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.