2:50-3:15                  Coffee

Steven Heilman:
Title: Strong Contraction and Influences in Tail Spaces.
Abstract: We study contraction under a Markov semi-group and influence bounds for functions all of whose low level Fourier coefficients vanish. This study is motivated by the explicit construction of 3-regular expander graphs of Mendel and Naor, though our results have no direct implication for the construction of expander graphs. In the positive direction we prove an $L_{p}$ Poincar\'{e} inequality and moment decay estimates for mean $0$ functions and for all $1<p<\infty$, proving the degree one case of a conjecture of Mendel and Naor as well as the general degree case of the conjecture when restricted to Boolean functions. In the negative direction, we answer negatively two questions of Hatami and Kalai concerning extensions of the Kahn-Kalai-Linial and Harper Theorems to tail spaces. For example, we construct a function $f\colon\{-1,1\}^{n}\to\{-1,1\}$ whose Fourier coefficients vanish up to level $c \log n$, with all influences bounded by $C \log n/n$ for some constants $0<c,C< \infty$. That is, the Kahn-Kalai-Linial Theorem cannot be improved, even if we assume that the first $c\log n$ Fourier coefficients of the function vanish. This implies there is a phase transition in the largest guaranteed influence of functions $f\colon\{-1,1\}^{n}\to\{-1,1\}$, which occurs when the first $g(n)\log n$ Fourier coefficients vanish and $g(n)\to\infty$ as $n\to\infty$ or $g(n)$ is bounded as $n\to\infty$.

Vilmos Prokaj
Title: On the ergodicity of the L\'evy transformation
Abstract: In the talk I will present an approach to the old standing open problem of the
ergodicity of the so called L\'evy transformation of the Wiener space.
This transformation $T$ sends a Brownian motion $B$ into another one by the formula
(T B)_t=\int_0^t \operatorname{sign}(B_t)dB_t.
The main result is that the question could be answered in the affirmative by investigating the sequence
\{(T^nB)_1:\, n\geq 0\}.
Roughly speaking, if the expected hitting time to small neighborhoods of zero is inversely proportional to the size of the neighborhood then $T$ is even exact, that is,
\cap_n\sigma(T^nB)
is trivial. Although the proof of the proposed sufficient condition is still missing, some simulation results supporting it will be presented.

Sneha Subramanian
Title: Zeros of the derivatives of random polynomials and random entire functions
Abstract: For random polynomials and random analytic functions in $\mathbb{C}$ that are constructed as per prescribed random zeros, what can we say about the zeros of the derivative (i.e. critical points) of these functions? First part of this talk we discuss the behavior of the critical points of a random polynomial whose zeros are IID any distribution on the unit circle. For the second part, we study the result of repeatedly differentiating a random entire function whose zeros are points of a Poisson process of intensity 1 on $\mathbb{R}$. Based on joint work with Robin Pemantle.
Title: The frog model on trees.