in the automorphism group of the binary tree of depthn and its generalizations. These groups have an important role in grouptheory, and they also arise in connection with complex dynamics, fractalsand finite automata. We use branching processes to determine theasymptotic order of a random element, answering an old question of Turan.We show that three random elements generate a large subgroup with highprobability, leading to the solution of a problem of Shalev.This is joint work with M. Abert.

##### Nonparametric Tail Index Estimation

In Politis (2002) a method of tail index estimation for heavy-tailed time series, based on examining the growth rate of the logged sample second moment of the data, was proposed and studied. This estimator has a slow rate of convergence to the tail index, which is due to the high dependence of the summands of the statistic. To ameliorate the convergence rate, this work proposes an estimator with reduced bias, computed over subblocks of the whole data set. The resulting estimator obtains a polynomial rate of consistency for the tail index, and in simulation studies shows itself decidedly superior to competing prior art, such as the above-mentioned estimator of Politis (2002), as well as the reknowned Hill estimator.

##### The McKay correspondence

Prof. Craw is a Recruitment candidate

##### Locally integrable solutions of D-modules and characters of semi-simple Lie groups

A famous theorem of Harish-Chandra asserts that all invarianteigendistributions on a semisimple Lie group are locally integrablefunctions. We show that this result and its extension to symmetricpairs are consequences of a general result about systems of PDEwhose solutions are all locally integrable.

##### Spaces of rational curves and rational connectivity

We will discuss results by Joe Harris, Jason Starr and otherson the spaces of rational curves on Fano varieties.

##### Discrete Morse functions on posets

Professor Patricia Hersh is a potential recruitment candidate.Forman introduced discrete Morse theory as a tool for studyingCW-complexes by collapsing them onto smaller, simpler-to-understandcomplexes of critical cells. Chari provided a combinatorialreformulation based on acyclic matchings for their face posets. Injoint work with Eric Babson, we showed how to construct a discreteMorse function with a fairly small (but typically not optimal)number of critical cells for the order complex of any finite posetfrom any lexicographic order on its saturated chains. I willdiscuss this construction as well as two more recent results abouthow to improve a discrete Morse function by cancelling pairs ofcritical cells. A key ingredient will be a correspondence betweengradient paths in poset "lexicographic discrete Morse functions" andreduced expressions for permutations.As an application, in joint work with Volkmar Welker, we construct adiscrete Morse function for graded monoid posets which yields upperbounds on Poincare' series coefficients for affine semigroup rings(by way of the Morse inequalities). These bounds are determined bythe degree of a Gr"obner basis for the toric ideal of syzygies andrelated data.I will begin with a brief review of discrete Morse theory.

##### Optimum Alphabetic Binary Trees

Given a sequence of leaf nodes with positive weights, the optimumalphabetic binary tree can be constructed in O( nlogn) time in the worstcase and in O(n) time in most cases.The open question is: if the weight distribution is random,what percentageof cases be solved in linear time?

##### The Brownian Forest

Harris discovered a corrrespondence between random walk excursions and random trees whose continuous analog relates a Brownian excursion to Aldous's concept of a continuum random tree. This idea has been developed and applied in various ways by Neveu, Le Gall and others.I will review these ideas in terms of a forest growth process, originallydevised by Aldous to describe the asymptotics of large finite trees, but nowrelated to the structure of a Brownian path exposed by sampling at the timesof points of an independent Poisson process.Reference: Chapter 6 of "Combinatorial Stochastic Processes", available viahttp://stat-www.berkeley.edu/users/pitman/bibliog.html

##### Revisiting the Gross-Koblitz formula

The Gross-Koblitz formula describes Gauss sums on finite fieldsof characteristic $p$ in terms of the $p$-adic Gamma function. Thisformula is a $p$-adic lifting of Stickelberger's congruence on Gauss sums.There have been several proofs of the formula (Gross-Koblitz,Dwork-Boyarsky, Katz, Coleman), which all involve some cohomologicalcalculations. Recently, a proof has been found by A. Robert which isconceptual but truly elementary, at the level of "freshman $p$-adicanalysis." We will discuss Robert's proof and describe possibleextensions

##### Sequences, Patterns and Coincidences - II

In this series of lectures, we will start with some basictheory, examples and techniques of studying occurences of patterns,and waiting times in sequences of independent Bernoulli or multinomialtrials. We will discuss some exact theory, and asymptotics, such ascentral limit theorems and Poisson approximations. We will then proceed tospecial types of coincidences, such as matching, and the problem oflook-alikes. We also hope to present some of the modern developments onlongest increasing subsequences, and applications, such as to genetics.

##### Change Point Problems for Composite Pre-Change Distribution

Potential Statistics Recruitment CandidateChange point problems have a variety of applications including industrial quality control, reliability, clinical trials, surveillance, and security systems. By monitoring data streams which are generated from the process, we are interested in quickly detecting malfunctioning once the process goes out of control, while keeping false alarms as infrequent as possible when the process is in control. Suppose that $f_ heta(x)$, the distribution of the data, is indexed by $heta$, a vector of one or more parameters. Most research has been done under the assumption that the value of $heta$ is known before a change occurs. In this talk, we investigate the situation where the value of $heta$ is composite before a change occurs. We present a new formulation of the problem by specifying required average time to detect after the value of $heta$ shifts to a specified $heta_1$ and trying to minimize the frequency of false alarms over a range of possible value $heta$ before a change occurs. Asymptotically optimal procedures will be presented.

##### Pascal Matrices

Put the famous Pascal triangle into a matrix. It could go intoa lower triangular $L$ or its transpose $L'$ or a symmetric matrix $S$: $$L=matrix{ [ 1 0 0 0 ] cr [ 1 1 1 1 ] cr [ 1 1 0 0 ] cr[ 1 2 1 0 ] cr[ 1 3 3 1 ] cr}quad L' =matrix{ [ 1 1 1 1]cr [ 0 1 2 3 ]cr [ 0 0 1 3 ] cr[ 0 0 0 1 ] cr}quad S = matrix{[ 1 2 3 4]cr [ 1 3 6 10]cr [ 1 4 10 20]cr}$$These binomial numbers come from a recursion, or from the formulafor $i$ choose $j$, or functionally from taking powers of $(1 + x)$. The amazing thing is that $L imes L' = S$. (OK for $4 imes 4$)It follows that $S$ has determinant 1. The matrices have otherunexpected properties too, that give beautiful examples in teachinglinear algebra. The proof of $L L' = S$ comes 3 ways: 1. By induction using the recursion formula for the matrix entries. 2. By an identity for the coefficients $i+j$ choose $j$ in $S$. 3. By applying both sides to the column vector $[ 1 x x^2 x^3 ... ]'$.The third way also gives a proof that $S^3 = -I$ but we doubt that result. The rows of the hypercube matrix" $L^2$ count corners and edgesand faces and ... in n dimensional cubes.

##### A recipe for combining classifiers, and a particular application to isolated (spoken) digit recognition

Recently there has been a fair amount of interest in combining several classification trees so as to obtain better decision rules. Techniques such as bagging, boosting, and randomized trees are particularly popular in statistics and computer science.The best PAC-learning theoretical bounds on the classification error rate achieved by these techniques do not offer any insight into howone should combine these classifiers in order to reduce the error rate. In this talk I will present the notion of weakly dependent classifiers, and show that when both the dependence between the classifiers is low, and the expected margins (a measure of confidence in the classifiers) are large, then exponential upper bounds on the classification error rate can be achieved. In particular, experiments with several data sets indicate that thereappears to be a trade-off between weak dependence and expected margins in the sense that to compensate for low expected margins there should be low mutual dependence between the classifiers.The results will be motivated and become more intuitive through an application of randomized relational decision trees to speech recognition.

##### The homotopy BV nature of arc operads and their relationto moduli spaces and string topology

Operads are a general tool which allows one to encodetopological and algebraic structures and their relations.Recently, we defined an operad based on arcs on surfacesand showed that this operad (or rather its chains)has an explicit structure of a homotopy BV operad. The spaceon which this operad is defined is closely related toRiemann's moduli space and can be thought of a kind ofcombinatorial model for it. From this point of viewit is natural that the operad governs manytopological and algebraic questions, some of whichare related to physics.For instance, there is a suboperad of our operad,which is related to Chas-Sullivans' string topology.In this and other examples, we will show howthe natural composition of arcs in the arc operad yields thesestructures, gives them a surface interpretationand generalizes them.

