# Colloquium, AY 2007-2008

Thursdays 4:10 pm in 5211 Stevenson Center, unless otherwise noted

Tea at 3:30 pm in 1425 Stevenson Center

 September 6, 2007 Franco Montagna, University of Siena, Italy Advances in normal GBL-algebras Abstract: Abstrac.pdf Contact person: Constantine Tsinakis John Lott, University of Michigan Long-time behavior of Ricci flow Abstract: Perelman proved Thurston’s geometrization conjecture using Hamilton’s Ricci flow. The first part of the talk will be an introduction to this work. Perelman gave enough information about the long-time behavior of a 3-dimensional Ricci flow to prove the validity of Thurston’s geometric decomposition. However, it is not known whether Ricci flow performs the decomposition for you, i.e. whether as time passes one sees the various geometries appearing. I will give some results in this direction. Contact person: Guoliang Yu Vladimir Manuilov, Moscow State University Almost representations and asymptotic representations of discrete groups Abstract: There are now several versions of weakening the notion of a group representation and keeping some of its properties. One of them, due to A. Connes, M. Gromov and H. Moscovici, is to consider, for a finitely presented group, maps from the set of generators to the (infinite) unitary group topologized by the operator norm, such that the relations are satisfied up to some $\varepsilon>0$. A continuous family of such $\varepsilon_t$-almost representations is an asymptotic representation of the group if $\lim_{t\to\infty}\varepsilon_t=0$. We shall discuss the relation between almost representations and asymptotic ones. Typically, asymptotic representations do not converge to any genuine representation and there is a topological obstruction for that: a construction by A. Mishchenko shows that almost representations can be used to define vector bundles over classifying spaces of groups similarly to genuine representations and that these vector bundles may represent non-trivial elements of the $K^0$-group of the classifying space. Contact person: Gennadi Kasparov Carl Sundberg, University of Tennessee, Knoxville Von Neumann’s inequality and model theory for several commuting operators Abstract: Let T be a contraction on a Hilbert space H, i.e. a bounded linear operator on H whose norm is less than or equal to 1. Von Neumann’s inequality says that if p is an analytic polynomial then the norm of p(T) is bounded by the supremum of the values of |p(z)| for z in the unit disk. Following Sz. Nagy-Foias this can be proved by developing a “model-theory” for contractions, i.e. one shows that every contraction can be modelled as the restriction to an invariant subspace of a particular nice kind of operator for which von Neumann’s inequality can be easily proved. An abstract approach to model theory was developed by Agler and this approach suggests far reaching applications to the study of a variety of classes of operators. There are many ways to generalize von Neumann’s inequality to several commuting operators, some of which work and some of which don’t. We will discuss these generalizations, focusing particularly on one due to Drury in 1978. This generalization is to “row-contractions” of commuting operators. Drury’s result was rediscovered and further developed by Arveson, who produced a model theory for row contractions. We discuss Drury and Arveson’s work along with recent joint work with Stefan Richter, in which we show how Arveson’s model theory can be produced using Agler’s approach. Contact person: Dechao Zheng, Guoliang Yu Uffe Haagerup, University of Southern Denmark Random matrices and the Ext-invariant for C*-algebras Abstract: In this talk I will discuss a surprising connection between two quite different areas of mathematics, namely “random matrices” and “operator algebras” (i.e. C*-algebras and von Neumann algebras), a connection that has been developed over the last 16 years. In 1991 Voiculescu introduced a random matrix model for a free semicircular system, which has led to the solution of a number of classical problems in von Neumann algebra theory. More recently, Steen Thorbjoernesn and I have developed methods which allowed us to apply random matrices to problems in C*-algebra theory as well. In particular, we proved (Annals of Math. 2005) that the Brown-Douglas-Fillmore Ext-invariant for the reduced C*-algebra C*r(F2) for the free group on two generators is not a group, but only a semi-group, a problem which had been open since 1978. Further results in this direction have been obtained in collaboration with Hanne Schultz and Steen Thorbjoernsen (Advances in Math. 2006). Contact person: Dietmar Bisch Zhilan Feng, Purdue University Influence of anti-viral drug treatments on evolution of HIV-1 pathogen Abstract: Mathematical models are used to study possible impact of drug treatment of infections with the human immunodeficiency virus type 1 (HIV-1) on the evolution of the pathogen. Treating HIV-infected patients with a combination of several antiretroviral drugs usually contributes to a substantial decline in viral load and an increase in CD4+ T cells. However, continuing viral replication in the presence of drug therapy can lead to the emergence of drug-resistant virus variants, which subsequently results in incomplete viral suppression and a greater risk of disease progression. As different types of drugs (e.g., reverse transcriptase inhibitors, protease inhibitors and entry inhibitors) help to reduce the HIV replication at different stages of the cell infection, infection-age-structured models