Events for the week of November 16 2009

Abstract. First I will present a geometric method, due to Kazhdan, of approximating representation theory of reductive groups over local fields of positive characteristic (like the field $\mathbf{F}_p((t))$ of Laurent power series with coefficients in a finite field) with representation theory of reductive groups over local fields of zero characteristic (like the field $\mathbf{Q}_p$ of p-adic numbers). Then I will present a generalization of this method, due to Gourevitch, Avni and myself, which approximates harmonic analysis over spherical varieties over local fields of positive characteristic with harmonic analysis over spherical varieties over local fields of zero characteristic. As an application we show that $(\text{GL}(n+1,F),\text{GL}(n,F))$ is a strong Gelfand pair for all local fields F of positive characteristic. This means that the restriction to $\text{GL}(n,F)$ of every irreducible smooth representation of $\text{GL}(n+1,F)$ "decomposes" with multiplicity one. We use our method to deduce this from the zero characteristic case, which was proven two years ago by Gourevitch, Rallis, Schiffmann and myself.

Abstract. http://www.math.ucla.edu/~ploh/ipamseminar/1117-peter-hegarty.pdf

Abstract. The critical threshold for the emergence of a giant  component in the random site subgraph of a $d$-dimensional  Hamming torus is given by the positive root of a polynomial.   This value is distinct from the critical threshold for the random  edge subgraph of the Hamming torus.  Most interestingly,  the proof uses a novel application of multitype branching  processes.

Abstract. The study of planar simple closed curves (or "2D shapes") and their similarities is a central problem in the field of computer vision. It arises in the task of characterizing and classifying objects from their observed silhouette. Defining natural distance between 2D shapes creates a metric on the infinite-dimensional space of shapes. In this talk I will describe one particular metric, which comes from the conformal mapping of the 2D shapes, via the theory of Teichm\"uller spaces. In this space every simple closed curve (or a 2D shape) is represented by a smooth self-map of a circle. I will talk about a specific class of soliton-like geodesics on the space of shapes, called teichons. Some numerical examples of geodesics and effects of the curvature will be demonstrated.

Abstract. Notions of communication and computation are most naturally formulated in the quantum arena. Unlike the stuff of conventional communication theory, quantum information cannot be copied, nor eavesdropped on without disturbance, and it can mediate the intense and private form of correlation known as entanglement. As in classical information theory, quantum capacity has to do with sphere packing, but in C_2^{\otimes n} rather than Z_2^{n}. This difference gives rise to a much richer theory. For example, in contrast to what happens classically, here we often find strong nonadditivity of capacity---the capacity of two channels used together can be much larger than the sum of the individual capacities.

Abstract. http://www.math.ucla.edu/~ploh/ipamseminar/1119-poshen-loh.pdf

Abstract. I will talk about my viewpoint at a method for proving property T developed by Dymara and Januszkiewicz. Their original motivation came from groups acting buildings, but the idea does not used anything more than angles between subspaces in a (finite dimensional) Euclidian space. The main result says that if a group G is generated by finite subgroups $G_i$ and each pair generates a group with property T and sufficiently large Kazhdan constant then the whole group also has property T. One can use this result to show that some groups like $SL_n(F_p[t_1,...,t_k])$ have property T, almost without using any representation theory. Another application allows us to compute the exact values of the Kazhdan constant and the spectral gap for the Laplacian for any finite Coxeter group with respect to its standard generating set. Parts of the talk are based on a work of M. Ershov and A. Jaikin.

Abstract. A continuous map $f : S^2 \to S^2$ on a 2-sphere $S^2$ is called a branched cover if near each point it can be written as $z \to z^n$ for some $n\in\mathbb{N}$ in suitable local coordinates in domain and image. A critical point of such a map $f$ is a point in $S^2$ where f is not a local homeomorphism, that is, where $n\geq2$. Thurston considered branched covers of $S^2$ for which the forward orbit of each critical point under iteration is finite. These maps are now called Thurston maps. Basic examples include maps on the Riemann sphere such as $z\mapsto z^2 + i$ or $z\mapsto 1 − 2/z^2$. The study of these maps provides links to areas such as dynamical systems, classical conformal analysis, hyperbolic geometry, geometric group theory, and analysis on metric spaces. In my talk I will give a survey on this subject and discuss some recent joint work with Daniel Meyer.

Abstract. Urysohn's metric space U is the unique (up to isometry) separable, complete metric space which is universal (i.e. all separable, complete metric spaces admit an isometric embedding into U) and ultrahomogeneous (i.e. any isometry between finite subsets of U can be extended to a self-isometry of U). The Urysohn space (as well as its group of self-isometries) has been of considerable interest to descriptive set theorists in recent years. In this talk, we describe some of the model-theoretic aspects of Urysohn's space in the context of continuous logic. The talk will begin with a primer on continuous logic using analogies with ordinary first-order logic.

Abstract. We study the moduli space of the McKay quiver representations associated to the binary polyhedral groups G< SU(2)< SU(3). The derived category of such representations is equivalent to the derived category of coherent sheaves on the corresponding ADE resolution Y = G-Hilb(C^3). By making particular choices of parameters in the space of stability conditions on the equivalent derived categories above, we recover Donaldson-Thomas (DT), Pandharipande-Thomas (PT) and Szendroi (NCDT) moduli spaces, and prove a wall crossing formula relating the corresponding invariants. We also compute the Gromov-Witten (GW) partition function of Y directly and verify the conjectural GW/PT/DT/NCDT-correspondence by assuming the DT/PT correspondence which has been proven recently. The NCDT invariants in this case are the same as the orbifold Donaldson-Thomas invariants for C^3/G. This allows us to verify the Crepant Resolution Conjecture for the orbifold Donaldson-Thomas theory in this case.

Abstract. Using the Fourier transform, solutions of a linear partial differential equation with constant coefficients can be written as oscillatory integrals whose long-time asymptotics can be studied using the stationary phase method. A Riemann-Hilbert problem is a factorization problem and it can be used to invert the one-dimensional scattering transform, which is a nonlinear Fourier transform for many nonlinear partial differential equations. In my talk, I will describe a nonlinear analogue of the classical stationary phase method for oscillatory Riemann-Hilbert problems, which can be used to obtain long-time asymptotics of solutions to such partial differential equations.