##### Modularity in Characteristic p

The connection between elliptic curves over the rational numbers${f Q}$ and modular forms for $SL_2({f Z})$ is now very well known.This fundamental relationship both establishes that the L-series of onesuch elliptic curve has an analytic continuation and functional equationand gives representatives of the isogeny class of the elliptic curve(inside the Jacobians of modular curves).Now let $A:={f F}_q[T]$, where ${f F}_q$ is the finite field with$q$-elements, and let $k:={f F}_q(T)$. Let $K:={f F}_q((1/T))$with associated algebraic closure $ar{K}$. Mimicking the classicaldefinition of $f Z$-lattices inside the complex numbers $f C$,one has the notion of ${f F}_q[T]$-lattices inside $ar{K}$.Rank one lattices correspond to analogs of the exponential functionand rank two lattices uniformize analogs of elliptic curves. Theserank two \"Drinfeld modules\"" give rise to modular curves in exact analogywith elliptic curves. Remarkably

##### Vanishing integrals and symmetric spaces

One particularly interesting consequence of the theory ofsymmetric spaces is that if one integrates a Schur function $s_lambda$(i.e., an irreducible character of the unitary group) over theorthogonal group, the integral vanishes unless all parts of thepartition $lambda$ are even (when the integral is 1). I'll discusssome quantum'' generalizations of this fact, in which the Schurfunctions are replaced by Macdonald polynomials; I'll also discussanalogous integrals related to other classical symmetric spaces.

##### The link between Cacti, Connes-Kreimer's Hopf algebraand Deligne's Conjecture

We will introduce several varieties of cacti operadswhich are interrelated by direct and semi-direct products.These operads can all be naturally realized as suboperadsof the arc operad. Furthermore the homotopy equivalence ofof these operads to the little discs and framed little discspoints the way to Deligne's conjecture. In this direction, wewill also consider operations on Hochschild cohomology.On the other hand, but astonishingly in the same spirit,we will show how the non-trivial coproduct of the renormalizationHopf algebra of Connes and Kreimer can also be undersood as beingnatural when viewed in terms of our arc operad composition.

##### Mass, quasi-local mass and static metric extension in general relativity

We will first discuss a generalized Positive Mass Theorem on a class ofpiecewise smooth asymptotically flat manifolds with broken mean curvatureacross a hypersurface. Then we will relate it to Bartnik's quasi-localmass definition and explain how Corvino's scalar curvaturedeformation theorem implies that a minimal mass extension, if exists,must be static. Finally, we will prove that, for any metric thatis close enough to the Euclidean metric on a ball and has reflectioninvariant boundary data, there always exists an asymptotically flat, scalar flat and static metric extension with Bartnik's geometric boundarycondition.

##### Maximum Likelihood and Rank Estimation for All-Pass Time Series Models

All-pass models are autoregressive-moving average models in which the roots of the autoregressive polynomial are reciprocals of roots of the moving average polynomial and vice versa. They generate uncorrelated (white noise) time series, but these series are not independent in the non-Gaussian case. Because all-pass series are uncorrelated, estimation methods based on Gaussian likelihood, least-squares, or related second-order moment techniques cannot identify all-pass models. Consequently, I use maximum likelihood and rank techniques to obtain parameter estimates. Maximum likelihood estimation has already been studied for autoregressive-moving average models. However, the parameters in the autoregressive polynomial of an all-pass model are functions of parameters in the moving average polynomial and vice versa, so the results for autoregressive-moving average models cannot be used for all-pass models. I discuss asymptotic properties of the two types of estimators, examine their behavior for finite samples via simulation, and consider an application for all-pass models--fitting noninvertible moving average models (known as nonminimum phase models in the engineering literature). I apply the results to stock market data. This is joint work with Jay Breidt and Richard Davis.

##### Genome Rearrangements in Mammalian Evolution: Lessons from Human andMouse Genomes

Although analysis of genome rearrangements was pioneered by Dobzhanskyand Sturtevant 65 years ago, we still know very little about therearrangement events that produced the existing varieties of genomicarchitectures. The genomic sequences of human and mouse provideevidence for a larger number of rearrangements than previouslythought. We describe a new algorithm for constructing synteny blocks,study arrangements of synteny blocks in human and mouse, derive a mostparsimonious human-mouse rearrangement scenario, and provide evidencethat intrachromosomal rearrangements are more frequent thaninterchromosomal. Our analysis is based on the human-mouse breakpointgraph, which reveals related breakpoints and allows one to find a mostparsimonious scenario. We also provide the first evidence that thewidely accepted Nadeau-Taylor model of chromosomal rearrangements mustbe revised, in view of details that were not visible prior to theavailability of high-resolution genomic sequences.Potential recruitment candidate for Bioinformatics

##### Obstacle and Stefan type problems with no sign restriction

We show how to prove the regularity of free boundaries in these classical problems even if one drops the nonnegativity assumption on the solution (and on its time derivative in the case of Stefan problem). This involves the application of two different kinds of monotonicity formulas: one due to L. Caffarelli to prove the regularity of the solution, and the other due to G. Weiss to classify the free boundary points by their homogeneity properties.

##### Detecting and Estimating Sparse Mixtures

Sparse Mixture Models have important applications in many areas, such as Signal and Image Processing, Genomics, Covert Communication, etc. In my talk, I will consider the problems of detecting and estimating sparse mixtures.Detection: Higher Criticism is a statistic inspired by a multiple comparisons concept mentioned in passing by Tukey (1976) (but as a term: Higher Criticism is invented by a German historian Johann Eichhorn (1787)). We are able to show that the resulting : Higher Criticism Statistic is effective at resolving a very subtle testing problem: testing whether $n$ normal means are all zero versus the alternative that a small fraction is nonzero; the subtlety of this sparse normalmeans' testing problem can be seen from work of Ingster (1999) and Jin(2002), who studied such problems in great detail. In their studies, they identified an interesting range of cases where the small fraction of nonzero means is so small that the alternative hypothesis exhibits little noticeable effect on the distribution on the $p$-values either for the bulk of the tests or for the few most highly significant tests. In this range, when the amplitude of nonzero means is calibrated with the fraction of nonzero means, the likelihood ratio test for a precisely specified alternative would still succeed in separating the two hypotheses. We show that the higher criticism is successful throughout the same region of amplitude vs. sparsity where the likelihood ratio test would succeed. Since it does not require a specification of the alternative, this shows that Higher Criticism is in a sense optimally adaptive to unknown sparsity and size of the non-null effects. While our theoretical work is largely asymptotic, we provide simulations in finite samples. We also show Higher Criticism works very well over a range of non-Gaussian cases.Estimation: False Discovery Rate (FDR) control is a recent innovation in multiple hypothesis testing, in which one seeks to ensure that at most a certain fraction of the rejected null hypotheses correspond to false rejections (i.e. false discoveries). The FDR principle also can be used in highly multivariate estimation problems, where it has recently been shown to provide an asymptotically minimax solution to the problem of estimating a sparse mean vector in the presence of Gaussian white noise. In effect, FDR provides an effective method of setting a threshold for separating signal from noise when the signal is sparse and the noise is Gaussian. In this talk we consider the application of FDR thresholding to non-Gaussian settings, in hopes of learning whether the good asymptotic properties of FDR thresholding as an estimation tool hold more broadly than just at the standard Gaussian model. We study sparse exponential model and sparse Poisson model, which are important models for non-Gaussian data, and have applications in many areas as well, such as Astronomy and Positron Emission Tomography (PET) etc. We show that the FDR principle also provide an asymptotically minimax solution to the problem of estimating a sparse mean vector even in the presence of exponential/Poisson noise, and in effect FDR provides an effective method of setting a threshold for separating signal from noise when the signal is sparse and the noise is exponential/Poisson. We compare our results with work in the Gaussian setting by Abramovich, Benjamini, Donoho, Johnstone (2000).Joint work with David L. Donoho.

##### Toric Surfaces, Gromov-Witten and tropical algebraic geomerty

The talk presents a new formula for the Gromov-Witten invariants ofarbitrary genus in the projective plane as well as for the relatedenumerative invariants in other toric surfaces. The answer is given interms of certain lattice paths in the relevant Newton polygon. Thelength of the paths turns out to be responsible for the genus of theholomorphic curves in the count. The formula is obtained by working interms of the so-called tropical algebraic geometry. This version ofalgebraic geometry is simpler than its classical counterpart in manyaspects. In particular, complex algebraic varieties themselves becomepiecewise-linear objects in the real space. The transition from theclassical geometry is provided by consideration of the "large complexlimit" (which is also known as "dequantization" or "patchworking" insome other areas of Mathematics).

##### The Nagata automorphism is wild

It is well-known that the automorphisms of polynomial rings and free associative algebras in two variables are "tame", that is, they admit a decomposition into a product of linear automorphisms and the automorphisms of the type $(x,y)mapsto (x,y+f(x))$. However, in the case of three or more variables the similar question was open and known as The generation gap problem" or The tame generators problem". In 1972 Nagata constructed a certain automorphism of the polynomial ring in three variables and conjectered that it is non-tame or "wild". The purpose of the present work is to confirm the Nagata conjecture. Our main result states that the tame automorphisms of the polynomial ring in three variables over a field of characteristic $0$ are algorithmically recognizable. In particular, the Nagata automorphism is wild.