are useful to more realistically model the effect of these drugs. The model analysis will be presented and the results are linked to the biological questions under investigation. By demonstrating how drug therapy may influence the within host viral fitness we show that while a higher treatment efficacy reduces the fitness of the drug-sensitive virus, it may provide a stronger selection force for drug-resistant viruses which allows for a wider range of resistant strains to invade. Contact person: Glenn Webb Vaughan Jones, UC Berkeley Random matrices and planar algebras Abstract: I will discuss a connection between random matrices and subfactors discovered through the planar algebra approach to subfactors and the fact that there is a genus expansion for certain integrals over large matrices with the leading term being of genus zero. This is joint work with Alice Guionnet and Dimitri Shlyakhtenko. Contact person: Dietmar Bisch Konstantina Trivisa, University of Maryland Multicomponent reactive flows Abstract: Multicomponent reactive flows arise in many practical applications such as combustion, atmospheric modelling, astrophysics, chemical reactions, mathematical biology etc. The objective of this work is to develop a rigorous mathematical theory based on the principles of continuum mechanics. Contact person: Gieri Simonett Constantine Dafermos, Brown University Hyperbolic conservation laws with involutions and contingent entropies Abstract: The lecture will discuss paradigms from the theory of hyperbolic conservation laws in which physics imposes special challenges to the analyst, but then also provides the means of overcoming these challenges. Contact person: Gieri Simonett, Mikhail Perepelitsa Catherine Yan, Texas A&M Crossings and nestings in combinatorics Abstract: We present results on the crossings and nestings for various combinatorial strucutures, in particular, for matchings and set partitions. By a variant of the RSK algorithm from algebraic combinatorics, matchings and set partitions are in one-to-one correspondence with oscillating/vacillating tableaux, which are certain random walks in the Hasse diagram of the lattice of integer partitions. We prove that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of [n], as well as over all matchings of [2n]. Many enumerative results about the noncrossing/nonnesting partitions will be discussed. At the end, we will extend the method to filling of Ferrer shapes, filling of Moon polyominoes, and pattern avoidence in general multi-graphs. Contact person: Mark Ellingham Special Colloquium, 4:10-5:00 pm, SC 5211 Special Colloquium, 4:10-5:00 pm, SC 1307 Special Colloquium, 4:10-5:00 pm, SC 1308 Special Colloquium, 4:10-5:00 pm, SC 5211 Special Colloquium, 4:10-5:00 pm, SC 5211 Special Colloquium, 4:10-5:00 pm, SC 1308 Special Colloquium, 4:10-5:00 pm, SC 5211 Special Colloquium, 4:10-5:00 pm, SC 1307 Carlos Cabrelli, University of Buenos Aires Optimal shift invariant spaces Abstract: Shift invariant spaces are spaces of functions that are invariant under integer translates. They are important in many areas, in particular in approximation theory, numerical analysis and wavelet theory. They also serve as models for signal and image processing applications as well as sampling theory. Their very rich and interesting mathematical structure can be successfully exploited in many problems. In this talk we will try to describe part of this structure and give and application to the problem of finding the “best fitting” shift invariant space for a given set of data. Contact person: Akram Aldroubi Denis Osin, CUNY Group theoretic Dehn surgery and its applications Abstract: In my talk I will introduce a group theoretic version of Dehn surgery in 3-manifolds. It turns out that many basic facts about ordinary Dehn surgery can be translated to algebraic language. The main result in this direction is a group theoretic analogue of the Thurston Hyperbolic Dehn Surgery Theorem. Although this result is strongly motivated by geometry, it also has some unexpected algebraic applications. For example, most results obtained by means of small cancellation theory over hyperbolic and relatively hyperbolic groups can be recovered using Dehn surgery of a very special type. Contact person: Alexander Ol’shanskiy Ian Agol, University of California, Berkeley Finiteness of arithmetic reflection groups Abstract: We show that there are only finitely many maximal reflection groups in hyperbolic spaces. This talk will define the terms in this theorem, and explain some of the ingredients of the proof. Contact person: John Ratcliffe Spring Break Serguei P. Novikov, University of Maryland and Landau Institute New discretization of complex analysis Abstract: A new discretization of complex analysis and geometric GL_n connections was found (in collaboration with I. Dynnikov) a few years ago as a by-product of the theory of integrable systems. We are using an equilateral triangle lattice instead of the standard square lattice for complex analysis, and simplicial complexes for GL_n connections. Contact person: Gennadi Kasparov Bruce Kleiner, Yale University A new proof of Gromov’s theorem on groups of polynomial growth Abstract: In 1981 Gromov showed that any finitely generated group of polynomial growth contains a finite index nilpotent subgroup. This was a landmark paper in several respects. The proof was based on the idea that one can take a sequence of rescalings of an infinite group G, pass to a limiting metric space, and apply deep results about the structure of locally compact groups