Analysis Seminar, Dept. of Mathematics and Applied Math. -- Σεμινάριο Ανάλυσης

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Analysis Seminar in Crete (2024-25)

Abstract: We prove that the fractional chromatic number of the unit distance graph of the Euclidean plane is greater than or equal to 4. This improves a series of earlier lower bounds edging closer to 4 over the past decades. A fundamental ingredient of the proof is the notion of geometric fractional chromatic number introduced recently in connection with the density of planar 1-avoiding sets. In the proof we also exploit the amenability of the group of Euclidean transformations in dimension 2.

Abstract: I shall present two refinements of Webb’s sharp upper bound on the volume of central slices of the regular simplex: stability as well as sharp bounds on $L_p$ norms.

Thu, 31 Oct 2024, 11:15, Room: Α303
Speaker: Natalia Tziotziou (NTUA-SEMFE)

Abstract: We provide extensions of geometric inequalities about sections and projections of convex bodies to the setting of integrable log-concave functions. Namely, we consider suitable generalizations of the affine and dual affine quermassintegrals of a log-concave function f and obtain upper and lower estimates for them in terms of the integral $\|f\|_1$ of $f$, we give estimates for sections and projections of log-concave functions in the spirit of the lower dimensional Busemann-Petty and Shephard problem, and we extend to log-concave functions the affirmative answer to a variant of the Busemann-Petty and Shephard problems, proposed by V. Milman.

Abstract: The power series of $\sin(x)$, $\exp(-\pi x^2)$, and $\exp(1-\exp(x))$, all converge to a bounded function on the real line. What do their coefficients have in common? In this talk, we explore this question from analytical, topological, and algebraic perspectives. For $\exp(1-\exp(x))$, the question relates to an open problem in analytic combinatorics.

Abstract: In a recent article, Donoso, Le, Moreira and Sun studied sets of recurrence for multiplicative actions of the natural numbers and provided some sufficient conditions for sets of the form $S=\{ (an+b)/(cn+d) : n\in \mathbb{N}\}$ to be sets of recurrence for such actions. A necessary condition for $S$ to be a set of multiplicative recurrence is that for every completely multiplicative function $f$ taking values in $\mathbb{S}^1$ we have that $\liminf_{n\to \infty} |f(an+b)-f(cn+d)|=0$. We fully characterise the integer quadruples $(a,b,c,d)$ which satisfy the latter property. Our result generalizes a result of Klurman and Mangerel concerning the pair $(n,n+1)$, as well as some results of Donoso, Le, Moreira and Sun. This is based on joint work with Dimitrios Charamaras and Konstantinos Tsinas.

Abstract: Take an interval $[t, t+1]$ on the $x$-axis together with the same interval on the $y$-axis and let $\rho$ be the normalized one-dimensional Lebesgue measure on this set of two segments. Continuing the work done by Lev (2018), Lai, Liu and Prince (2021) as well as Ai, Lu and Zhou (2023) we examine the spectrality of this measure for all different values of $t$ (being spectral means that there is an orthonormal basis for $L^2(\rho)$ consisting of exponentials $e^{2\pi i (\lambda_1 x + \lambda_2 y)}$). We almost complete the study showing that for $-\frac12 \lt t \lt 0$ and for all $t \notin \QQ$ the measure $\rho$ is not spectral. The only remaining undecided case is the case t=-1/2 (plus space). We also observe that in all known cases of spectral instances of this measure the spectrum is contained in a line and we give an easy necessary and sufficient condition for such measures to have a line spectrum.

Exponential polynomials and identification of polygonal regions from Fourier samples

Abstract: Consider the set $E(D, N)$ of all bivariate exponential polynomials $$ f(\xi, \eta) = \sum_{j=1}^n p_j(\xi, \eta) e^{2\pi i (x_j\xi+y_j\eta)}, $$ where the polynomials $p_j \in \CC[\xi, \eta]$ have degree $ We use this in order to prove some uniqueness results about polygonal regions given a small set of samples of the Fourier Transform of their indicator function. If the number of different slopes of the edges of the polygonal region is $\le k$ then the region is determined from a predetermined set of Fourier samples that depends only on $k$ and the maximum number of vertices $N$ and is of size $O(k^2 N \log N)$. In the particular case where all edges are known to be parallel to the axes the polygonal region is determined from a set of $O(N \log N)$ Fourier samples that depends on $N$ only.

New algorithms for two-player zero-sum games: from games to optimization and learning

Abstract: By the seminal paper of John von Neumann on the Minimax Theorem for two-player zero-sum games, this class of games became one of the most fundamental class in game theory. It is well-known that the computation of a pair of optimal solutions, Nash equilibrium, in this class can be done in polynomial-time by a number of different algorithms such as algorithms for Linear programming, first-order methods and learning algorithms. However, many of these algorithms are centralized and/or converge to an optimal solution on average. On the other hand, recent applications of zero-sum games in machine learning, particularly Generative Adversarial Networks (GANS), have created the need of new algorithms with better properties such as last-iterate convergence to an optimal solution. In this presentation, we present our two recent algorithms for computing a Nash equilibrium in these games: one gradient-based algorithm and one learning algorithm that improve the state of the art from different perspectives.