Events for the week of May 14 2012

Abstract. We begin by reviewing the definition and basic properties of arithmetic Hecke characters. Then we explain that the Galois representation attached to a modular form of weight at least two is absolutely irreducible and if it is non CM, the image of the representation is full. The last notion will be made precise in this talk.

Abstract. Uncertainty is everywhere. It inhabits in quantum physics, biology genetics, control systems and default testing, human decision making processes, and more typically in financial markets. A. N. Kolmogorov?s ?Foundations of the Theory of Probability (1933)? has established the modern axiomatic foundations of probability theory. Since then this theory has been profoundly developed and widely applied to situations where uncertainty cannot be neglected. But in 1921 Frank Knight has been already clearly classified two types of uncertainties: the first one is for which the probability is known; the second one, now called Knightian uncertainty, is for cases where the probability itself is also uncertain. The situation with Knightian uncertainty has become one of main concerns in the domain of data processing, economics, statistics, and specially in measuring and controlling financial risks. A long time challenging problem is how to establish a theoretical framework comparable to the Kolmogorov?s one, to treat these more complicated situations with Knightian uncertainties. The theory of nonlinear expectations, rapidly developed in recent years, is to solve this problem. This is an important program, some fundamental results have been well-established such as law of large numbers, central limit theorem, martingales, G-Brownian motions, G-martingales and the corresponding stochastic calculus of It?o?s type, nonlinear Markov processes, as well as the calculation of measures of risk in finance. But still so many deep problems are still to be explored. Nonlinear expectation is naturally and deeply linked to partial differential equations (PDE) and, especially, to a new type of ?path-dependent PDEs? which relates interesting problems of statistics, econometrics, stochastic controls and risk measures in finance.

Abstract. Let $S$ be a connected, closed, oriented surface of negative Euler characteristic. We consider the $\PSL_n(\R)$--character variety $\mathrm{Rep}_{\PSL_n(\R)}(S)$. An interesting connected component of the latter space is the Hitchin space $\mathcal{H}(S)$: it contains a copy of the Teichm\"uller space $\mathcal{T}(S)$, and hence is regarded as the higher rank Teichm\"uller space in the case of $\PSL_n(\R)$. In order to study the elements in the space $\mathcal{H}(S)$, F. Labourie introduced the notion of Anosov representation. In particular, he proved that every Anosov representation is discrete and injective, some properties already shared by Teichm\"uller representations. The purpose of this talk is to extend to Anosov representations some classic tools from hyperbolic geometry designed to study Teichm\"uller representations: we generalize Thurston's length function and cataclysm deformation, and we analyse how they relate to each other. We shall then discuss how these techniques provide crucial information about a new system of coordinates on the Hitchin space $\mathcal{H}(S)$.

Abstract. Semiprojectivity of a C*-algebra A is one formulation of the idea that approximate homomorphisms from A should be close to exact homomorphisms. For example, the algebra of continuous functions on the circle is semiprojective, essentially because an approximate homomorphism to B corresponds to an approximately unitary element in B, and the nearby exact homomorphism comes from the unitary part of its polar decomposition.

We investigate a version of semiprojectivity which requires equivariance with respect to an action of a compact group. We outline the proof that every action of a compact group on a finite dimensional C*-algebra is equivariantly semiprojective, and we describe several other results. Unconventional methods seem to be required. A number of open problems will be stated.

The main application so for is to the classification of actions of finite groups on purely infinite simple separable nuclear C*-algebras.

Abstract. We consider a model of a pure-jump process on a bounded open interval perturbed by Brownian Motion. The jumps occur according to a continuous and spatially dependent rate. We study the model under the assumption that the jump rate vanishes at the boundary. We provide sharp asymptotic bounds on the principal eigenvalue for the generator of the process as the diffusion coefficient of the Brownian Motion tends to zero. Probabilistically, the principal eigenvalue gives the exponential rate of decay of the probability of not exiting the interval for a long time. Our results show non-trivial dependence of the principal eigenvalue on the behavior of the jump rate near the boundary, including a phase transition. This work answers a question posed by Arcusin and Pinsky who studied the multi-dimensional setting with jump rate that is bounded below by a positive constant.

Abstract. We prove modularity of some two dimensional 2-adic Galois representations over a totally real field that are nearly ordinary at all places above 2 and that are residually dihedral. We do this by employing the strategy of Skinner and Wiles using Hida families together with the 2-adic patching method of Khare and Wintenberger. As an application we deduce modularity of some elliptic curves over totally real fields that have good ordinary or multiplicative reduction at places above 2.

Abstract. The structure of lozenge tilings of a hexagon has beautiful combinatorial and probabilistic properties. Random large tilings exhibit a "frozen region" behavior outside the so called "arctic circle". This is related to symmetric (Schur) functions and "Schur processes", determinantal point processes, (discrete) orthogonal polynomials and random matrices, and other statistical mechanical and combinatorial objects. In the talk, we first survey some of the aforementioned known results. We then generalize these results to natural many parameter deformations of known probability measures which we call "elliptic distributions" (elliptic standing for elliptic curves). They provide a combinatorial and probabilistic interpretation of some of Rains' elliptic special functions (generalizing Macodnald and Koornwinder polynomials). We focus on an efficient sampling algorithm based on quasi commutation of elliptic difference operators. We finish by defining the Schur process analogue in the elliptic case, generalizing some of Okounkov and Reshetikhin's work (and Borodin and Corwin's on Macdonald processes). The talk is aimed at a general mathematical audience and assumes no background.

Abstract. The importance of allowing the p-adic variation of modular forms is now well understood. However, in the non-ordinary case, until recently the only method for changing the weight was by multiplication by Eisenstein series of various weights. In a recent work, Andreatta-Iovita-Pilloni found a more conceptual way by varying the sheaf of differentials whose sections define the modular forms. Brinon, Mokrane and myself found another conceptual way which involves a new Igusa tower "the overconvergent Igusa tower". It provides a natural bridge between Katz p-adic modular forms and classical forms. We show it is compatible with Andreatta-Iovita-Pilloni's construction.

Abstract. We will prove that the Borel chromatic number of a graph generated by two Borel functions (which need not commute) is either $\omega$ or at most 5. This is an optimal result. We will also discuss the case of graphs generated by three Borel functions, and prove that their Borel chromatic number, if finite, is at most 8. It is possible that the optimal number in this case is 7.

Abstract. The Patlak-Keller-Segel (PKS) equation models the collective motion of cells which are attracted by a self-emitted chemical substance. While the global well-posedness and finite-time blow-up criteria are well understood, the asymptotic behaviors of solutions are not completely clear. In this talk I will present some results on the asymptotic behavior of solutions, which are obtained by maximum-type arguments. I will also introduce another model which is defined via a gradient flow structure, and discuss its connection with the PKS equation. This is joint work with Inwon Kim.