to draw conclusions about the original group G. In the process, the paper introduced Gromov-Hausdorff convergence, initiated the subject of geometric group theory, and gave the first application of the Montgomery-Zippin solution to Hilbert’s fifth problem (and subsequent extensions due to Yamabe). The purpose of the lecture is to give a new, much shorter, proof of Gromov’s theorem. The main step involves showing that any infinite group of polynomial growth admits a finite dimensional linear representation with infinite image. We establish this using harmonic maps, thereby avoiding the Montgomery-Zippin-Yamabe theory of locally compact groups which was used in Gromov’s original proof. I will explain the proof in a manner accessible to a broad audience. Contact person: Mark Sapir Petar Markovic, University of Novi Sad, Serbia Computational complexity of the constraint satisfaction problem Abstract: I will give an overview of the current state of knowledge on the following conjecture: Given the class of all constraint satisfaction problems with a fixed template, the complexity of deciding if there is a solution of an instance of such a constraint satisfaction problem is either polynomial-time solvable, or NP-complete (depending on the template). In the first part of the talk I will provide all definitions needed for understanding the problem. In the second part I will overview the best partial results known so far using the approaches of logic, complexity theory, universal algebra and graph theory. In the third part I will concentrate on a subcase of the problem when the template is a directed graph, as the dichotomy conjecture for computational complexity is equivalent to the restricted conjecture for this subcase. I will present the most recent results which prove the conjecture in the case that the template has no sources and no sinks, and some extensions of this obtained in collaboration with R. McKenzie this Fall. This approach uses a combination of graph-theoretic and universal algebraic methods, and we feel that it is the most promising one of all that have been attempted so far. Contact person: Ralph McKenzie Yair Minsky, Yale University Geometry and rigidity of mapping class groups Abstract: The mapping class group of a compact surface S is the group of homeomorphisms of S modulo isotopy. Via its Cayley graph it can be viewed as an infinite-diameter metric space, whose large-scale geometry is strongly connected with the algebraic properties of the group. Â In joint work with Behrstock, Kleiner and Mosher, we study this large-scale geometry and prove in particular that its quasi-isometries are bounded perturbations of the action of the group. This implies (as was also independently shown by Hamenstadt) that the group is quasi-isometrically rigid. Contact person: Mark Sapir Sergei Ivanov, University of Illinois at Urbana-Champaign On Dehn functions of infinite presentations of groups Abstract: We discuss two new types of Dehn functions of group presentations which seem more suitable (than the standard Dehn function) for infinite group presentations and indicate the fundamental equivalence between the solvability of the word problem for a group presentation defined by a decidable set of defining words and the property of being computable for one of the newly introduced functions (this equivalence fails for the standard Dehn function). Elaborating on this equivalence and making use of this function, we present a characterization of finitely generated groups for which the word problem can be solved in nondeterministic polynomial time. We also give upper bounds for these functions, as well as for the standard Dehn function, in the case of the Grigorchuk 2-group of intermediate growth and free Burnside groups of sufficiently large exponents, defined by some minimal systems of defining words. This talk is based on joint with R.I. Grigorchuk work. Contact person: Alexander Ol’shanskii Yasuyuki Kawahigashi, University of Tokyo Moonshine and operator algebras Abstract: ”Moonshine” is a name for mysterious relations between elliptic modular functions and the largest finite simple group among the 26 sporadic groups, the Monster. The Moonshine vertex operator algebra is a mathematical object to understand these relations, and I will present its operator algebraic counterpart in the framework of conformal field theory. Contact person: Pinhas Grossman, Dietmar Bisch Anna Gilbert, University of Michigan (Fast) algorithms for compressed sensing Abstract: “Compressed sensing” captures a new paradigm which connects sparse representations, high-dimensional geometry, probability, and algorithms. It suggests a new paradigm in information acquisition and processing of compressible signals. These signals can be approximated using an amount of information much smaller than the nominal dimension of the signal. Traditional approaches acquire the entire signal and process it to extract the information. The new approach acquires a small number of nonadaptive linear measurements of the signal and uses sophisticated algorithms to determine its information content. Emerging technologies can compute these general linear measurements of a signal at unit cost per measurement. I will discuss the connections to randomized algorithms and signal processing. In particular, I will focus on extremely fast algorithms and measurement designs for compressed sensing, including some prototype hardware designs. Contact person: Alex Powell

Colloquium Chair (2007-2008): Gieri Simonett

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