Tuesday April 6 |
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| 3:00-3:50pm Ramon van Handel (Princeton) The extremals of the Alexandrov-Fenchel inequality It is a basic fact of convexity that the volume of convex bodies is a polynomial, whose coefficients (mixed volumes) define a large family of natural geometric parameters. A fundamental result of convex geometry, the Alexandrov-Fenchel inequality, states that these coefficients are log-concave. This result proves to have striking connections with other areas of mathematics, such as combinatorics and algebraic geometry. There is a long-standing problem surrounding the Alexandrov-Fenchel inequality that has remained open since the original works of Minkowski (1903) and Alexandrov (1937): in what cases is equality attained? This question corresponds to the solution of certain unusual isoperimetric problems, whose extremal bodies turn out to be numerous and strikingly bizarre. With Y. Shenfeld, we recently succeeded to settle this problem completely in the setting of convex polytopes, as well as to develop new tools for the study of general convex bodies. In this talk, I aim to sketch what the extremals look like and to indicate some combinatorial, analytic, and geometric issues that arise in their characterization. |
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| 4:00-4:50pm Pablo Shmerkin (University of British Columbia) Explicit and nonlinear variants of Bourgain's projection theorem, slides Bourgain's projection theorem is a significant extension of his celebrated discretized sum-product theorem. After reviewing the original formulation of the projection theorem, I will present an explicit version, an extension to parametrized families of smooth maps, and applications to the Falconer distance set problem. Partly based on joint work in progress with Hong Wang. |
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Tuesday April 20 |
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| 3:00-3:50pm Nathan Keller (Bar Ilan University) The mysteries of low-degree Boolean functions, slides Analysis of Boolean functions studies functions on the discrete cube {-1,1}n, aiming at understanding what the structure of the (discrete) Fourier transform tells us about the function. In this talk we focus on the structure of low-degree functions on the discrete cube, namely, on functions whose Fourier coefficients are concentrated on low degrees. While such functions look very simple, we are surprisingly far from understanding them well, even in the most basic first-degree case. We shall present several results on first-degree Boolean functions, including the recent proof of Tomaszewski's conjecture (1986) which asserts that any first-degree function (viewed as a random variable) lies within one standard deviation from its expectation with probability at least 1/2. Then we shall discuss several core open questions, which boil down to understanding, what does the knowledge that a low-degree function is bounded, or is two-valued, tell us about its structure.
Based on joint work with Ohad Klein
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| 4:00-4:50pm Oleksiy Klurman (University of Bristol) On the zeros of Fekete polynomials, slides Since its discovery by Dirichlet in the nineteenth century, Fekete polynomials (with coefficients being Legendre symbols) and their zeros attracted considerable attention, in particular, due to their intimate connection with putative Siegel zero and small class number problem.
The goal of this talk is to discuss what we knew, know and would like to know about zeros of such (and, time
permitting, related) polynomials.
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Tuesday May 4 |
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| 10:00-10:50am Dorin Bucur (Université de Savoie) Rigidity results for measurable sets, slides Let Ω ⊂ ℝd be a set with finite Lebesgue measure such that, for a fixed radius r>0, the Lebesgue measure of Ω ∩ Br(x) is equal to a positive constant when x varies in the essential boundary of Ω. We prove that Ω is a ball (or a finite union of equal balls) provided it satisfies a nondegeneracy condition, which holds in particular for any set of diameter larger than r which is either open and connected, or of finite perimeter and indecomposable. This is a joint work with Ilaria Fragala. |
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| 11:00-11:50am Stefanie Petermichl (Universität Würzburg) The matrix-weighted Hardy-Littlewood maximal function is unbounded, slides In a joint work with Nazarov, Skreb and Treil, we highlight a marked difference in the presence of a matrix weight between the Doob type maximal operator in the dyadic setting (with absolute values outside) and the dyadic Hardy-Littlewood type maximal operator (with absolute values inside). The former is L2 bounded while the latter is not. First, it will be discussed how to interpret these operators in a space with matrix weight. For this, we will use convex bodies to replace absolute values (equivalent to the more familiar Christ-Goldberg type definition). We will also discuss the Carleson Embedding Theorems that are the natural partners of these maximal operators and observe a different behaviour as well. |
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Tuesday May 18 |
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| 3:00-3:50pm Izabella Laba (University of British Columbia) Tiling the integers with translates of one tile: the Coven-Meyerowitz tiling conditions for three prime factors, slides It is well known that if a finite set of integers A tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A+B=ZM of a finite cyclic group. Coven and Meyerowitz (1998) proved that when the tiling period M has at most two distinct prime factors, each of the sets A and B can be replaced by a highly ordered "standard" tiling complement. It is not known whether this behaviour persists for all tilings with no restrictions on the number of prime factors of M. In joint work with Itay Londner, we proved that this is true when M=(pqr)2 is odd. (We are currently finalizing the even case.) In my talk I will discuss this problem and introduce the main ingredients in the proof. |
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| 4:00-4:50pm Søren Fournais (Aarhus University) Energy of the Dilute Bose Gas in 3D, slides In this talk, we will review recent progress on the energy of the 3 dimensional dilute Bose gas. It has recently become possible to verify the old prediction by Bogoliubov and Lee-Huang-Yang of the first correction term to the ground state energy of the interacting gas in the thermodynamic limit. If time permits, I will also discuss the relation of these energy results to proofs of "Bose-Einstein condensation” on density dependent length scales. This is joint work with Jan Philip Solovej. |
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Tuesday June 8 |
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| 10:00-10:50am Guofang Wei (UCSB) Fundamental Gap Estimate for Convex Domains In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap conjecture that difference of first two eigenvalues of the Laplacian with Dirichlet boundary condition on convex domain with diameter D in the Euclidean space is greater than or equal to 3π2/D2. In several joint works with X. Dai, Z. He, S. Seto, L. Wang (in various subsets) the estimate is generalized, showing the same lower bound holds for convex domains in the unit sphere. In sharp contrast, in recent joint work with T. Bourni, J. Clutterbuck, X. Nguyen, A. Stancu and V. Wheeler, we prove that the product of the fundamental gap with the square of the diameter can be arbitrarily small for convex domains of any diameter in hyperbolic space. Very recently, jointed with X. Nguyen, A. Stancu, we show that even for horoconvex domains in the hyperbolic space, the product of their fundamental gap with the square of their diameter has no positive lower bound. |
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| 11:00-11:50am Tamar Ziegler (The Hebrew University of Jerusalem) Some applications of analysis over finite fields We describe how one can use equidistribution properties of families of polynomials defined over finite fields to derive some interesting effective results in algebra. For example : given an ideal J generated by m complex homogeneous polynomials of degree < d, we show that J is contained in an ideal J' generated by C(m) homogeneous polynomials of degree < d that form a regular sequence, where C(m) is polynomial in m. All terms will be defined and explained in the talk. |
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Last update: June 9, 2021 | |