##### Multiscale Stochastic Volatility Asymptotics

We consider the problem of pricing derivative securities in an environment of uncertain and changing market volatility. The popular Black-Scholes model relates derivative rices to current stock prices through a constant volatility parameter. The natural extension of this approach is to model the volatility as a stochastic process. In a regime with a multiscale or bursty stochastic volatility we derive an generalized pricing theory that incorporates the main effects of a stochastic volatility. We consider high frequency S&P 500 historical pricing data and analyze these with a view toward identifying important time scales and systematic features. The data shows a periodic behavior that depends on both maturity dates and also the trading hour. We examine the implications of this for modeling and option pricing.

##### Stark-type Conjectures "over Z"

In the 1970s and early 1980s Stark developed a remarkableconjecture aimed at interpreting the first non-vanishing derivative of anArtin L-function $L_{K/k, S}(s, chi)$ at $s=0$ in terms of arithmeticproperties of the Galois extension of global fields K/k. Work of Tate,Chinburg, and Stark himself has revealed far reaching applications ofStark's Conjecture to Hilbert's 12-th Problem and the theory of Galoismodule structure of groups of units and ideal-class groups. In his searchfor new examples of Euler Systems, Rubin has formulated in 1994 a strongversion ("over Z", in Tate's terminology) of Stark's Conjecture forabelian L-functions of arbitrary order of vanishing at s=0. Our study ofthe functorial base-change behavior of Rubin's Conjecture led us toformulating a seemingly more natural Stark-type conjecture "over Z". Wewill discuss and provide evidence for this new statement, as well asbriefly describe the main goals of the conjectural program initiated byStark.

##### Long arithmetic progressions in sumsets and the number of zero-sum-free sets

Let n be a large prime. A set A of residues modulo n is zero-sum-free if no subsetsum of A is divisible by n. Zero-sum-free sets have been studied for a long time but little was know about the following fundamental question: How many zero-sum-free sets are there ?In this talk, we shall present a sharp answer to this question, using new results about long arithmetic progressions in sumsets. In fact, we are able to characterize zero-sum-free sets: the main (and natural) reason for a set to be zero-sum-free is that the sum of its elements is less than n. (joint work with E. Szemeredi)

##### Analogs to the Mass and the Positive Mass Theorem on Spheres

We describe two mass-like quantities arising from the Green's function for the Laplacian operator on surfaces. The Robin's mass is obtained by regularizing the logarithmic singularity of the Green's function. We show that the Robin's mass is connected to a spectral invariant. On spheres, we introduce a "geometrical mass", which is, a priori, a smooth function on the sphere. The goemetrical mass is shown to be independent of the point on the sphere, and it is also a spectral invariant. Moreover, a connection to a Sobolev-type inequality reveals that it is minimized at the standard round metric. The definition of the geometrical mass is inspired by the roles played by the Green's function for the conformal Laplacian and the Positive Mass Theorem in the solution to the Yamabe Problem.

##### Linearization coefficients for orthogonal polynomials using stochastic processes

A family of polynomials ${P_n}$ such that $P_n$ has degree n is a basis for the polynomial ring. A product $P_{n_1}$ $P_{n_2}$ ... $P_{n_k}$ can be expanded in this basis, and the coefficients in this expansion are called linearization coefficients. If the basis consists of orthogonal polynomials, these coefficients are generalizations of the moments of the measure of orthogonality. Just like moments, these coefficients have combinatorial significance for many classical families. For instance, for the Hermite polynomials they are the numbers of inhomogeneous matchings. I will describe the linearization coefficients for a number of classical families. The proofs are based on the relation between the polynomials and certain stochastic processes. They involve the machinery of combinatorial stochastic measures, introduced by Rota and Wallstrom. The number of examples treated by this method is increased significantly by using non-commutative stochastic processes, consisting of operators on a q-deformed full Fock space.

##### Statistical methods for identifying transcription factor binding sites

The completion of the genomes of model organisms represents just the beginning of a long march toward in-depth understanding of biological systems. One challenge in post-genomic research is the detection of functional patterns from full-length genomic sequences. This talk focuses on statistical methods in finding patterns with functional or structural importance in biological sequences, in particular the identification of transcription factor binding sites (TFBSs). Some of the underlying mathematical theories will be discussed as well.TFBSs are often short and degenerate in sequence. Therefore they are often described by position- specific score matrices (PSSMs), which are used to score candidate TFBSs for their similarities to known binding sites. The similarity scores generated by PSSMs are essential to the computational prediction of single TFBSs or regulatory modules. We develop the Local Markov Method (LMM), which provides local p-values as a more reliable and rigorous alternative. Applying LMM to large-scale known human binding site sequences in situ, we show that compared to current popular methods, LMM can reduce false positive errors by more than 50% without compromising sensitivity.

##### Spectra of fractal groups and the Atiyah Conjecture

After a quick introduction to the spectral theory of groups and graphswe take a more careful look at the so called lamplighter group Land show that the discrete Laplace operator on the Cayley graph of L(with respect to a certain generating set) is pure point spectrum andthe spectral measure is discrete (and explicitly computed). This is thefirst example of a group with discrete spectral measure.We take an unusual point of view and realize the lamplighter group L asa group generated by a 2-state automaton. This approach, along with someC* arguments, provides a crucial tool in our computations.The above result is applied to answer a question of Michael Atyiah onthe possible range of $L^2$ Betti numbers. Namely, we construct a 7dimensional closed manifold whose third $L^2$ Betti number is not aninteger (it is 1/3). The manifold also provides a counterexample to theso called strong Atiyah Conjecture concerning a relation between therange of $L^2$ Betti numbers and the orders of the finite subgroups of thefundamental group of the manifold.

##### Nonlinear Dispersive Waves:MultiChannel Large Time behaviour

The Nonlinear Schrodinger equation, is an example of a dispersive wave equation which has many different asymptotic states depending on the initial data. Such time dependent equations play a central role in many of the latestscientific advances,such as Bose-Einstein condensates and optical devices .I will discuss the solutions of such equations,including the large time behavior:first rigorous proof of the phenomena of ground state selection, asymptotic instability of the excited states and more.These results are obtained by deriving a novel Nonlinear Master equation and multitime scale analysis of its properties.The talk will be general.

##### Invariant and coinvariant polynomials for the generalized symmetric groups

Symmetric polynomials are the invariants of the classical action of the symmetric group Sn on the space Q[Xn] of polynomials by permutation of the variables. It is well known that the dimension of the quotient of Q[Xn] by the ideal generated by symmetric, constant-free polynomials is n!. When we consider other actions or other groups, we have different spaces of invariants, and different quotients (coinvariants). We will discuss some examples, in particular quasi-symmetrizing actions, whose coinvariants have dimensions given by the Catalan numbers. We shall give explicit Grobner bases for the ideals generated by the invariants .

##### A friendly competitor to the Thompson group

We will discuss the properties of a 2-generator 2-relator group whichshares many properties with the famous Richard Thompson group F. Thegroup arises as an ascending HNN extension of a certain group which isgenerated by a 3-state automaton. The latter group will be in the focusof our considerations. In particular we will explain that it is theiterated monodromy group of the map z --> $z^2-1$.

##### Ihara zeta function of infinite graphs and groups

We will define the analog of the Ihara zeta function for a Cayley graphof a finitely generated group. This will also be done for an infiniteregular graph which is a limit of a sequence of finite graphs.Interesting examples of computations based on spectral theory of fractalgroups will be considered. Some aspects of the computations are relatedto dynamical systems.

##### The Sound of Fluid Flowing

We will discuss flows of fluids for which the flow velocity is much lessthan the sound speed. For such flows we isolate that aspect of the flowwhich creates sound.As an application we study flow which is nearly steady-state and show howto compute the amplitude and frequencies of the sound that it generates.Using this computation we show how to construct a flow meter which canmeasure the rate of flow of a fluid through a given vessel.

##### Some Problems in Feedback Control Theory Motivated by Systems Molecular Biology

I firmly believe that, in the coming years, molecular biology will play a rolein shaping mathematics research analogous to that played by physics during thepast three centuries. New areas in mathematics will arise, and novel problemsin established areas will be formulated. These will provide the conceptualframework needed in order to analyze, and eventually help solve, many of thequestions of central importance to biology.After a quick introduction of some basic concepts in biochemistry andgenetics, as well as some basic notions of feedback control, the emphasis ofthe talk will be on "systems molecular biology" as a generator of new problemsin mathematics, and particularly in control theory.Rather than generalities, the talk will specifically describe several researchvignettes (picked from my own recent work), including (and subject to timelimitations): the development of monotone i/o systems, in order to analyzesignaling cascades under negative feedback (small gain theorems) and positivefeedback (multi-stability and hysteresis); the formulation of an "internalmodel principle", motivated by E-coli chemotaxis; the study of robustly stablechemical reactions, motivated by kinetic proofreading and receptor-ligandmodels, and associated new types of global nonlinear state estimators; a noveltechnique for unraveling gene and protein network structure; and mathematicalobstructions to regular feedback laws, giving rise to the need for hybridcontrollers.

