# Colloquium 2007-08

 September 6 Skip Garibaldi (Emory University) Host: Andrei Rapinchuk Title: Cohomological Invariants A popular strategy for proving that two things are not isomorphic is to show that an invariant takes different values on the two objects. But sometimes all the invariants in one's toolkit agree on the two objects, which leads to the problem of classifying all of the possible invariants. In algebra, recent results of Serre and Rost -- exploiting the fact that many algebraic objects can be viewed as torsors under an algebraic group -- have made this problem tractable in some cases. Current work focuses on the invariants of G-torsors, for various connected simple groups G. September 14* Mathematics Symposium for Loren Pitt Milne Anderson (University College, London) Title: The Logarithmic Derivative of a Polynomial Daniel Stroock (MIT) Title: Some Queer Diffusions Several years ago, David Williams and I studied a simple looking diffusion equation on the right half line. What made the problem challenging was our imposition of a boundary condition which invalidated the minimum principle. As a consequence, non-negative initial data does not necessarily stay non-negative under the evolution. Our search for the necessary and sufficient condition on the initial data which guarantees that the solution stays non-negative involved an interesting and novel connection between classical analysis and probability theory. After explaining the simplest case, I will report on some more recent progress on this type of problem. September 20 Brian Sutton (Randolph-Macon College) Host: Don Ramirez Title: Computing the complete CS decomposition The CS decomposition (CSD) is a matrix decomposition reminiscent of the eigenvalue and singular value decompositions. The existence of the CSD for any partitioned unitary matrix was proved by Stewart in 1977, but since then, no algorithm for computing the decomposition has been available. (Existing CSD algorithms only solve a reduced version of the problem.) We present a solution to the problem of computing the complete CS decomposition. September 27 Francis Su (Harvey Mudd College) Host: IMS Title: Combinatorial Fixed Point Theorems and Fairness The Brouwer fixed point theorem is a well-known classical theorem in topology with important applications. Less well-known is an equivalent combinatorial formulation known as Sperner's lemma. We survey some recent applications of variants and relatives of Sperner's lemma, including an extension to polytopes and combinatorial equivalents of other topological theorems. These have some striking applications in "fair division" problems in mathematical economics: cake-cutting, rent-splitting among housemates, and resource allocation. Proofs exhibit interesting connections between combinatorics, topology and the social sciences. Research with undergraduates has played a big role. October 4 David Sherman (UVa) Title: Symmetries of noncommutative Banach spaces   Many Banach spaces are described most naturally as spaces of functions. Motivated partially by quantum mechanics, their noncommutative versions are built out of operators. Examples are C*-algebras, von Neumann algebras, and noncommutative Lp spaces.    I'll begin this talk by showing how the symmetry group of a square underlies Banach's classical results on isometries of C(K) and Lp spaces. Then I will explain how these results have gradually evolved into fully noncommutative theorems. Although new phenomena have appeared and the proofs have gotten more complicated, there is a surprising unity (and simplicity) in the results: all of these isometries should be viewed as "noncommutative weighted composition operators." October 11 Michael Desai (Princeton) Host: IMS Title: How Large Asexual Populations Adapt We often think of beneficial mutations as being rare, and of adaptation as a sequence of selected substitutions: a beneficial mutation occurs, spreads through a population in a selective sweep, then later another beneficial mutation occurs, and so on. This simple picture is the basis for much of our intuition about adaptive evolution, and underlies a number of practical techniques for analyzing sequence data. Yet many large and mostly asexual populations -- including a wide variety of unicellular organisms and viruses -- live in a very different world. In these populations, beneficial mutations are common, and frequently interfere or cooperate with one another as they all attempt to sweep simultaneously. This radically changes the way these populations adapt: rather than an orderly sequence of selective sweeps, evolution is a constant swarm of competing and interfering mutations. I will describe some aspects of these dynamics, including why large asexual populations cannot evolve very quickly and the character of the diversity they maintain. I will explain how this changes our expectations of sequence data, how sex can help a population adapt, and the potential role of "mutator" phenotypes with abnormally high mutation rates. I will also describe ways to study these dynamics directly using experimental yeast populations. October 18 Benjamin Wells (UVa) Title: Fourier Transforms Vanishing at Infinity The talk will survey some old and some more recent results on the behavior of the Fourier transform of functions and measures defined on the circle and on the real line as values of their arguments become large. In the case of the circle one can ask whether a transform vanishing at infinity on a subset of integers E must necessarily vanish on the complement of E. My interest in the subject arises from an old question concerning subsets E of integers having the property that the only measures whose transforms vanish on E are absolutely continuous with respect to Lebesgue measure. October 25 Jason Papin (UVa) Host: IMS Title: Interrogating emergent properties of biochemical networks The reconstruction and mathematical analysis of genome-scale biochemical networks is a pressing challenge for making the connection between genotype and phenotype of biological systems. Three topics will be discussed: (1) the development of novel computational approaches for interrogating properties of mathematical representations of these networks; (2) the discovery of fundamental biology with these systems-level models; and (3) the application of such network analysis tools to address clinical problems. These network reconstructions and analyses facilitate the integration of high-throughput datasets to characterize properties that arise from the biochemical networks and are thus beginning to drive fundamental discoveries in biology. November 1 Yiftach Barnea (University of London) Host: Mikhail Ershov Title: Pro-p groups with few normal subgroups During the 80's Charles Leedham-Green and Mike Newman came with five striking conjectures on the structure of finite p-groups. Some of these conjectures were formulated as a structure theory on the class of pro-p groups of finite coclass. In combined efforts of many mathematicians including Donkin, Leedham-Green, Shalev, and Zelmanov the conjectures were proved. The main tool was the theory of p-adic analytic pro-p groups developed by Lubotzky and Mann based on Lazard's solution of Hilbert's fifth problem over the p-adics. I will discuss various generalizations of the notion of coclass in pro-p groups and relations between them. I will then show that all the known examples of pro-p groups with few normal subgroups, not only have few normal subgroups, but strikingly have periodicity in the lattice of normal subgroups. Following these examples I will pose several problems. This talk is self contained, there is no need to know what a pro-p group is, and I will give all the necessary background. This is joint work with N. Gavioli, A. Jaikin-Zapirain, V. Monti, and C. Scoppola. November 8 Larry Smith (University of Göttingen) Host: Bob Stong Retirement Conference Title: On a Theorem of Chevalley about Algebras of Coinvariants  Let rho: G --> GL(n, F) be a faithful representation of a finite group G over the field F. By means of rho the group G acts on the algebra F[V] of polynomial functions on the representation space V = F^n, fixing the subalgebra F[V]^G (the invariant algebra) and hence also on the algebra F[V]_G = F tensor_{F[V]^G} F[V] of coinvariants. Chevalley's Theorem states that, in the case that F=R and G is a real reflection group, the coinvariant algebra as a G-representation is isomorphic to the regular representation of G.   In this talk I will describe joint work with Abraham Broer of the Universite de Montreal, Victor Reiner and Peter Webb of the University of Minnesota, in which we generalize Chevalley's Theorem in three ways.   (1) we need no assumption on the ground field F, in particular it can be of any characteristic, even one dividing the order of G,   (2) we need no assumption on the representation rho, so in particular G need not be a reflection group, nor must F[V]^G be a polynomial algebra, and finally   (3) we can deal with a relative situation of H < G being a subgroup of G and dealing with the algebra F \tensor_{F[V]^G} F[V]^H as a representation of the group W_G(H) = N_G (H) /H where N_G (H) is the normalizer of G in H.   Our methods are a mixture of homological algebra, representation theory, and invariant theory. November 15 Ronald Fintushel (Michigan State University) Host: Tom Mark Title: The countdown to CP2 One of the key problems in 4-manifold topology is to understand whether "standard manifolds" admit exotic smooth structures, i.e. given a smooth 4-manifold, if there are manifolds homeomorphic but not diffeomorphic to it. In the last several years dramatic progress has been made in understanding this problem for the manifolds obtained by blowing up CP2 at a small number of points. I will describe the problems in this area, the techniques that have been used to study them, and the results that have been obtained, starting with work of Donaldson from the 1980's and leading to some outstanding results of young mathematicians in this last year. November 29 David Blecher (University of Houston) Host: David Sherman Title: Operator spaces and algebras, and their duality We describe some beautiful and accessible ideas from the new theory of linear spaces or algebras of continuous linear operators between Hilbert spaces. We will also describe some of our results on the duality of such spaces, and on the relations between the metric structure' and the underlying algebraic structure. --Cancelled-- December 6 Gianluca Guadagni (UVa) Title: Exact Renormalization Group on the lattice I will give a description of the RG transformation for functional integrals of weakly perturbed Gaussian measures. The method is based on polymer expansions and it will be applied to a simplified model on a lattice. December 13 Paul Kirk (Indiana University) Host: Tom Mark Title: Constructing symplectic 4-manifolds with prescribed fundamental groups

January 17 (Postponed due to weather)
January 18* Philip Gressman (Yale University) Special Colloquium
Title: Lebesgue and Sobolev estimates for geometric averaging operators A geometric averaging operator is any mapping which sends a function on a manifold to its integral over some family of submanifolds, the canonical examples being the X-ray and Radon transforms. For general families of submanifolds, little is known about the boundedness properties of these operators. This talk will discuss the motivations for studying such objects and recent sharp results (in terms of $L^p$-improving and Sobolev smoothing estimates) for "generic" degenerate averaging operators.
