Every physical theory has to confront two issues: 1) What are the meaningful questions ("observables")? 2) What do they mean?i.e., what part of the theory purports to correspond to facts in the universe, and how? In quantum gravity (QG) these issuessometimes become particularly entangled. This talk will reviewthe causal set approach to QG and then describe recent advances that go some way toward answering these questions, showing how they find a clear formulation and solution in the case of a toy stochastic model of causal set cosmology, and giving brief prospects for the extension to a quantum theory. The toy stochastic model defines an evolution for posets that is analogous to the dynamical models for the growth of massive random graphs which are of current interest.

We define an infinite sequence of new invariants, $delta_n$, of agroup G that measure the size of the successive quotients of the derivedseries of G. In the case that G is the fundamental group of a 3-manifold,we obtain new 3-manifold invariants. These invariants are closely relatedto the topology of the 3-manifold. They give lower bounds for theThurston norm which provide better estimates than the bound establishedby McMullen using the Alexander norm. We also show that the $delta_n$ giveobstructions to a 3-manifold fibering over $S^1$ and to a 3-manifold beingSeifert fibered. Moreover, we show that the $delta_n$ give computablealgebraic obstructions to a 4-manifold of the form $X x S^1$ admitting asymplectic structure even when the obstructions given by theSeiberg-Witten invariants fail. There are also applications to theminimal ropelength of knots and links in $S^3$. In addition, wediscuss the applications to the cut number of a 3-manifold (this is alsoknown as the corank of the fundamental group of the 3-manifold)

The most famous zeta function is Riemanns. We will discuss its basicproperties, for example its expression as a product over primes due toEuler. Then we consider the analog for a finite connected graph X. Thatmeans we must discuss primes in X. They will be closed paths. Then it iseasy to figure out what the Iharas zeta function of X is. We use thiszeta function to obtain a graph analog of the prime number theoremcounting the number of primes of a certain length in X. When X isregular, the poles of the Ihara zeta function of X satisfy an analog ofthe Riemann hypothesis iff the graph is Ramanujan (meaning that itprovides a good communication network). We will give examples of graphsfor which the Riemann hypothesis is true and other examples for which itis false.

Let $V$ be a representation of a (complex) reductive group $G$. A classical theorem due to Weyl says that the simultaneous invariants (and covariants) of any number of copies of $V$ can be seen in $dim V$ copies. This means that the invariants of more than $dim V$ copies are given by polarization, those for less by restriction. (There are stronger results in case the representation is orthogonal or symplectic.)We are studying the question whether certain geometric objects associated to a representation have a similar behavior. A trivial example is the structure of orbits and their closures which can also be seen in $dim V$ copies. Another example is the rationality of the quotients $V^m/G$ in the sense of geometric invariant theory. Again, it is easy to see that if the quotient is rational for $m$ copies then it is also rational for more than $m$ copies.In the talk we are mainly interested in the set of unstable vectors, the so-called nullcone of the representation. We show that for a certain number $m$, calculated from the weight system of $V$, the number of irreducible components of the nullcone is the same for $geq m$ copies, and that all these components have a nice resolution of singularities. Another interesting question here is if the polarizations define the nullcone, since this has important applications to the computation of invariants. This, in turn, is related to the problem of linear subspaces in the nullcone which is widely open, even in very classical situations.(This is joint work with Nolan Wallach from UCSD. It arose from the study of the representations $C^2otimes C^2otimes cdots otimes C^2$ under $SU_2 imes SU_2 imes cdots imes SU_2$ which play a role in some mathematical aspects of quantum computing.)

In this talk we make a (perhaps) unexpected link between the combinatorialnotion of partially ordered sets (posets) and certain families ofmatrices. Using this link, we give a simple, matrix-based proof of theMoebius Inversion Formula on a poset and a variant of it. It is thisvariant which is really at the heart of the talk, for it will allow us tosimultaneously analyze certain properties (determinant, inertia, etc.) ofentire families of matrices via their $LDL^T$-Factorizations.Refreshments will be provided

Let $G$ be a group and let $V$ be a finitedimensional $G$-module. Let $B$ denote the algebra of $G$-invariantpolynomial differential operators on $V$. It is natural to pose thefollowing questions:1) What is the representation theory of $B$? What are the primitiveideals of $B$?2) Does $B$ have finite dimensional representations? If so, are theycompletely reducible?medskip
oindentLittle is known about these questions when $G$ is noncommutative. Wegive answers for the adjoint representation of SL$_3(C)$, alreadyan interesting and difficult case.