##### Discrete Analytic Functions and Global Information from Local Observation

We observe a certain random process on a graph "locally", i.e., in theneighborhood of a node, and would like to derive information about"global" properties of the graph. For example, what can we know about agraph based on observing the returns of a random walk to a given node?This can be considered as a discrete version of "Can you hear the shapeof a drum?"Our main result concerns a graph embedded in an orientable surface withgenus g, and a process, consisting of random excitations of edges andrandom balancing around nodes and faces. It is shown that by observingthe process locally in a "small" neighborhood of any node sufficiently(but only polynomially) long, we can determine the genus of the surface.The result depends on the notion of "discrete analytic functions" ongraphs embedded in a surface, and extensions of basic results onanalytic functions to such discrete objects; one of these is the factthat such functions are determined by their values in a "small"neighborhood of any node.This is joint work with Itai Benjamini.

##### A processor sharing queue with timing requirements

Many important models of stochastic networks exhibit congestion and delay. This implies that the total time a job spends in the system (the "sojourn time") is typically longer than the actual amount of service time needed by the job. The sojourn time is a classical measure of the performance of a queueing system. Recently, various authors have begun to consider more general measures of the delay experienced in a queueing system. One such measure is the "lead time," a dynamic quantity describing the time until expiration of some deadline which the job may have. The initial lead time of a job could be random and different from the service time of the job. In this talk, we will discuss recent results for the GI/GI/1 processor sharing queue when jobs have timing requirements represented as lead times. Our primary tools are fluid model and state space collapse techniques involving a measure valued process that jointly keeps track of residual service times and lead times of individual jobs in the system. The main result is a heavy traffic diffusion approximation for the (appropriately rescaled) measure valued process. This is joint work with Lukasz Kruk.

##### Calculus and Hodge theory in characteristic p

Calculus means the differentiation and integration of functions and differential forms, and leads naturally to the notion of de Rham cohomology. Classical Hodge theory provides harmonic representatives for de Rham cohomology classes. Its more recent nonabelian version associates a Higgs bundle to a representation $ho$ of the fundamental group and Higgs classes to $ho$-twisted cohomology classes. Calculus and de Rham cohomology make sense algebraically and in any characteristic, but Hodge theory is profoundly analytic. Nevertheless I will describe recent attempts to construct an analog of nonabelian Hodge theory in characteristic $p$ (joint work in progress with V. Vologodsky).

##### Surgeries on manifolds with almost positive isotropic curvature

It is known that a simply connected manifold with positive isotropic curvature (PIC) is homeomorphic to a sphere. The observation that the product metric on $S^1 imes S^n$ has PIC and the fact that the class of manifolds with PIC is closed under connected sums led to the conjecture that the fundamental group of manifolds with PIC is almost free. Significant progress on this conjecture has recently been made by Ailana Fraser. In this talk I will describe the notion of almost PIC (which still implies positive scalar curvature) and I will indicate why this class of manifolds is closed under surgery over a circle. In particular, there is no restriction on the fundamental group for manifolds with almost PIC. By considering higher dimensional surgeries, there is even reason to believe that the class of simply connected manifolds with almost PIC coincides with that of positive scalar curvature. This is joint work in progress with Ingi Petursson.

##### Parametric Tail Index Estimation

We consider a symmetric $alpha$-stable model for heavy tailed time series, which allows for some dependence structure (or memory) in the data. In this context, an estimator for the tail index is $alpha$ is presented, which has a rapid rate of convergence -in particular $Op(n^{-1/2})$ -which is robust under intermediate memory. There is no need for blocking or tuning parameters for this estimator. Small sample results and full asymptotics are provided in this paper, and simulation studies on various $alpha$-stable data sets are given as well.

testing

##### Large Convolutional Codes with Maximal or near-Maximal Distance

In comparison to the literature on linear block codes there existonly relatively few algebraic constructions ofconvolutional codes having some good designed distance. There areeven fewer algebraic decoding algorithms which are capable ofexploiting the algebraic structure of the code.Convolutional codes are typically decoded via the Viterbialgorithm which has the advantage that soft information can beprocessed. This algorithm has however the disadvantage that it istoo complex for codes with large degree or large memory or whenthe block length is large. The algorithm is also not practicalfor convolutional codes defined over large alphabets. There aresome alternative sub-optimal algorithms such as sequentialdecoding and feedback decoding. All these algorithms do not in generalexploit the algebraic structure of the convolutional code.In this talk some good classes of algebraic convolutional codeswill be introduced. These codes are particularly suited for applicationswhere large alphabets are involved. The free distance of these codesis the maximal possible distance a convolutional code of a certainrate and degree can have. It is shown that these codes can decode amaximum number of errors per time interval when compared with otherconvolutional codes of the same rate and degree.These codes have also a maximum or near maximum distance profile.A code has a maximum distance profile if and only if the dual code hasthis property.Professor Smarchande will give a TUTORIAL at 12.30 in APM 7218 on communications

##### Diffusion, advection, and geometry of population habitats

We will discuss the effects of advection along environmental gradients on logistic reaction-diffusion models for population growth. The local population growth rate is assumed to be spatially inhomogeneous, and the advection is taken to be a multiple of the gradient of the local population growth rate. We show that the effects of such advection depend crucially on the gemeotry of the habitats of population: if the habitat is convex, the movement in the direction of the gradient of the growth rate is beneficial to the population, while such advection could be harmful for certain non-convex habitats.

##### Using functions of Brownian motion to test for regression effectin survival studies

Regression models in survival can be expressed in great generality as non proportional hazards models. The particular case of proportional hazards has seen wide application in practice. Inference for these models is difficult and appeals to non standard statistical techniques such as the partial likelihood. One purpose of this talk is to show that standard methods can be used following from simple applications of Donsker's theorem. Brownian motion, Brownian motion with drift, Integrated Brownian motion and Ornstein-Uhlenbeck processes can all be used to throw light on real problems arising in survival analysis.

##### Is there a one-sided quantum group

There exist bialgebras with a left antipode but no right antipode(J.A.Green,W.D.Nichols,E.J.Taft,J.Algebra 65,399-411). We try toconstruct such a left Hopf algebra in the framework of quantum groups.We start with $3$ of the $6$ relations defining quantum $GL(2)$,plus inverting the quantum determinant. In asking that the left antipode, with itsstandard action on the $4$ generators, be an algebra antiendomorphism, weare forced to add new relations. The process stops at a Hopf algebra( two-sided) which seems to be new. It has the unusual feature that itremains non-commutative when $q=1$. Recently, we have dropped thecondition that the left antipode be an algebra antiendomorphism, but try to make it reverse the product only on irreducible words in thegenerators( there is a Birkhoff-Witt type basis). This almost works,but causes trouble on one nasty irreducible word. We hope to overcome this. ( Joint work with Suemi Rodriguez-Romo)

##### Tchebotarev Density Theorem for Prime Cycles in Graphs

I will compare some examples of density of sets of primes in numberfields and function fields with those in graphs. The results require ArtinL-functions of Galois coverings of graphs.

##### Comparing multiple populations

Brief Biography: Dr. Good was born in Montreal, educated at McGill, and received his degrees in mathematical statistics at Berkeley. He has taught anatomy, biology, computer science, mathematics, and physics at the college level. He is the author of five statistics texts, 600+ popular articles on sports andcomputers, sixteen published short stories, and fifteen unpublished novels.

##### Synchronized Permutations, Similarities, and the Exact Analysis of Experimental Designs

Recently, the class of experimental designs that are analyzable by permutation means (thus yielding exact p-values) was extended to the two-factor casethrough the use of synchronized permutations (Salmaso, 2003 and Pesarin 2001). The new tests achieve exact p-values through weak exchangeability(Good, 2002). By recognizing that synchronized permutations can be identified with similarities, these results can be extended to k-factor designs, balancedor not, complete or not.[full paper at: http://users.oco.net/drphilgood/synch02a.htm ]

##### Representation theorems for historical interacting Fisher-Wright diffusions

We consider spatially interacting Moran models and their diffusion limit which are interacting Fisher-Wright diffusions. For both models the historical process is constructed, which gives information about genealogies. For any fixed time, particle representations for the historical process of a collection of Moran models with increasing particle intensity and of the limiting interacting Fisher-Wright diffusions are provided on one and the same probability space by means of a look-down process. It will be discussed how this can be used to obtain new results on the long term behavior. In particular, we give representations for the equilibrium historical processes. Based on the latter the behavior of large finite systems in comparison with the infinite system is described on the level of the historical processes. The talk is based on joint work with Andreas Greven and Vlada Limic.

##### Estimating the binomial parameter n on the basis of known and unknown proportions

Estimation of the number of stem cells involved in repopulationof the bone marrow following marrow transplantation usuallyrelies on methods of cell biology. The amount of stem cellsinvolved in this process may be a determinant of a numberof potential complications of the disease.Statistical methods relying upon some simple assumptionsconcerning cell dynamics can also be used. The problem thenbecomes one which can be framed in a context of binomialsampling. However, unlike the more classic problem where wecondition upon some value n and make inference on a populationparameter p based on the observed number of successes, hereinference is focussed upon n. We are unable to observe thenumber of successes directly. Bayesian methods are also possible.

