Leírás
A Magyar Tudomány Ünnepe eseménysorozat keretében Intézetünk fiatal kutatói tartanak húszperces előadásokat friss eredményeikből. Mivel az idei esemény mottója "Határtalan tudomány", ezért külföldi kollégáink adnak elő.
A részeletes program:
14:00 - 14:20 Chau Ngoc Huy: On stochastic gradient Langevin dynamics with stationary data
streams in the logconcave case
Abstract: Stochastic Gradient Langevin Dynamics (SGLD) is a combination of a Robbins-Monro type
algorithm with Langevin dynamics in order to perform data-driven stochastic optimization. In this
paper, the SGLD method with fixed step size $\lambda$ is considered in order to sample from a logconcave
target distribution $\pi$, known up to a normalisation factor. We assume that unbiased estimates of
the gradient from possibly non i.i.d., non Markovian observations, are available. The Wasserstein-2
distance of the $n$th iterate of the SGLD algorithm from $\pi$ is derived. Joint work with M. Rasonyi (Alfred Renyi Institute)
M. Barkhagen, S. Sabanis, Y. Zhang (University of Edinburgh)
E. Moulines (Ecole Polytechnique)
14:25 - 14:45 Mikolaj Fraczyk: Growth of mod-$p$ homology in sequences of locally symmetric spaces.
Abstract: Let $\Gamma_n\backslash X$ be a sequence of finite volume quotients of a symmetric space $X$ by lattices in its isometry group and let $F$ be a field. We are interested in the growth of the Betti numbers $b_i(\Gamma_n\backslash X, F)=\dim_F H_1(\Gamma_n\backslash X, F)$ as the volume of $\Gamma_n\backslash X$ tends to infinity. If the characteristic of $F$ is equal to $0$ a very satisfactory answer is provided by L\"uck approximation theorem and its refinements. They predict that under some natural conditions imposed on the sequence $\Gamma_n$ the limit $\lim_{n\to\infty} \frac{b_i(\Gamma_n\backslash X,F)}{{\rm Vol}(\Gamma_n\backslash X)}$ exists and equals so called $L^2$-Betti number $\beta_i(X)$ which depends only on $X$ and can be explicitly calculated. When $F$ is of positive characteristic much less is known. In my talk I will describe a new method designed to study the growth of Betti numbers in positive characteristic. It is based on the observation that if all the homology classes can be represented by cycles of "small support" then $\lim_{n\to\infty} \frac{b_i(\Gamma_n\backslash X,F)}{{\rm Vol}(\Gamma_n\backslash X)}=0.$ As an application we will show that $\lim_{n\to\infty} \frac{b_1(\Gamma_n\backslash X,\mathbb{Z}/2\mathbb{Z})}{{\rm Vol}(\Gamma_n\backslash X)}=0$ for every higher rank symmetric space $X$ and every sequence of lattices $\Gamma_n$ such that $\Gamma_n\backslash X$ converges Benjamini-Schramm to $X$.
14:50 - 15:10 Deniz Kaptan: Large Gaps Between Primes in Arithmetic Progressions
Abstract: We apply the recent developments in the study of prime gaps to the investigation of large gaps between primes in progressions. Over a range of the moduli $M$ with certain restrictions, we seek uniform lower bounds for the largest gap between consecutive primes $a \mod{M}$ up to $X$. As one would expect, we obtain the bound from the ordinary case expanded by a factor of $\phi(M)$, updating a result due to Zaccagnini.
Kávészünet
15:40 - 16:00 Fabio Gironella: The group of contactomorphisms: from Sophus Lie to modern topological techniques.
Abstract: I will start by recalling the notion of contact transformation, introduced by Lie in 1896 as a geometrical tool to study differential equations, and I will (very briefly) describe how, thanks to the work of several authors, contact geometry has since then become a field of research on its own, also acquiring a much more topological flavor starting from mid 1980s. The aim of the second part of the talk is then to describe how this modern topological point of view can be fruitfully adopted to study the group of contact transformations.
16:00 - 16:20 Nika Salia: Which HyperGraphs, without Long Berge Paths and Cycles, are Extremal?
Abstract: A well-known theorem of Erdős and Gallai from 1959, asserts that a graph with no path of length k contains at most $\frac{(k−1)n}{2}$ edges and a graph with no cycle of length at least k contains at most $\frac{(k-1)(n-1)}{2}$ edges. We will discuss extensions of this results for hypergraphs.