We give a quantitative proof that forsufficiently large N=N(c), every subset of$[N]^2$ of size at least $cN^2$ contains a square, i.e. four pointswith coordinates {(a,b),(a+d,b),(a,b+d),(a+d,b+d)}.

Carl Boyer, in the introduction of his 1949 calculus textbook,remarked that "Mathematics is as much an aspect of culture as it is acollection of algorithms." Indeed, any collection of human beings whichone might identify as having the rudiments of society -- a spokenlanguage, a sense of spirituality, a rule of law, etc. -- have also alwayshad some sense of number as a means of codifying the world around them.In this talk, we will examine how different cultures at different times havedeveloped their sense of number, space and form. We will also look therecurring roles basic tools --- such as rope, rods, and rocks --- haveplayed in their development and refinement --- be it ancient Incaspreadsheets in South America, stone altars built by rope in India, orbroken notched sticks on which the British treasury depended well into the1800s! This talk is aimed at anyone with an interest in culture,language, philosophy, history, ... or, of course, mathematics.REFRESHMENTS WILL BE SERVED!

Analogies between number fields and function fields have inspired newdevelopments in number theory for a long time. We will discuss asurprising non-analogy related to the distribution of primes. Forinstance, it is expected that any irreducible in ${mathbf Z}[t]$ havingat least two relatively prime values will take prime values infinitelyoften. (An example is $t^2+1$, while a nonexample is $t^2+t+2$, since$n^2+n+2$ is always even.) The analogue in ${mathbf F}_p[x][t]$ isfalse, e.g., $t^8+x^3$ is irreducible in ${mathbf F}_2[x][t]$ but$g(x)^8+x^3$ is reducible in ${mathbf F}_2[x]$ for every $g(x)$.

Whole-genome expression experiments provide the building blocks forunderstanding cellular control mechanisms underlying health anddisease, from cardiac function to learning and memory to oncogenesis.To put post-genomic information to work requires buildingbiophysically detailed, quantitative models for the biochemicalinteractions controlling cell behavior. Such biochemical signalingnetworks form a class of high-dimensional, stochastic nonlineardynamical systems, analytical and numerical techniques for the studyof which are only in the early stages of development. I will presentsome approaches to reducing the complexity of cellular signalingnetworks and to extending the classical deterministic approximationsfor these systems to include fluctuation effects. I will focus onapplications to biophysical systems currently under investigation at UCSD, Scripps and Salk Institute.

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We discuss applications of the Milnor-Thom Theorem, a result from AlgebraicGeometry concerning signs of polynomials, to the combinatorics of geometricconfigurations. In particular, we define an {it abstract lineconfiguration}, modelled on line configurations in $mathbb{R}^3$, and showthat there are many forbidden sub-configurations. We discuss theimplications for unavoidable "stacked" sets of lines via theErdH{o}s-Hajnal conjecture, and pose several open questions. Joint workwith Uli Wagner of ETH Z"{u}rich.

Baseball is a sport whose statistics are ingrained in the collectiveconsciousness of their fans. Ask any baseball fanatic to identify thesenumbers, and the response should be immediate: 56, 2130, 2632, 61 in 61,73, 4191, 4256, .406, 714, 755, 5714, 511, 7. Baseball fans also like toargue about which player or team is better, and the extensive history andrich landscape of numbers in the sport lends itself to spirited chats overa hot stove as well as mathematical analyses. It is not quite the HotStove season, but we will chat about some of the mathematics used toanalyze baseball statistics -- the study of baseball statistics is knownas "sabermetrics." Mathematical ideas like Markov chains, Monte Carlosimulation, and linear regression are used to shed new light on Americas sporting pastime. Some rudimentary knowledge of baseball and baseballstatistics is probably helpful, but not absolutely necessary.