##### A Bernstein problem for special Lagrangian equations

In this talk, we derive a Bernstein type result for the specialLagrangian equation, namely, any global convex solution must be quadratic.In terms of minimal surfaces, the result says that any global minimalLagrangian graph with convex potential must be a hyper-plane.

##### The Worm Turns

We study the classical worm domain of several complex variables.We develop new results about the Bergman kernel and projection.

##### On the global structure of spacetimes

I will survey some known facts and open questions concerning the global properties of 3+1 dimensional spacetimes, with emphasis on the Cauchy problem. According to the BKL proposal generic singularities with non-stiff matter are space-like and oscillatory while in the presence of stiff matter or scalar fields the singularity is quiescent. I will discuss the picture that emerges from the BKL proposal and review some recent results on nonlinear stability and the Cauchy problem with rough initial data.

##### Invariants of Symmetric Bilinear Forms

%\textfont\bffam=\tenbf \scriptfont\bffam=\tenbf
products on $\Bbb R^n$. Since the dot product is essentially the only
one, they are not very interesting. People actually study a broader
class of objects --- {\it symmetric bilinear forms} --- which include
what physicists call Minkowski space.

The desire to classify symmetric bilinear forms leads to investigating
their invariants. This talk will discuss symmetric bilinear forms as a
case study to motivate the general theory of cohomological invariants,
as presented in a forthcoming book by Serre, et al. A good portion of
this talk will be classical in nature, and so accessible to a more
general audience.

##### Towards a geometric interpretation of the path model

We present a new version of the path model for representations of a semisimple group G. In the new setting the LS-paths are replaced by certain galleries in the affine Coxeter complex. Using the corresponding affine Tits building, we associate in a canonical way to the galleries some finite dimensional projective varieties in the affine grassmannian associated to G. It turns out that these are precisely the intersection homology cycles investigated by Mirkovic and Vilonen.

##### Bode\'s Law and The Neptune Hole : Coincidence or Something More ?

A long time ago, astronomer Johann Bode noticed a remarkable regularity
in the interplanetary spacings in our solar system. It is not just the
regularity in the spacings, but the fact that a few planets previously
unknown were discovered by using the Bode law.

Is it a coincidence ? Is it a semi-physical law ? Or, could it just be
that the mathematical formula characterzing the interplanetary spacings
is embeddable in a parametric class of sequences that will give great fit
to a lot of empirical sequences of small length ? (The suggestion has been
made that it COULD be a sign from a creator too)

We will present the story of Bode\'s law, the story of discovery of the
planets, and the resultant mathematical questions, with some calculations,
and many examples.

##### A survey of crystal graphs

There is a rich and highly-developed combinatorial theory for Schur functions (Young tableaux, the Littlewood - Richardson Rule, etc), but one can argue that it suffers from a few too many seemingly arbitrary choices and miracles.
On the other hand, Kashiwara\'s theory of crystal bases for quantum groups comes close to subsuming this theory, and at the same time is (a) canonical and (b) has a much greater range of applicability (namely, to the representations of semisimple Lie groups and algebras and their quantum analogues).
The main goal of our talk will be to explain that Kashiwara\'s theory can be developed at a purely combinatorial level, and need not rely on any of the representation theory of quantum groups. Even in type A, this leads to a more natural understanding of the combinatorics of Schur functions.

##### Principal-Agent Problems in Continuous Time

Motivated by the problems of optimal compensation of executives and of investment fund managers, we consider principal-agent problems in continuous time, when the principal\'s and the agent\'s risk-aversion are modeled by standard utility functions. The agent can control both the drift (the mean\") and the volatility (the variance\"") of the underlying stochastic process. The principal decides what type of contract/payoff to give to the agent. We use martingale/duality methods familiar from the theory of continuous-time optimal portfolio selection. Our results depend on whether the agent can control the drift independently of the volatility

##### Power

When does one Fibonacci number divide another? Let $F_0 = 0$, $F_1 = 1$, and for $n\\geq 2$, $F_n = F_{n-1} + F_{n-2}$. It is well known
that for $F_m > 1$

This last result was used in Yuri Matijasevi\\u{c}\'s solution of
Hilbert\'s 10th problem.

Using simple combinatorial arguments, we derive previuosly unknown
necessary and sufficient conditions for the following question: For
any $L \\geq 1$

When does $F_m^L$ divide $F_n$?

Our method allows us to answer this same question for any Lucas
sequence of the first kind, defined by $U_0 = 0$, $U_1 = 1$, and for
$n\\geq 2$, $U_n = aU_{n-1} + bU_{n-2}$. This talk is based on joint
work with Harvey Mudd College undergraduate Jeremy Rouse, while
attending the 10th International Conference on Applications of
Fibonacci Numbers.

##### Solvability of Convolution Equations and A Model for Energy Redistributing among Stars

Let X be a fixed random variable and U,Z,W three others, mutually
independent. We ask if the convolution equation X = U(Z + W) can be solved
in U for given Z,W or in W for given U,Z. We also look at certain
generalizations. A particular version arises in a model for energy
redistribution among stars. We present some general theorems, and some
special examples.

##### Congruences between modular forms

The old subject of congruences between elliptic modular forms has lately become relevant to contemporary arithmetic issues, notably by its role in Wiles\' arguments related to the Taniyama-Shimura conjecture. Some classical results on such congruences will be discussed and an approach based on the modular representation theory of reductive groups will be proposed.

##### Elliptic curves mod p and analytic number theory

The reductions modulo primes of a fixed elliptic curve defined over the
rational numbers provide an interesting and useful generalization of the
finite field Z/pZ. The study of Z/pZ as p varies has been a fertile
source of problems in classical analytic number theory. Similar problems
about elliptic curves are providing new challenges for modern analytic
number theory.

In this talk I will first review some of the analogies and especially the
insights of Serre concerning these reductions. In order to understand them
it is helpful to identify the Frobenius in explicit terms. This has some
nice side applications for "non-abelian reciprocity" . It also leads to a
set of problems where the classical sieve techniques break down and new
ones must be found. These problems are often so difficult that the
generalized Riemann hypothesis must be assumed in order to prove realistic
results.

##### Estimating the number of unseen species in a population

The problem of estimating the number of unseen species in a
population based on the results of a single sample of animals
is a familiar one in the statistical literature. In a related
problem associated with genome sequencing the goal is to design
a sampling strategy for finding a specified proportion of the
total number of species. A generalized multinomial model is
applied to estimate the number of unseen species; the model also
forms the basis for a Monte Carlo simulation approach to determing
the sample size required to guarantee that a specified proportion
of the total species are collected. The methods are demonstrated
on simulated data and data from a DNA sequencing application.

##### A simple proof of an (Gittens) index theorem for graphs

We consider the problem which informally can be formulated as follows. Initially a finite set of independent trials is available. If a Decision Maker (DM) chooses to test a particular trial she receives a reward depending on the trial tested. As a result of testing a random finite set (possibly empty) of new independent trials is added to the set of available trials, and so on, but the total number of potential trials is finite. A DM knows the rewards and transition probabilities of all trials. On each step she can either stop testing or continue. Her goal is to select a testing order and a stopping time to maximize the expected total reward. This problem has a long history and is related to the Multi-armed Bandit Problem with independent arms. We prove that an index can be assigned to each possible trial, and the optimal strategy is to use on each step a trial with maximal index among those available. We give a simple procedure for constructing this index.

This is joint work with Isaac Sonin.

##### The Mobius function in characteristic p

For an irreducible polynomial $f(T)$ in ${\\bf Z}[T]$ whose
values are not all multiples of a common prime, the sequence $\\mu(f(n))$
is not expected to have any periodicity properties. In contrast, there
can be periodicity when $f(T) \\in {\\bf F}[u][T]$ with $\\bf F$ a
finite field. That is, the sequence $\\mu(f(g))$ can be periodic as $g$
runs over ${\\bf F}[u]$. This is based on peculiarities of
characteristic p.

We will briefly discuss the case of odd characteristic, and then focus on
the extra subtleties of characteristic 2, where we make an interesting
application of the residue theorem for a certain rational differential
form. The ideas will be made explicit by treating a concrete case: as $g$
runs over ${\\bf F}_2[u]$, $\\mu(g^8+(u^3+u)g^4+u) = 1$.

Remark: Note the unusual day, time, and room for the number theory
seminar this week.

##### New obstructions to doubly slicing knots

A knot is doubly slice if it is the intersection of a three sphere with a
trivially embedded two sphere in a four sphere. The resulting knot splits
the two sphere into two distinct slicing disks for the knot. Thus, the
term \"doubly slice\"".

##### Symmetric functions in superspace

We will discuss how the theory of symmetric functions can be extended to the supersymmetric case, involving anti-commuting variables in addition to those
that commute. We shall see how beautiful combinatorics arises from the super-extension of classical symmetric functions.

##### Semi-classical limit for the first eigenfunction on Riemannian Compact

Recently, we developped an approach to study the concentration of the
first eigenfunction of a positive second order operator on Riemannian
Compact manifolds. The set of limit measures will be described and can be
characterized explicitly. In particular in some cases, the first
eigenfunction sequence concentrates along manifolds of dimension
k(=0,1,2). Explicit formula can be given for the restriction of the
invariant measure on the invariant manifolds.

