This is the 7th lecture in the series.

Abstract: 

The theory of influences in product measures has profound applications in theoretical computer science, combinatorics, and discrete probability.

This deep theory is intimately connected to functional inequalities and to the Fourier analysis of discrete groups.
Originally, influences of functions were motivated by the study of social choice theory, wherein a Boolean function represents a voting scheme, its inputs represent the votes,
and its output represents the outcome of the elections. Thus, product measures represent a scenario in which the votes of the parties are randomly and independently
distributed, which is often far from the truth in real-life scenarios.

We begin to develop the theory of influences for more general measures under mixing or spectral independence conditions.
More specifically, we prove analogues of the KKL and Talagrand influence theorems for Markov Random Fields on bounded degree graphs
when the Glauber dynamics mix rapidly.
We thus resolve a long standing challenge, stated for example by Kalai and Safra (2005).
We show how some of the original applications of the theory of in terms of voting and coalitions extend to these general dependent measures.
Our results thus shed light both on voting with correlated voters and on the behavior of general functions of Markov Random Fields (also called “spin-systems”) where the Glauber dynamics mixes rapidly.

We consider several problems pertaining to low and high-dimensional data, and their relation to the approximation power of Neural Networks. Given a noisy point cloud in a high-dimensional space, we will address the question of denoising and reconstructing a low-dimensional manifold in a high-dimensional space. To address this challenge, we introduce a framework named “Manifold Locally Optimal Projection (MLOP)” and provide its accompanying theoretical analysis.

In the second part of my talk, we will delve into the theoretical aspects of Neural Networks through the lens of approximation theory. Refinable functions, which are the solutions of refinement equations, are the building stones in many constructions; including subdivision schemes used in computer graphics, wavelets, B-splines, as well as several fractals. Even though our earlier work proved that all refinable functions can be implemented, up to arbitrary high precision, by ReLu-based Neural Networks, it was far from clear how such functions could be learned from data. We propose a different type of refinement that involves not only translation and rescaling but also mirroring; functions satisfying the resulting reflecto-refinement equations still generate multiresolution hierarchies that provide an excellent approximation for many functional spaces of interest, yet are also adapted to ReLu networks. We will illustrate the proposed methodology to create new function families.

The talk will be based on joint works with David Levin (TAU) and Ingrid Daubechies (Duke)

This is the 8th lecture in the series. 

Recently, Francis Bonahon, Helen Wong and Tian Yang constructed a quantum invariant for surface diffeomorphism using representation theory of Kauffman Bracket Skein Algebra. They proposed a conjecture that relates this invariant with the volume of mapping torus coming from the diffeomorphism. We will talk about this construction, some explicit computation techniques and recent results.  

We give a description of stationary probability measures on projective spaces for
an iid random walk on GLd(R) without any algebraic assumptions. This is done
in two parts. In a first part, we study the case (non-critical or block-dominated
case) where the random walk has distinct deterministic exponents in the sense of
Furstenberg-Kifer-Hennion. In a second part (critical case), we show that if the
random walk has only one deterministic exponent, then any stationary probability
measure on the projective space lives on a subspace on which the ambient group of
the random walk acts completely reducibly. This connects the critical setting with
the work of Guivarc’h-Raugi and Benoist-Quint. Combination of all these works
allow to get a description of stationary probability measures. Joint works with Richard Aoun.

Transverse subgroups of semi-simple Lie groups include all discrete subgroups of rank one Lie groups and Anosov and relatively Anosov subgroups of higher rank Lie groups. Every transverse group comes equipped with a  family of “geodesic flows’’ associated to linear functionals on the Cartan subspace. The assumption of transversality allows for the construction of well-behaved Patterson-Sullivan measures on the limit set and associated Bowen-Margulis-Sullivan measures on the flows.  We discuss a Hopf-Tsuji-Sullivan dichotomy in this context. Finally, we show that relatively Anosov groups live on the divergent side of this dichotomy (Joint work with Tengren Zhang and Andy Zimmer)

A relaxed-pace seminar on impromptu subjects related to the interests of the audience.

Everyone is welcome.

The subjects are geometry, probability, combinatorics, dynamics, and more!

We present an efficient high-order discretization scheme for the evaluation of Laplace and Helmholtz layer potentials on smooth surfaces in three dimensions. The scheme is panel based and applies an analytical surface to line integral conversion on each panel to evaluate single layer, double layer, and their normal derivatives accurately. A new basis approximation scheme tailed for these layer potential kernels is proposed. Both nearly singular and singular cases are supported via a unified recursive framework. The Laplace part of the scheme is joint work with Shravan Veerapaneni, and the Helmholtz part is joint work with Shidong Jiang.

Let F be a family of finite coverings of a hyperbolic surface S. A spectral gap of F is an interval I = [0, epsilon] such that the eigenvalues in I  (counted with multiplicity) of the Laplacian \Delta_S of S and \Delta_X, any X ∈ F, are the same. I will present a joint work with M. Magee where we give a spectral gap for congruence coverings when S is the surface associated to a Schottky subgroup of SL(2, Z) with thick enough limit set. The proof exploits the link between eigenvalues of the Laplacian and zeros of dynamical zeta functions attached to S via the thermodynamic formalism.