January 21* Alexey Cheskidov (University of Chicago) Special Colloquium
Title: On the regularity of weak solutions to the 3D Navier-Stokes equations in critical Besov spaces In this talk, I will make a brief introduction to the regularity problem for the 3D Navier-Stokes equations. Even thought it is far from been solved, numerous regularity criteria have been proved in critical spaces since the work of Leray. However, the regularity problem remains completely open in supercritical spaces. We will discuss a new regularity criterion in the largest critical space.
January 22* Michael Goldberg (Johns Hopkins University) Special Colloquium
Title: Strichartz Estimates for a Wealth of Schrödinger Operators   The Strichartz estimates are a fundamental family of Lp inequalities governing solutions to the free Schrödinger equation. They decribe the dispersive nature of the evolution -- local concentrations of mass can only exist for a specified finite period of time. The associated function spaces can be used to assess well-posedness and scattering properties of perturbed or nonlinear Schrödinger equations.
I will review the efforts to identify other Schrödinger operators on Rn that satisfy the same range of Strichartz estimates as the Laplacian. On the technical side, recent progress is driven by improvements in the underlying harmonic and functional analysis. On the motivational side, specific applications often give rise to operators that fall outside (or at the edge) of our current understanding. I will present the orbital stability of solitons in NLS as one such application.
January 28* Marcin Bownik (University of Oregon) Special Colloquium
Title: How to construct multidimensional wavelets with good time-frequency localization? In this talk I will discuss the problem of constructing orthogonal wavelets in higher dimensions. In general this is a difficult problem if we require some special properties on wavelets, such as regularity or fast decay. I will present some positive results on the existence of regular wavelets. I will also describe certain inherent limitations on the existence of wavelets with good time-frequency localization.
January 31
February 7
February 14 Olivier Pfister (UVa) Host: IMS
Title: Towards quantum computing with graphs and light Quantum computing and quantum information have attracted much attention over the past decade because they predict spectacular enhancements of computational performance for historically (if not provably) hard problems such as factoring. Quantum computing has fundamental overlaps with group representation theory, topology, and graph theory, and the physical implementation of nontrivial quantum computing is an exciting, if daunting, challenge to physicists devoted to the experimental study of quantum systems. In this talk, I will introduce an interesting flavor of quantum computing, called one-way quantum computing, which interfaces with mathematical graph theory. I will then outline how the power and elegance of graph quantum states translate into "physical reality" in our experimental setup, next door.
February 21 Jaydeep Chipalkatti (University of Manitoba) Host: Malek Abdesselam
Title: Polar polyhedra, and the polynomial Waring's problem The polynomial version of Waring's problem (which originated in the 19th century) asks under what conditions a homogeneous form may be expressed as a sum of powers of linear forms. There are geometrically peculiar cases where such a representation is impossible even though a priori one has enough parameters available. I will explain the solution to the general problem by Alexander and Hirschowitz, and a little of what remains to be done. The talk should be generally accessible, and no special knowledge of algebraic geometry will be assumed.
