# Faculty Activities

*Abstract:*

I will discuss some recent results on minimal actions of general countable groups. In particular I will describe a new property of such minimal actions called the DJ property which is defined in terms of the notion of disjointness of actions and explain how it is related to an old question of Furstenberg on the algebra spanned by the minimal functions on a group. All concepts above will be explained.

*Abstract:*

The octopus lemma states that certain operators on the symmetric group are positive semi-definite. Its original application was to resolve a long-standing conjecture of Aldous related to the spectral gap of interacting particle systems. Since then it has found other applications. We will survey this new topic, perhaps some proofs will be involved.

*Abstract:*

The octopus lemma states that certain operators on the symmetric group are positive semi-definite. Its original application was to resolve a long-standing conjecture of Aldous related to the spectral gap of interacting particle systems. Since then it has found other applications. We will survey this new topic, perhaps some proofs will be involved.

*Abstract:*

Mathematical epidemiology uses modelling to study the spread of contagious diseases in a population, in order to understand the underlying mechanisms and aid public health planning. In recent years there is growing interest in applying similar models to the study of `social contagion': the spread of ideas and behaviors. It is of great interest is to consider the ways in which social contagion differs from biological contagion at the individual level, and to use mathematical modelling to understand the population-level consequences of these differences. In this talk I will present simple `two-stage' contagion models motivated by social-science literature, and study their dynamics. It turns out that these models give rise to some interesting and non-intuitive nonlinear phenomena which do not arise in the `classical' models of mathematical epidemiology, and which might have relevance to understanding some real-world observations.

*Abstract:*

A main goal of geometric group theory is to understand finitely generated groups up to a coarse equivalence (quasi-isometry) of their Cayley graphs. Right-angled Coxeter groups, in particular, are important classical objects that have been unexpectedly linked to the theory of hyperbolic 3-manifolds through recent results, including those of Agol and Wise. I will give a brief background of what is currently known regarding the quasi-isometric classification of right-angled Coxeter groups. I will then describe a new computable quasi-isometry invariant, the hypergraph index, and its relation to other invariants such as divergence and thickness.

*Abstract:*

T.B.A.

*Abstract:*

**Advisor: **Emanuel Milman

**Abstract: **We establish new sharp inequalities of Poincare or log-Sobolev type, on weighted Riemannian manifolds whose (generalized) Ricci curvature is bounded from below. To this end we establish a general method which complements the 'localization' theorem which has recently been established by B. Klartag. Klartag's theorem is based on optimal transport techniques, leading to a disintegration of the manifold measure into marginal measures supported on geodesics of the manifold. This leads to a reduction of the problem of proving a n-dimensional inequality into an optimization problem over a class of measures with 1-dimensional supports. Our method is based on functional analytic techniques, and leads to a further reduction of the optimization problem into a simpler problem over a sub-class of model-space measures. By employing ad-hoc analytical techniques we solve the optimization problems associated with the Poincare and the log-Sobolev inequalities. Quiet unexpectedly the solution to the problem of characterizing the sharp Poincare constant reveals anomalous behavior within a certain domain of the generalized-dimension parameter, hinting on a new phenomena.

*Abstract:*

In 2008 Agol showed that a 3-manifold with a certain condition on its fundamental group is virtually fibered, i.e. it has a finite covering that is a surface bundle over the circle. A few years later it was shown by Agol and Wise that the fundamental groups of most 3-manifold satisfy Agol's condition, i.e. most 3-manifodls are virtually fibered. We will outline a proof of Agol's theorem following an approach taken by myself and Kitayama.

*Abstract:*

T.B.A.

*Abstract:*

The Boltzmann equation without angular cutoff is considered when the initial data is a perturbation of a global Maxwellian with algebraic decay in the velocity variable. Global solution is proved by combining the analysis in moment propagation, spectrum of the linearized operator and the smoothing effect of the linearized operator when initial data in Sobolev space with negative index.

This is a joint work with Ricardo Alonso, Yoshinori Morimoto and Weiran Sun.

*Abstract:*

Nonholonomic mechanics concerns with mechanical systems whose velocity is constrained. If these velocity constraints are linear, they define k-planes at every point of the configuration space of the system. In more complex situations further constraints appear: the movement of the system not only has to be tangent to these k-planes, but must obey conditions in which tangent vectors to the trajectories have constant length, or satisfy other, in general nonlinear, relations. This equips kinematics of nonholonomic mechanical systems with various geometric structures. These are: vector distributions on manifolds, their symmetry groups, differential invariants, associated exterior differential systems, Cartan connections, etc.

In the lectures we will discuss these geometric structures in simple examples of existing (or possible to exist) mechanical systems. We will concentrate on systems whose kinematics is described by a low dimensional parabolic geometry i.e. a geometry modeled on a homogeneous space G/P, with G being a simple Lie group, and P being its parabolic subgroup. The considered systems will include a movement of ice skaters on an ice rink, rolling without slipping or twisting of rigid bodies, movements of snakes and ants, and even movements of flying saucers. Geometric relations between these exemplary nonholonomic systems will be revealed. An appearance of the simple exceptional Lie group G2 will be a repetitive geometric phenomenon in these examples.

*Abstract:*

T.B.A.

*Abstract:*

T.B.A.

*Abstract:*

TBA