We consider a two-dimensional weakly dissipative dynamical system withtime-periodic coefficients. Their time average is governed by a degenerateHamiltonian whose set of critical points has an interior. The dynamics ofthe system is studied in the presence of three time scales. Using themartingale problem approach and separating the involved time scales, weaverage the system to show convergence to a Markov process on a stratifiedspace. The corresponding strata of the reduced space are a two-sphere, apoint, and a line segment. Special attention is given to the domain of thelimiting generator, including the analysis of the gluing conditions at thepoint where the strata meet. The gluing conditions resulting from thehierarchy of the time scales are similar to the conditions on the domain ofskew Brownian motion.

In this talk I will present joint work with John Etnyre on classificationand structure theorems for Legendrian knots in the standard contact3-space. I will describe some of the unexpected phenomena that arise alongthe way.

We compute the structure of the Lie algebras associated to twoexamples of branch groups, and show that one has finite width whilethe other, the "Gupta-Sidki group", has unbounded width(Corollary~
ef{cor:gamma:rk}). This answers a question by Sidki.We then draw a general result relating the growth of a branch group,of its Lie algebra, of its graded group ring, and of a naturalhomogeneous space we call emph{parabolic space}, namely thequotient of the group by the stabilizer of an infinite ray. Thegrowth of the group is bounded from below by the growth of itsgraded group ring, which connects to the growth of the Lie algebraby a product-sum formula, and the growth of the parabolic space isbounded from below by the growth of the Lie algebraFinally we use this information to explicitly describe the normalsubgroups of $G$, the "Grigorchuk group". All normal subgroupsare characteristic, and the number $b_n$ of normal subgroups of$G$ of index $2^n$ is odd and satisfies${limsup,liminf}b_n/n^{log_2(3)}={5^{log_2(3)},frac29}$.

We compute the structure of the Lie algebras associated to twoexamples of branch groups, and show that one has finite width whilethe other, the \"Gupta-Sidki group\", has unbounded width(Corollary~
ef{cor:gamma:rk}). This answers a question by Sidki.We then draw a general result relating the growth of a branch group,of its Lie algebra, of its graded group ring, and of a naturalhomogeneous space we call emph{parabolic space}, namely thequotient of the group by the stabilizer of an infinite ray. Thegrowth of the group is bounded from below by the growth of itsgraded group ring, which connects to the growth of the Lie algebraby a product-sum formula, and the growth of the parabolic space isbounded from below by the growth of the Lie algebraFinally we use this information to explicitly describe the normalsubgroups of $G$, the \"Grigorchuk group\". All normal subgroupsare characteristic, and the number $b_n$ of normal subgroups of$G$ of index $2^n$ is odd and satisfies${limsup,liminf}b_n/n^{log_2(3)}={5^{log_2(3)},frac29}$.

The permutation $pi$ of $1,2,ldots,n$ that satisfies> $$0 < { pi(1) alpha } < { pi(2) alpha } < cdots < { pi(n)alpha} < 1$$ ($alpha$ is any irrational) has been studied from a combinatorialviewpoint (S—s, Boyd) and from an analytic viewpoint (Schoi§engeier). I will present some results on algebraic properties of this permutation, the most significant being a mysterious appearance in the study of nearly periodic binary words (a.k.a., Sturmian words) with the representation theory of symmetric groups.

The Hilbert series of the graded ring associated to a projective variety contains a lot of information about the variety, {it e.g./}, dimension, degree, arithmetic genus etc. If the projective variety is nice then the Hilbert series can be (uniquely) writtenin the form $h(t)=f(t)/(1-t)^n$, where $f(t)$ is a polynomial with non-negative integer coefficients and $f(1)
ot =0$.In this talk we consider the Hilbert series of determinantal varieties (including symmetric and skew-symmetric determinantal varieties). These varieties arise naturally in many branches of mathematics, {it e.g.}, in classical invariant theory. We give an interpretation of the coefficients of the numerator $f(t)$ of the Hilbert series as dimensions of representations of certain compact Lie groups. (The presented work is joint work with Enright and is an extension of previous work by Enright and Willenbring.)