February 28 Dave Witte Morris (University of Lethbridge) Host: Andrei Rapinchuk
Title: Some arithmetic groups that cannot act on the line It is known that finite-index subgroups of the arithmetic group SL(3,Z) have no (orientation-preserving) actions on the real line. This naturally led to the conjecture that most other arithmetic groups (of higher real rank) also cannot act on the line. This problem remains open, but joint work with Lucy Lifschitz verifies the conjecture for many examples. This includes all finite-index subgroups of SL(2,Z[alpha]), where alpha is any irrational, real algebraic integer. The proof is based on the fact, proved by D.Carter, G.Keller, and E.Paige, that every element of these groups is a product of a bounded number of elementary matrices. No familiarity with arithmetic groups will be assumed.
March 6 Spring break
March 13 Andrei Jaikin-Zapirain (Universidad Autonoma de Madrid) Host: Mikhail Ershov
Title: On the p-gradient of finitely presented groups    Let Γ be a finitely presented group and Γ = Γ0 > Γ1 > … a nested sequence of normal subgroups of Γ of finite index. W. Lueck proved that if {Γi} has trivial intersection, then
 limi→∞ dimC H1 (Γi, C) |Γ : Γi|
2-Betti number of Γ. In particular, this limit does not depend on the sequence Γ = Γ0 > Γ1 > …. The L2-Betti numbers were introduced first by M. Atiyah for compact Riemann manifolds and later extended by J. Dodziuk to general CW-complexes of finite type. In my talk I consider an analogous problem in which instead of dimC H1i,C) we take dimFp H1i,Fp):

Conjecture 1: Let Γ be a finitely presented group and Γ = Γ0 > Γ1 > … a nested sequence of normal subgroups of Γ of finite index with trivial intersection. Then
 limi→∞ dimFp H1 (Γi, Fp) |Γ : Γi|
0 > Γ1 > ….

In a particular case when the profinite completion of Γ with respect to {Γi} is a p-adic analytic group, Conjecture 1 is reformulated as a problem about embeddings of the group ring Fp[Γ] into division rings. Among the examples we consider are lattices in SL2C. We will establish a relation between Conjecture 1 and the structure of the congruence kernel of arithmetic lattices in SL2C.
March 20 Charles Akemann (UC Santa Barbara) Host: David Sherman
Title: The continuum hypothesis as a tool for constructing examples in functional analysis It is well-known that the Axiom of Choice is widely used in functional analysis for the construction of examples that would not otherwise exist. Examples include subsets of the real line that are not Lebesgue measurable, discontinuous linear functionals on Banach spaces and the like. If these were the only kind of examples, then the Axiom of Choice might have been left in the trash can. After all, who really needs things like these? On the other hand, non-principal ultrafilters on the integers, the Hahn-Banach Theorem, various fixed point theorems, and Tychonoff's Theorem (the product of compact spaces is compact) and the like would also be unavailable. The loss of these items would reduce the beauty of the subject substantially. Consequently, analysis without the Axiom of Choice is almost reduced to numerical analysis. At first glance the Continuum Hypothesis seems like the opposite of the Axiom of Choice. The latter is an existence axiom, while the former is a non-existence axiom. Functional analysts have adopted a "take it or leave it" attitude toward the Continuum Hypothesis. There never seemed to be a use for mysterious uncountable cardinals smaller than the continuum. Perhaps unexpectedly, the Continuum Hypothesis has allowed the construction of analysis examples, rather than showing the non-existence of interesting examples. In several cases the question of the need for that axiom is still open, thus leaving open questions for set theorists to answer. This talk will focus on the value of the Continuum Hypothesis in constructions. The key point is that many, many analysis objects have cardinality c (the continuum). Despite this, countability and sequences play a key role. It turns out to be very convenient to maintain countability for as long as possible in a transfinite construction. With the Continuum Hypothesis assumed, some constructions that require countability can go forward at every ordinal less that c. While some of these constructions are old, this method recently led Nik Weaver and me to the solutions of two problems from the 1950s.