##### The holomorphic continuation of generalized Jacquet integrals

In the representation theoretic interpretation of the theory of
automorphic forms Fourier transforms at cusps are products of two
quantities. The first (under a multiplicity one condition) is a scalar
containing all of the arithmetic information. The second is a (generalized)
Whittaker model for the representation associated with the form. In this
lecture we will analyze the integrals involved in the second part of this
factorization. These integrals are paramaetrized by points in a complex
vector space and converge and are holomorphic in a half space. The main
result gives an algebraic condition that guarantees a holomorphic
continuation to the entire space. This result generalizes or implies every
known case of a holomorphic continuation of a generalized Jacquet integral.

##### Permutation enumeration via symmetric functions I

An innovative connection between symmetric functions and the study of
permutation enumeration will be described. Generating functions will be
produced which enumerate permutations (and other Coxeter groups) by natural statistics. These techniques consolidate many classic results and give new information about interesting subsets of the symmetric group such as 321-avoiding and alternating permutations. This is the first of two talks
on the subject.

##### Infinitesimals in Combinatorial Game Theory

A combinatorial game is a perfect information game with no chance played
by two players who take turns making moves -- the last player to move wins
the game. There is an algebraic system associated with combinatorial game
theory that features bizarre objects such as infinitesimals -- things that
are positive, yet so small that any sum of them, no matter how many, is
not bigger than any positive number. The real numbers do not have
infinitesimals, but combinatorial game theory is rife with them. While
their structure can be baffling at times, the ideas are very simple.
We\'ll play some games -- and very TINY ones at that! You need to know
nothing to understand the majority of this talk -- we\'ll introduce
everything we need during the talk.

Refreshments will be served.

##### Solving Equations: Tschirnhaus Transformations and Covariants of Finite Groups

In a short paper in 1861 Hermite showed that a general equation of degree 5,
$$x^5 + a_1x^4 + a_2 x^3 + a_3x^2 + a_4x + a_5=0,$$
can be reduced to the form $x^5 + ax^3 + bx + b = 0$. Since then, this
result and related questions have been studied from different viewpoints,
by Felix Klein, David Hilbert, Richard Brauer, Jean-Pierre Serre, Yu I.
Manin and others. More recently, Joe Buhler and Zinovy Reichstein found a
very interesting connection of these problems with the study of rational
covariants of the symmetric group.
We will explain this approach and show how it is related to some classical
invariant theory. (This is joint work with G.W. Schwarz.)

##### Instability in many copies of a representation

The set of unstable vectors of a representation $V$ of a reductive
group $G$, the so-called {\\it nullcone\\/} $N_V$, contains a lot of information about the geometry of the representation $V$. E.g. if $N_V$ contains finitely many orbits then this holds for every fiber of the quotient morphism $\\pi_V\\colon V \\to V/\\!\\!/ G$.

The Hilbert-Mumford criterion allows to describe the nullcone as a union
$\\bigcup G V_\\lambda$, using maximal unstable subspaces of $V_\\lambda \\subset V$ annihilated by a 1-parameter subgroup $\\lambda$ of $G$. They
correspond to maximal unstable subsets of weights which allows some
interesting combinatorics.

We will give some methods how to determine the irreducible components
$GV_\\lambda$ of the nullcone and will describe their behavior if one
considers several copies of a given representation $V$. A rather complete
picture is obtained for the so-called $\\theta$ representations studied by
Kostant-Rallis and Vinberg. E.g. we were able to show that for the 4-qubits
$Q_4:={\\bf C}^2\\otimes {\\bf C}^2\\otimes {\\bf C}^2 \\otimes {\\bf C}^2$ the
nullcone has four irreducible components all of dimension 12 for one copy
and 12 irreducible components for $k\\geq 2$ copies. These 12 components
decompose into 3 orbits under the obvious action of $S_4$ on $Q_4$, each one
consisting of 4 elements, of dimensions $8k+4$, $8k+3$ and $8k+1$.

(This is joint work with Nolan Wallach.)

##### Permtation enumeration via symmetric functions II

We will show how the combinatorics of symmetric functions can give many
new and classic results in the theory of permutation enumeration. Among
other things, a well known result of Garsia and Gessel will be derivied
through this perspective.
This is the second of two talks on the subject; however, this talk is almost
self contained. For those not attending the first talk, we only assume
familiarity with the elementary and homogeneous symmetric functions and the
notions of descents and inversions for permutations.

##### Finite Quantum Chaos

Physicists have long studied spectra (or eigenvalues) of Schroedinger
operators and random matrices, thanks to the implications for quantum
mechanics. Analogously number theorists and geometers have investigated
the spectra of the differential operators known as Laplacians associated
to certain surfaces with a Riemannian distance. For surfaces with
symmetries coming from number theory, this has been termed \"arithmetic

##### Group Gradings on Simple Algebras

In this talk we present some results of our recent efforts (jointly with S.
Sehgal, I. Shestakov and M. Zaicev) to classify all possible abelian group
gradings of finite-dimensional simple algebras from various classes,
including those of associative, Lie and Jordan algebras. In a number of
cases this classification is essentially complete.

##### Geometric properties of mappings between CR manifolds of higher codimension

A classical result in SCV is the fact that a nonconstant holomorphic map sending a piece of the unit sphere in $\\ C^N$ into itself is necessarily locally biholomorphic (and, in fact, extends as an automorphism of the unit ball). Generalizations and variations of this result for mappings between real hypersurfaces have been obtained by a number of mathematicians over the last 30 years. In this talk, we shall discuss some recent joint work with L. Rothschild along these lines for mappings between CR manifolds of higher codimension.

Joint Seminar with UCI

##### Line diagrams, reduced words and Tableaux correspondences

The study of a tree introduced by Lascoux-Schutzenberger
yields a family of tableaux correspondences which can be
used to enumerate the number of reduced factorizations
of a permutation. Representing reduced words by certain line diagrams,
these corresponences can be obtained by an algorithm of D. Little
in a remarkably simple manner. The talk uses a beautiful Applet
also created by D. Little which illustrates the algorithm and
should be helful in any further studies of reduced words.

##### THE SECRET THOUGHTS OF ARCHIMEDES

The ancient Greek mathematicians did not use limits. We will
examine their proofs of some propositions for which our proofs would
involve limits. The discussion will show that their proofs helped the
reader gain insight neither into how these theorems were discovered nor
into how new results might be found. Some historians thought that calculus
was being used to make the discoveries, even though it was not used in
their proofs. Indeed, in about 1906, a copy of a book of Archimedes\'s
book, \"The Method\"" was discovered. Archimedes revealed his process of

##### DEQUANTIZATION OF MATHEMATICS

This is a talk on heuristic aspects of Idempotent Mathematics
in the spirit of current works of V.P. Maslov and his collaborators.
Idempotent Mathematics can be treated as a result of a dequantizati-
on of the traditional Mathematics as the Planck constant tends to
zero taking pure imaginary values. For example, the field of real
numbers can be treated as a quantum object whereas idempotent semi-
rings can be examined as \"classical\"" or \""semiclassical\"" objects

##### L-FUNCTIONS AND GALOIS THEORY

I will first recall some examples of L-functions and indicate some of the
ways they have been important in algebraic number theory. I will then
describe what appears to be their intimate connection with Galois theory
(eg the Fontaine-Mazur conjectures), as well as touching on their
relationship with algebraic geometry and automorphic forms. Finally, I will
discuss what can be proved in this direction.

##### Positivity aspects of the Fantappie transform in C\\^n

The main problem addressed in the talk is the
characterization of Fantappie transform of positive
measures in the unit ball of C\\^n. The analogous real
results were obtained by G.M.Henkin and A.A. Shananin,
in the line of the classical theorem of Bernstein on the
line. I will propose an approach based on Hilbert spaces
of analytic functions. This will provide, among other things,
a novel proof of Martineau\'s duality theorem.
Based on joint work in progress with John McCarthy.

##### Initial-boundary Value Formulation for Einstein\'s Equations in Weyl Form

The initial-boundary value formulation for the Einstein equations has
a number of special features when compared with that for other partial
differential equations. These issues are briefly discussed, and an
approach to prove local in time existence is presented. The main idea
is to rewrite Einstein\'s equations into an equivalent form, called Weyl
system. This work follows the main idea in, though is simpler than,
the work by Friedrich and Nagy, Comm. Math. Phys. 201, 619, (1999).