In model theory, we study the definable sets in a structure.This becomes applicabe to another field of mathematics ifthe definable sets are the objects of study in that field.For example, the definable sets in an algebraically closed field are precisely the constructible sets, and hence thetools of model theory can be used in algebraic geometry.Mathematically, one also studies quotients, but this canbe a problem model-theoretically, as the quotient by a definable equivalence relation in general cannot be expected to be definable. If every quotient can be identified with a definable set, we say that the structure eliminates imaginaries.Algebraically closed fields do eliminate imaginaries, but valued fields in general do not, at least in the simplest language for studying them. In this talk, I will explain the above ideas more precisely, discuss the obstacles to eliminatingimaginaries in valued fields, and describe the richer languagein which valued fields do eliminate imaginaries, as provedin recent work by myself, Ehud Hrushovski and Dugald Macpherson.

I will describe an existence theorem for traveling waves in water. Thisis aproblem of the dynamics of a free surface of an incompressible fluid.The first suchresult in two dimensional settings is due to T. Levi-Civita and D.Struik in the 1920's.In a recent paper we prove a general result for three dimensions (well,for anynumber of dimensions), when there is surface tension. The approach issurprisinglyclose to the Lyapunov center theorem of A. Weinstein, using the fact due toV. E. Zakharov that the water waves problem is a Hamiltonian system.Withoutsurface tension the problem exhibits small divisors, and is more difficult.

We give a criterion for the hypoellipticity of a class of differentialoperators of Hormander type$$L = 1/2sum_{i = 0}^n X_i^2 + X_0.$$Our criterion is weaker than the usual finite-type Hormander condition and allows for decencies of $L$ of exponential order on hypersurfaces in the ambient space. The methods of proof are probabilistic, being based upon an analysis of the regularity of the diffusion process driven by $L$. These methods are also used to study the Dirichlet problem associated with exponentially degenerate operators $L$.

We consider the decomposition rules for the tensor powers of thesmallest representations of quantum groups of type E. For the caseswhere the rank is not 9, H. Wenzl has found uniform combinatorialbehavior for decomposing certain summands of the tensor powers usingLittelmann paths. From this he describes generators of part of thecentralizer algebras of these summands in terms of R-matrices acting onpath spaces and thus obtains a 2-parameter family of representations ofArtin's braid group that generalizes the BMW-algebras. In currentresearch, with an eye towards extending the results to the affine (rank9) case, we find a submodule of the tensor power whose simpledecomposition follows the same pattern as in the cases considered byWenzl. The summands we define have a finite decomposition, and we showthat the inclusion rules for these summands behave well with respect tothe Littelmann path formalism. We also note that the eigenvalues of theaction of the R-matrices are predicted by Wenzl's formulas and we can safely follow his program for describing the centralizer algebra.

A crystal is a shape which tessellates the plane (or space) in a periodicway, so that the pattern repeats at regular intervals in all directions.The group of symmetries (translational, rotational, reflectional) of sucha tessellation is called a crystallographic group. In two dimensions thereare exactly 17 different kinds of symmetry, the so-called "wallpapergroups", which I'll describe. I'll also talk about what happens in threedimensions, in hyperbolic space, and how you can make "quasiperiodic"tessellations (Penrose tilings) with five-fold symmetry.

Our good calculus students know that $sum_{n=1}^infty frac 1n$diverges and that $sum_{n=1}^infty frac{(-1)^n}{n}$ converges. Ourvery good students can even explain why $sum_{n=1}^infty frac{(-1)^{lfloor n /3
floor}}{n}$ converges. Our stellar calculusstudents may even be able to explain why $sum_{n=1}^infty frac{(-1)^{lfloor log n
floor}}{n}$ diverges. In joint work withBruce Reznick and Monika Serbinowska, we show that $$sum_{n=1}^infty frac{(-1)^{lfloor n sqrt{2}
floor}}{n}$$converges. Our proofs rely on Diophantine properties of $sqrt{2}$, and donot apply (for example) if $sqrt{2}$ is replaced by$frac{sqrt{5}+1}{2}$.

