Let S be surface of hyperbolic type and T(S) be the Teichmüller space of S. We would like to study the group of quasi-isometries of T(S). As a first step we determine the rank of T(S), that is, the largest dimension N where a large box B in RN can be embedded quasi-isometrically into T(S). The main tool we use is the coarse differentiation lemma. This states that, any quasi-isometric embedding, in the correct scale, is nearly an affine map.

In this talk, we will start with an overview of the asymptotic Plateau problem, and then study the relation between being least area and being properly embedded in hyperbolic 3-space. In particular, we show that if P is an embedded least area plane in hyperbolic 3-space whose asymptotic boundary is a simple closed curve with at least one smooth point, then P is properly embedded. Remark: Note the room change from the 2pm talk.

Abstract: I will provide more details on one or both of the following: - The construction of topological invariants from SU(N) representation spaces. - Calculations of Floer homology using the "strip shrinking" isomorphism

Note the new room for the seminar this semester.

The group number function gnu(n) is defined as the number of groups of order n, up to isomorphism. In general, gnu(n) eludes coherent description, yet for square-free integers, it is given by a simple formula due to Hölder. We will explain this classical result using tools such as the Fitting subgroup and the transfer homomorphism. The enumeration of finite groups abounds with conjectures concerning the behavior of gnu(n), its restrictions, and related functions. The talk will highlight some major conjectures and developments.

Abstract: In many biological systems, we observe the emergence of self-organized dynamics (flock of birds, school of fish, aggregation of bacteria...). To understand the phenomenon of flocking, a simple model has been proposed, the so-called Vicsek model. This model describes how particles interact in order to align with their neighbors. In this talk, we first briefly explain how we can derive a large-scale limit of this model when the number of particles becomes very large. This derivation leads to a non-conservative hyperbolic system referred to as the "macroscopic model". In contrast, the original Vicsek model is called the "microscopic model". In a second part, different numerical methods are proposed to solve the macroscopic model. Since the model is non-conservative, the different methods lead to different solutions. However, the comparison between the numerical simulations and the microscopic model reveals that only one numerical method, called the "splitting method", is able to reproduce the large scale dynamics of the Vicsek model. Understand this result analytically is still an open problem.

Note the new room for the seminar this semester.

This talk will be an introduction to the deformation theory of infinite volume hyperbolic 3-manifolds. It will focus on the case where is a closed hyperbolic surface.

Let be a sequence of measurable sets in a probability space such that . The classical Borel--Cantelli lemma states that if the sets are independent, then . Suppose is a dynamical system and is a sequence of sets in . If infinitely often for a.e. we call the sequence a Borel--Cantelli sequence. If the sets are nested balls about a point then the question of whether infinitely often for a.e. is often called the shrinking target problem. If it is assumed that then we give conditions in terms of return time statistics which imply that for a.e. sequences of nested balls are Borel--Cantelli. Corollaries of our results are that for planar dispersing billiards and Lozi maps a.e. sequences of nested balls are Borel--Cantelli. We also give applications of these and related results to a variety of non-uniformly hyperbolic dynamical systems.

In this talk, I¡¯ll introduce the modulated Fourier expansion approach to highly oscillatory ODE problems. Mechanism of HMM for ODE systems and error analysis will be given.

Iwasawa theory provides a mysterious connection between the analytic and algebraic worlds. For elliptic curves, the two objects in each world were constructed in the 1970's by Mazur and Swinnerton-Dyer on the analytic side, and by Mazur on the algebraic side - but under an important assumption: that we be working with a prime of good ordinary reduction. As a corollary, Mazur gave a growth formula for (the p-primary part of) the Tate-Shafarevich group in towers of number fields, which is an analogue of a classical result by Iwasawa on the growth of (the p-primary part of) the class group in certain towers of number fields. Much less was known about the supersingular (= good, but not ordinary) case until the 2000s. Pollack constructed pairs of analytic objects, while these were related to pairs of algebraic objects by Kobayashi - although they both made the assumption that the trace of Frobenius a_p vanished. In this talk, I would like to explain how to remove the vanishing assumption on a_p and present some growth patterns of the Tate-Shafarevich group.

Understanding Pedestrian and crowd dynamics is an important issue in modern societies. Recent crowd disasters (e.g. Duisburg love parade (July 2010) or Cambodia water festival (November 2010)) have put the safety issues in the forward scene. Another important issue the is efficiency of public arenas such as subway corridors, train or airport terminals, or shopping malls. Many different kinds of models of pedestrian and crowd dynamics exist, from the discrete (individual-based) ones to the continuum ones. The talk will review the sallient features of these models. However, most of the existing models suffer from a poor validation against experimental data. There are many difficulties ranging from ethical to technical ones which explain why the data are scarce. We will report on a series of experiments that have been conducted in the 'Pedigree' project in France and the preliminary findings that have resulted from these data. Future modeling directions and challenges will be discussed.

This paper addresses the question of how to invest in a robust growth-optimal way in a market where the instantaneous expected return of the underlying process is unknown. The optimal investment strategy is identified using a generalized version of the principal eigenfunction for an elliptic second-order differential operator which depends on the covariance structure of the underlying process used for investing. The robust growth-optimal strategy can also be seen as a limit, as the terminal date goes to infinity, of optimal arbitrages in the terminology of Fernholz and Karatzas. (Joint Work with Kostas Kardaras).

Chaotic systems have become of increasing interest in applied math because of their seemingly unpredictable nature: a small change in initial data can lead to surprising consequences very quickly. We would like to be able to model such systems because it turns out they occur quite frequently in nature - examples include fluid dynamics, medicine, and weather forecasting. In this talk we will focus on the role of chaos in population growth; we will be looking at the discrete logistic equation and will ultimately prove something very cool about its behavior.