##### ASYMPTOTIC IMPROVEMENT OF THE GILBERT-VARSHAMOV BOUND

Given positive integers $n$ and $d$, let $A_2(n,d)$ denote the maximum
size of a binary code of length $n$ and minimum distance $d$. The well
known Gilbert-Varshamov bound asserts that $A_2(n,d) \\geq 2^n/V(n,d-1)$,
where $V(n,d) = \\sum_{i=0}^d {n \\choose i}$ is the volume of a Hamming
sphere of radius $d$. We show that, in fact, there exists a positive
constant $c$ such that
$$A_2(n,d) \\geq c {2n \\over (n,d-1)}$$

whenever $d/n \\le 0.499$. The result follows by recasting the Gilbert-
Varshamov bound into a graph-theoretic framework and using the fact that
the corresponding graph is locally sparse. Generalizations and extensions
of this result will be briefly discussed.
*joint work with T. Jiang, Math Department, University of Miami

##### The mathematics of Internet graphs

Many very large graphs that arise in Internet and telecommunications
applications share various properties with random graphs (while some
differences remain). We will discuss some recent developments and mention
a number of problems and results in random graphs and algorithmic design
suggested by the study of these \"massive\"" graphs.

##### Counting embeddings of cubic rings and Fourier coefficients

A classical problem in number theory is to count the number of ways a
quadratic form can be represented by another. The generating function for
these numbers turn out to be a modular form (the so-called theta
functions). In this talk, I will discuss an analogous problem involving
cubic forms, and what sort of modular forms it leads to.

##### The Word Problem and Relations in Rings

The Word Problem is known to be undecidable. Many interesting problems in computational mathematics, however, lead to variants of the Word Problem. This talk will examine the problem of producing an algorithm to decide if a relation in a ring is a consequence of some given relations.

##### Traffic, Entropy and the World Wide Web

It is reasonable to expect that some of the techniques
used in road traffic theory would apply to modeling of
traffic on the World Wide Web. We review the
derivation and use of entropy maximizing models for the
traffic distribution problem, which calls for the
solution of a matrix balancing problem, and then apply
a similar approach to estimating traffic on the WWW,
which results in a hybrid matrix balancing
model. Recent work has shown that a more general
non-linear interior-point optimization algorithm is
also surprisingly efficient for these very large-scale
problems.

##### The minimum area convex lattice $n$-gon

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, and explain what the shape of
the minimizing convex lattice $n$-gon is. This is joint work with
Norihide Tokushige.

##### The fractional Helly number for convex lattice sets

A set of the form $C\\cap\\bf{Z}^d$, where $C\\subseteq R^d$ is convex
and $Z^d$ denotes the integer lattice, is called a {\\it convex
lattice set}. I will explain that the Helly number
of $d$-dimensional convex lattice sets is $2^d$.
However, the {\\it fractional Helly number\\/} is only $d+1$:
For every $d$ and every $\\alpha\\in (0,1]$ there exists $\\beta>0$
such that whenever $F_1,\\ldots,F_n$ are convex lattice sets in $\\bf{Z}^d$
such that $\\bigcap _{i\\in I} F_i\\neq\\emptyset$
for at least $\\alpha{n\\choose d+1}$ index sets $I\\subseteq\\{1,2,\\ldots,n\\}$
of size $d+1$, then there exists a (lattice) point common to
at least $\\beta n$ of the $F_i$. This implies a $(p,d+1)$-theorem
for every $p\\geq d+1$; that is, if $H$ is a finite family
of convex lattice sets in $\\bf{Z}^d$ such that among every $p$ sets of $H$,
some $d+1$ intersect, then $H$ has a transversal of size
bounded by a function of $d$ and $p$. This is joint work with J.
Matousek.

##### \"How to keep a secret using only what you learned in math class\"""

We will show how number theory can be used to make communication secure.

Refreshments will be served!

##### Solving the Erdos-Folkman conjecture

For a sequence A of integers, S(A) denotes the collection of partial sums
of A. About forty years ago, Erdos and Folkman made the following
conjecture: Let A be an infinite sequence of integers with density at
least $Cn^{1/2}$ (i.e., A contains at least $Cn^{1/2}$ numbers between $1$ and n
for every larger n), then S(A) contains an infinite arithmetic
progression. Partial results have been obtained by Erdos (1962), Folkman
(1966), Hegyvari (2000), Luczak-Schoen (2000). Together with Szemeredi,
we have recently proved the conjecture. In this talk, I plan to survey
this development.

##### Representations of classical groups: tensor products and minimal orbits

We consider tensor products $V_{\\lambda}\\otimes V_{\\mu}$ of
irreducible representations of a classical group $G$.
In general, such a tensor product decomposes in irreducible
components. It is a fundamental question how the components
are embedded in the tensor product.
Of special interest is the so-called Cartan component
$V_{\\lambda+\\mu}$. It appears exactly once in the decomposition.

On the other hand, one can look at decomposable tensors
(tensors of the form $v\\otimes w$) in the tensor product.

A natural question arising here is the following: are the
decomposable
tensors in the Cartan component given as the closure of
the minimal orbit in $V_{\\lambda+\\mu}$? If this is the
case we say that the Cartan component is small.

We give a characterization and a combinatorial description
of tensor products with small Cartan components. In particular,
we show that for general $\\lambda$, $\\mu$, Cartan components
are small.

##### Multiplicity formula for cubic unipotent Arthur packets

Arthur has given rather precise conjectures on the
decomposition of the regular representation L^2(G(F)\\G(A)),
where G is a simple Lie group over a number field F,
with adele ring A. In particular, the irreducible constituents are
partitioned into classes called Arthur packets. I will discuss the
construction of some of these packets when G is the exceptional group
$G_2$ and how one can justify that the constructed packets are the right
ones.

##### Closed $G_2$ Structures

Jointy Sponsored by UCI - This seminar will be held at UCI

##### The classical Dirichlet problem with rational data

The Dirichlet problem (DP) for the Laplace operator can be used to model a
number of different physical situations. For instance, if the surface of
a ball is kept at a given constant temperature f, then the steady state
temperature inside the ball is given by the solution of the Dirichlet
problem in the ball with data f on the sphere. A curious fact is that the
solution of the DP with rational data on the unit disk in the plane is
rational, whereas the corresponding statement is not true in 3-space (or
in any dimension greater than or equal to 3 for that matter). This can be
explained by quite elementary methods.

Refreshments will be served.

##### Random fields, brains and manifolds

I shall start by discussing some statistical problems related to mapping the brain, both the cerebrum (a 3-dimensional object) and the cerebral cortex, or \"brain surface\"" (a 2-dimensional manifold in 3-dimensional space). This problem has motivated recent deep results describing the geometry of Gaussian random fields on manifolds

##### Optimizing Stability and Controllability via Pseudospectra

A matrix is stable if its eigenvalues are in the left half of the complex
plane. More practical stability measures include the pseudospectral abscissa
(maximum real part of the pseudospectrum) and the distance to instability
(minimum norm perturbation required to make a stable matrix unstable).
Likewise, the classical definition of controllability is not as useful
as a measure of the distance to uncontrollability.

Matrices often arise in applications as parameter dependent.
Optimization of stability or controllability measures over parameters is
challenging because the objective functions are nonsmooth and nonconvex.
We solve such optimization problems, locally at least, via a novel method
based on gradient sampling. One of our stability optimization examples is a
difficult problem from the control literature: finding stable low-order
controllers for a model of a Boeing 767 at a flutter condition. We also give
a controllability optimization example and explain its connection with an
interesting open question: how many connected components are possible for
pseudospectra of rectangular matrices?

joint work with
James V. Burke, University of Washington, Seattle, WA

##### An Elementary Introduction to Juggling and Juggling Mathematics

Have you ever seen someone juggle and wonder how he or she does
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 explain how juggling sequences can be used to
describe simple juggling patterns and will address some of the
mathematical questions related to juggling sequences. I will also
illustrate some juggling patterns by juggling them (when I\'m not picking
up the balls off the floor) and by using a juggling animator program to
juggle patterns that are too difficult for me.

Refreshments will be served!

##### \"Values of Zeta Functions at Negative Integers and Toric Varieties\"""

In this talk, we will explore the connections between
special values of zeta functions, invariants of toric
varieties, and generalized Dedekind sums. We use invariants
arising in formulas for the Todd class of a toric variety
to give formulas for the zeta function of a real quadratic
number field at nonpositive integers.

##### Evolutionary Game Dynamics in Finite Populations

We study the evolutionary game dynamics of a two-strategy game. In infinite populations, the well-known replicator equations describe the deterministic evolutionary dynamics. There are three generic selection scenarios. The dynamics of a finite group of players has received little attention. We provide a framework for studying stochastic evolutionary game dynamics in finite populations. We define a Moran process with frequency dependent fitness. We find that there are eight selection scenarios. And for a given payoff matrix, a number of these sceanrios can occur for different population size. Our results have interesting applications in biology and economics. In particular,
we obtain new results on the evolution of cooperation in the classic repeated Prisoner\'s Dilemma game. This is joint work with Drew Fudenberg and Martin Nowak.