March 27 Gerda de Vries (University of Alberta) Host: IMS
Title: Understanding Bursting Oscillations through Bifurcation Analysis Bursting oscillations are commonly seen to be the primary mode of electrical behavior in a variety of nerve and endocrine cells, and have also been observed in some biochemical and chemical systems. Bursting oscillations are characterized by an alternation of silent and active phases. An observable of a system (for example, the electrical potential across the membrane of a cell) remains relatively constant during the silent phase, while it undergoes rapid oscillations during the active phase. In this talk, I will begin by reviewing a well-studied bursting system, namely the electrical behavior of pancreatic beta cells. A minimal model of bursting oscillations in beta cells consists of three ordinary differential equations, with variables operating on different time scales. A decomposition of the system into a fast and a slow component facilitates a bifurcation analysis that reveals the mechanism underlying bursting. The main focus during the remainder of my talk will be to review efforts to understand the role of electrical coupling between individual cells in the modification of the bursting phenomenon and, possibly, the genesis of the bursting phenomenon in a network of cells.
April 3 Alexander Volberg (Michigan State University) Host: Don Ramirez
Title: Buffon Needle Probability and Analytic Capacity In 1733 count de Buffon asked the question: what is the probability for a needle of length L<1 to intersect a grid of parallel lines on the plane having distance 1 between each other? In 1898 Paul Painlevé asked another question: how to describe geometrically the compact sets on the plane such that the only functions analytic and bounded in the complement of these sets are constants. At the end of 20th century it became clear that these two questions are closely related. Moreover, they are closely related to a wide variety of problems: from percolation on graphs to electrostatics. In our talk we will explain some of these relationships.
April 10 Igor Kukavica (University of Southern California) Host: Zoran Grujic
Title: Partial regularity for the Navier-Stokes equations In this talk we will review the conditional and partial regularity of the Navier-Stokes equations in dimension three. A classical result of Caffarelli, Kohn, and Nirenberg states that the one dimensional Hausdorff measure of singularities of a suitable weak solution of the Navier-Stokes system is zero. We will review known results and different proofs of the partial regularity results including a recent short proof which reduces the assumption on the force term.
April 17 Alireza Salehi Golsefidy (IAS and Princeton) Host: Mikhail Ershov
Title: Lattices with small covolume Abstract: In this talk, I will discuss lattices with small" covolume in almost simple algebraic groups over non-Archimedean local fields. In the positive characteristic case, I will quickly recall my result, saying that up to isomorphism G(F_q[1/t]) is the only lattice of minimum covolume in G(F_q((t))), where G is a Chevalley group of classical type or of type E_6. Then I will give a partial answer to Lubotzky's question by showing that in `most" of the cases in characteristic p, a lattice of minimum covolume is non-uniform. I will also give a very short proof of Siegel-Klingen theorem on rationality of values of certain L-functions and zeta functions, using covolume of lattices. In the characteristic zero case, in a joint work with A. Mohammadi, we study discrete transitive actions on the Bruhat-Tits building, and prove that there is no lattice in PGL(n,K) which acts transitively on the vertices of the Bruhat-Tits building if n>8, in contrast to the positive characteristic where Cartwright and Steger constructed such an action for any dimension. For 9>n>4, we give a list of 14 lattices which are the only potential such examples, and show that at least one of them in dimension 5 actually acts transitively.
April 24 Xuhua He (Stony Brook) Host: Weiqiang Wang
Title: Introduction to G-stable pieces The notion of G-stable pieces was introduced by Lusztig in the study of parabolic character sheaves. It has a rich structure, combining combinatorics, group theory, geometric representation theory and other fields. In this talk, we will discuss the G-stable pieces from the point of view of combinatorics of Weyl groups. We will also talk about some connection with algebraic geometry and Poisson geometry.
May 1 Weimin Chen (University of Massachusetts, Amherst) Host: Thomas Mark
Title: Symmetries of 4-manifolds Consider the following question: is every smooth Z/pZ-action on the 4-dimensional sphere conjugate to an orthogonal action? In this talk we will discuss the current state of affairs of this problem and some of the ideas and techniques used to attack it. (The talk is intended for a general audience.)
May 8 Kevin Wortman (University of Utah) Host: Kai-Uwe Bux
Title: Finiteness properties of S-arithmetic groups over function fields Finiteness properties for groups can be thought of as generalizations of properties such as finite generation and finite presentability, and S-arithmetic groups over function fields can be thought of as generalizations of the groups SL(n,F[t]) where F is a finite field. In this talk I'll describe how the geometry of Euclidean buildings has been used to analyze the topic from the title. I'll talk about the progress that has been made, and discuss some work that remains to be done.