Lattice polytopes are polytopes with lattice vertices (points with integral coordinates). One can associate with lattice polytopes algebraic and geometric structures such as affine semigroup rings and toric varieties. The aim of this talk is to explain this relationship between combinatorics, algebra and geometry by means of triangulations which involve only the lattice points of the polytop

Bussemaker et al. (2000, PNAS) proposed the simple idea ofmodeling DNA non coding sequence as a concatenation of words and gavean algorithm to reconstruct deterministic words from an observedsequence. Moving from the same premises, we consider words that canbe spelled in a variety of forms (hence accounting for varying degreesof conservation of the same motif across genome locations).These ``words'' correspond to binding sites of regualtory proteins. Theoverall frequency of occurrence of each word in the sequence and theparameters describing the random spelling of words are estimated in amaximum-likelihood framework using an E-M gradient algorithm. Once these parameters are estimated, it is possible toevaluate the probability with which each motif occurs at a givenlocation in the sequence. These conditional probabilities can be used to predict whichgenes experience similar transcription regulations. Gene expression data can be used tovalidate/refine such predictions.

Given a semisimple Lie algebra and one of its involutions, it is possible to construct a coideal subalgebra B in the Êquantized enveloping algebra U which is a quantum analog of the classical enveloping algebra of the fixed Lie subalgebra. We study the space of B bi-invariants inside the associated quantized function algebra. Under the obvious restriction map, the space of bi-invariants is proved isomorphic to the Weyl group invariants of the character group ring associated to the restricted roots. The quantum Peter-Weyl decomposition and the classification of finite-dimensional spherical modules associated to U,B implies that this space of bi-invariants is a direct sum of one-dimensional eigenspaces for the action of the center of U. When the restricted root system is reduced, we show that the zonal spherical functions, i.e. representations of each eigenspace, correspond to Macdonald polynomials under a standard

Groups 32 was developed to illustrate an approach to writing mathematical research software. It has also proved useful in helping students understand group theory and learn to become better at proving theorems.

Large-scale similarity searches of genomic DNA are of fundamental importance in annotating functional elements in genomes. To perform large comparisons efficiently, BLAST and other widely used tools use seeded alignment, which compares only sequences that can be shown to share a common pattern or "seed" of matching bases. The choice of seed substantially affects the sensitivity of seeded alignment, but designing and evaluating optimal seeds is computationally challenging. In this talk I will address some of the computational and mathematical problems arising in seed design.The talk will rely on joint work with:- Ming Li, Bin Ma and John Tromp- Jeremy Buhler and Yanni Sun

The fundamental group of a graph of groups is a concept thatgeneralizes both amalgamated free products and HNN extensions, twofundamental constructions in geometric group theory. Understanding thestructure of these constructions based on the structure of the componentgroups is an important topic. The centralizer of an element $gin G$ is the subgroup $Z_{G}(g) = { g'in G mid gg'=g'g}$, the set of all elements which commute with $g$. In this talk, I will introduce both algebraic methods (based on normal forms of elements) and geometric methods (based on actions ofthe group on graph theoretic trees) for computing the centralizers in thecases of amalgamated free products and HNN extensions.I also hope to indicate what we might expect to happen in the generalcase of fundamental groups of graphs of groups and to talk about somepossible generalizations of my work.

Time compression affects the interpretation of data as either noise ortrend; hence by examining multiple time compressions of data streamssimultaneously, one may smooth out noise. This talk discusses some topicsrelated to time compression: noise versus trend, speed, fractality, nonzero-sum game trading, a calculus of risky assets, and stochasticityversus complexity perspectives.

Turan's theorem is probably one of the most well-known theorem in graph theory. Let k be a small integer, say 5, and n be a large integer. Turan showed that one must delete at least a $1/(k-1)$-fraction of the edges of the complete graph $K_n$ in order to destroy all cliques of size k. For instance, one must delete at least half of the edges to destroy all triangles. It is natural to study the problem for graphs other than $K_n$. A graph G on n vertices is called k-Turan if one needs to delete at least a $1/(k-1)$- fraction of the edges of G in order to destroy all k-cliques in G. Which graphs are k-Turan ? For instance, is the Paley graph 5-Turan ? Despite the long and rich history of Turan's theory, which spans several decades, not much has been known about this question. Recently, Sudakov, Szabo, and Vu, based on earlier results of the last two researchers, found a general sufficient condition for a regular graph to be k-Turan. We discovered a surprising connection between the k-Turan property and the spectral of the graph: if the second eigenvalue of G is sufficiently small compared to the degree, then G is k-Turan. In particular, good expanders have the Turan properties. The proof is completely elementary.