##### TQFT\'s in Dimension 2+1 over the Cyclotomic Integers

A Topological Quantum Field Theory (TQFT) is a functorial extension of invariants of 3-manifolds to manifolds with boundaries. They are thus highly structured and imply, for example, nontrivial representations of the mapping class groups. A large family of such TQFT\'s is given by the Witten-Reshetikhin-Turaev TQFT\'s. Assuming a mild modification of the TQFT axioms it is possible to define them over the cyclotomic integers (rather than just the complex numbers). The rich ideal structure of this ring combined with the modified functoriality yields a new and quite subtle tool to investigate various properties of the mapping class groups, specific 3-manifolds, and some of their classical invariants. In the talk I will give several examples of such applications.

##### D4 modular forms

I will discuss modular forms on two algebraic groups of type
D4. These two groups are naturally associated to the two octonion
algebras over the rationals. After introducing the basic properties of
modular forms on these two groups, I will discuss a theta-correspondence
between them. This can be thought of as an octonionic generalization of
the Jacquet-Langlands correspondence.

##### Statistical zero-knowledge proofs with efficient provers: lattice problems and more

We construct several new statistical zero-knowledge proofs
with$_efficient_provers_,$ i.e. ones where the prover strategy
runs in probabilistic polynomial time given an NP witness for
the input string.

Our first proof systems are for approximate versions of the
Shortest Vector Problem (SVP) and Closest Vector Problem (CVP),
where the witness is simply a short vector in the lattice or a
lattice vector close to the target, respectively. Our proof
systems are in fact proofs of knowledge, and as a result,
we immediately obtain efficient lattice-based identification
schemes which can be implemented with arbitrary families of
lattices in which the approximate SVP or CVP are hard.

We then turn to the general question of whether all
problems in SZK intersection NP admit statistical zero-knowledge
proofs with efficient provers. Towards this end, we give
a statistical zero-knowledge proof system with an efficient prover
for a natural restriction of Statistical Difference, a complete
problem for SZK. We also suggest a plausible approach to resolving
the general question in the positive.

Joint work with Salil Vadhan (Harvard University).
Talk based on a paper presented at CRYPTO $2003.$

##### The mixing time for simple exclusion

We obtain a tight bound of $O(L^2 log r)$ for the mixing time of the exclusion process in $Z^d/LZ^d$ with $r <= L^d/2$ particles.

##### 1-Motives, Etale Cohomology and Equivariant Iwasawa Theory

The classical conjectures of Gross and Brumer-Stark seem to describe two
completely unrelated properties of special values of Galois equivariant
global L-functions. In this talk, we will develop a general Equivariant
Main Conjecture in Iwasawa Theory which captures the Brumer-Stark and
Gross phenomena simultaneously and works equally well in characteristics $0$ and p. The characteristic p side of the theory draws its main ideas from
Deligne\'s construction of $1-motives$ associated to smooth, projective
curves over finite fields. The characteristic $0$ side of the theory is
based on our new construction of number field analogues of the l-adic
realizations (i.e. l-adic etale cohomology groups) of Deligne\'s $1-motives$
and is deeply rooted in earlier work of Tate and Ritter - Weiss on the
theory of multiplicative Galois module structure. Time permitting, we will
also provide evidence in support of this new equivariant Iwasawa theoretic
Rubin - Stark - type conjectures on special values of global L-functions.

##### An iterative algorithm for ill-posed inverse problems where the object has a sparse wavelet expansion

Many approaches exist to compute the (approximate) inverse of an operator K to recover an approximation to f from a dataset that represents a noise-corrupted version of Kf. Several approaches have been proposed that are adapted to the special case where f has a sparse wavelet expansion, a case that applies to many types of images or other types of signals; an example of the operator K in this context is, e.g., blurring, the convolution with a known function.
The talk will present an iterative approach to solve this problem, which can be used with respect to arbitrary orthonormal bases. The algorithm is similar to the Landweber algorithm, except that the prior information incorporated into the variational functional uses a weighted $l^p-norm$ of the wavelet coefficients instead of the $l2-norm$, standard for Landweber methods. This iterative approach converges in norm and is stable; some applications will be shown.

This is joint work with Michel Defrise (Vrije Universiteir Brussel) and Christine De Mol (Universite Libre de Bruxelles)

##### A generalization of the Cesaro integral operator to Hardy $(H^p )$ spaces

We introduce the Cesaro operator and trace its history. We define a generalization of this operator to all Hardy spaces $H^{p}$ (disc). We discuss the boundedness and compactness of such operators. We improve a result of Hardy and Littlewood on primitives of $H^{P}$ functions. One can compute the essential spectrum of a subspace of these operators .

##### Periods of modular forms on Shimura curves

We study the relation between the Petersson norm of a holomorphic GL(2) form $f$ and that of its (suitably normalized) Jacquet-Langlands lift $g$ to a Shimura curve. The ratio of these two norms was previously shown to be algebraic by Shimura and rational by Harris-Kudla. We prove an integrality result for this ratio and explain the arithmetic significance of this ratio in terms of certain congruences satisfied by $f$

##### Irrational Coin Tossing

This talk was inspired by a problem on the $1989$ Putnam Mathematical
Competition:

If p is an irrational number such that $0 < p < 1$, is there a finite game
with an honest coin such that the probability of one player winning the
game is p ?

We consider several solutions of this problem. An extension of the
problem leads naturally to an example of a continuous strictly increasing
function with derivative equal to zero almost everywhere.

Refreshments will be served.

##### Towards an analog of Stein theory in infinite dimensions

Simply put, a Stein manifold is a closed complex submanifold of
complex Euclidean space. Stein manifolds have rich function theory and
many of their global properties can be understood with the help of sheaf
cohomology. As a long term goal we would like to prove vanishing theorems
for closed submanifolds of Hilbert spaces, and give a cohomological
characterization of these submanifolds. In this talk we survey the
available results in this direction and look at the possible near future.

##### Lie Algebras and Superalgebras graded by Root Systems

I will discuss examples and classification of Lie algebras and superalgebras graded by root systems focusing on exceptional algebras (The Freudenthal-Tits Magic Square) and the recently discovered exceptional superconformal algebra.

##### Degeneracy loci, Quiver coefficients and Schubert calculus

In this talk, we describe some interactions between the combinatorics and algebraic geometry of the following setting studied by A. Buch and W. Fulton: Let $X$ be a smooth complex algebraic variety and $E_{0}to E_{1}to cdots to E_{n}$ be a sequence of vector bundles and maps over $X$. This gives rise to a degeneracy locus'' in $X$.
What are formulas for this locus in the $K$-theory? This question was originally answered in terms of combinatorially defined Quiver coefficients'', which were conjectured to alternate in sign according to codimension.
Combinatorial formulas for the Quiver coefficients, especially those that explain the alternating signs, are of interest since they provide, e.g.,:

(1) new `Giambelli-type'' formulas for Schubert classes, for both
classical and quantum cohomology of partial flag varieties;
(2) generalizations of the classical and K-theory Littlewood-Richardson
rules for Grassmannians;
(3) generalizations of the classical Giambelli-Thom-Porteous determinantal
formulas.

We present explanations for the alternating sign phenomenon and give new combinatorial formulas for the Quiver coefficients. Also, we suggest a geometric rationale for the alternating signs: the Quiver coefficients are Schubert structure constants for flag varieties. Combinatorial generalizations of the Quiver coefficients to the other classical Lie types will also be iscussed.

Our answers will involve semistandard tableaux, reduced words of
permutations and the combinatorics of Schubert polynomials.

This talk is based on math.AG/0211300, math.CO/0306389,
math.CO/0307019 and math.CO/0311390.

##### Coloring, quantum mechanics, and Euclid

Can you color the plane with three colors so that every equilateral
triangle with sides of length 1 has one vertex of each color? In the talk
I'll answer this question and describe several generalizations. Some of
these have implications for the foundations of quantum mechanics (the
Kochen-Specker theorem). I'll explain these using only linear algebra and
some elementary number theory, without assuming any knowledge of quantum
Postulates. I'll conclude with a brief discussion of this connection.

Refreshments will be served.

##### \"Normal forms of holomorphic vector fields\"""

We will give an introduction and a review of some
known results about normal forms of holomorphic singular vector fields.

##### Singular complete integrability

We will focus on the study of families of commuting (singular) vector
fields. We shall give sufficient conditions which will ensure that the family
can holomorphically normalized.

##### \"How to keep a secret using only what you learned in math class\"""

We will show how number theory can be used to make communication secure.

Refreshments will be served!

##### Abelian extensions of number fields with constant local degrees

Let $K$ be a number field. Given a positive integer $n$, does there exist
an algebraic extension $L/K$ with local degree $n$ at all finite places of
$K$, and degree two at the real places if $n$ is even?

This problem comes from Brauer groups of fields: given a field $K$ and a
positive integer $n$, is there an algebraic extension $L/K$ such that the
relative Brauer group $Br(L/K)$ is equal to the $n$-torsion subgroup of
the Brauer group $Br(K)$ of $K$?

In general the answer to the latter question is no, a counterexample
coming from two dimensional local fields. The first problem is
essentially equivalent to the second when $K$ is a number field, in which
case no counterexample has been found as yet. In fact, the answer is
affirmative when $K$ is the rationals $\\Bbb Q$, and for general global
fields under certain hypotheses.