I will start with an introduction to hyperbolic space and hyperbolictetrahedra. I will then show how to construct an arbitrary 3/4-idealhyperbolic tetrahedron out of 10 ideal tetrahedra. There will be lots ofpictures and not many equations in this talk!

Non-commutative conditional expectations and martingales arise inthe setting of von Neumann algebras, which are the naturalframework for non-commutative measure theory and integration.Analogues of classical martingale inequalities such asBurkholder-Gundy's square function inequalities and Doob'sinequality have recently been established for martingales innon-commutative $L_p$-spaces by Junge, Pisier and Xu. They alsoproved the analogue of the classical duality between $H^1$ and$BMO$ of martingales. We will discuss interpolation properties ofnon-commutative $BMO$ and show that it is a natural substitute for$L_infty,.$ As an application we establish boundedness ofnon-commutative martingale transforms.

If a group G acts on a tree X we may determine the precise structure of G byapplying machinery of Bass and Serre for reconstructing the group action. We use this to give structure theorems for certain subgroups of $SL_2 $ over a on-archimedean local field, and of groups associated to Kac-Moody Lie algebras over finite fields. Of special interest are lattice subgroups and their congruence subgroups.

We explore the geometric, algebraic, and computational implications of thepresence of continuous and discrete symmetries in semidefinite programs(SDPs). It is shown that symmetry exploitation allows a significantreduction in both matrix size and number of decision variables. To this end,we define a class of SDPs, that are invariant under the action of a finiteor continuous symmetry group. Using group averaging and linearrepresentation theory, it is shown that the feasible set can be restrictedto a specific invariant subspace, thus reducing the problem to a collectionof coupled semidefinite programs of smaller dimensions.We focus particularly on SDPs arising in the sum ofsquares/Positivstellensatz framework, where the group representation isinduced by an action on the space of monomials. It is shown that thecomplexity is significantly reduced, and the techniques are illustrated withnumerous examples. The results, reinterpreted from an invariant-theoreticviewpoint, provide a novel representation of nonnegative symmetricpolynomials. This alternative approach has as attractive features itscomputational efficiency and the natural connections with therepresentation-based approach developed earlier.Finally, the computational savings of the techniques are demonstrated insome large-scale problems. It is shown how the symmetry reduction techniquesenable the numerical solution of complicated instances, otherwisecomputationally infeasible to solve.The material in the talk is based on joint work with Karin Gatermann (ZIBBerlin).

We explore the group of 2x2 determinant 1 matrices with coefficients inthe ring of integers Z as well as an analog obtained by replacing Z by thering of polynomials F[t] where F is a finite field. Decomposition theoremsfor these groups will be obtained using the actions of the groups ontrees, which are connected graphs having no cycles, such as that of degree3 a part of which is pictured above.REFRESHMENTS WILL BE SERVED!

The binary space partition (BSP) is a recursive cutting scheme for aset of n disjoint polygonal scenes in the Euclidean space. We split thespace along a plane into two parts and then we partition recursivelyboth parts until the interior of every space fragment is disjointfrom the polygonal scenes. The size of a BSP is the number of splitsmade. The efficiency of applications (in computer graphics andcomputational geometry) vitally depends on the size of the BSPs.Paterson and Yao proved that the size of the smallest BSP for nrectangles is $Theta(n^2)$ in the worst case; and $Theta(n^{3/2})$ if allrectangles are orthogonal. Later, Agarwal et al. showed that for northogonal fat rectangles there is a BSP of size n $2^{(log n)^{1/2}}$.(A rectangle is fat if its aspect ratio is bounded by a constant.) Inthis talk, I show that one can generate a BSP of size $O(n log^8 n)$ forn orthogonal fat rectangles, improving the bound of Agarwal et al.,using the new technique of overlays of BSP

We explore the group of 2x2 determinant 1 matrices with coefficients inthe ring of integers Z as well as an analog obtained by replacing Z by thering of polynomials F[t] where F is a finite field. Decomposition theoremsfor these groups will be obtained using the actions of the groups ontrees, which are connected graphs having no cycles, such as that of degree3 a part of which is pictured above.REFRESHMENTS WILL BE SERVED!