Tuesday October 6 |
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| 3:00-3:50pm Stefan Steinerberger (University of Washington) Roots of polynomials under repeated differentiation: a nonlocal evolution equation with infinitely many conservation laws (and some universality phenomena), slides Suppose you have a polynomial of degree pn whose n real roots are roughly distributed like a Gaussian (or some other nice distribution) and you differentiate t×n times where 0<t<1. What's the distribution of the (1-t)×n roots of that (t×n)-th derivative? How does it depend on t? We identify a relatively simple nonlocal evolution equation (the nonlocality is given by a Hilbert transform); it has two nice closed-form solutions, a shrinking semicircle and a family of Marchenko-Pastur distributions (this sounds like random matrix theory and we make some remarks in that direction). Moreover, the underlying evolution satisfies an infinite number of conservation laws that one can write down explicitly. This suggests a lot of questions: Sean O'Rourke and I proposed an analogous equation for complex-valued polynomials. Motivated by some numerical simulations, Jeremy Hoskins and I conjectured that t=1, just before the polynomial disappears, the shape of the remaining roots is a semicircle and we prove that for a class of random polynomials. I promise lots of open problems and pretty pictures. |
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| 4:00-4:50pm Bjoern Bringmann (UCLA) Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity, slides In this talk, we discuss the construction and invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree-nonlinearity.
In the first part of the talk, we construct the Gibbs measure and examine its properties. We discuss the mutual
singularity of the Gibbs measure and the so-called Gaussian free field. In contrast, the Gibbs measure for one or
two-dimensional wave equations is absolutely continuous with respect to the Gaussian free field. |
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Tuesday October 21 |
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| 3:00-3:50pm Khang Huynh (UCLA) A geometric trapping approach to global regularity for 2D Navier-Stokes on manifolds, slides We use frequency decomposition techniques to give a direct proof of global existence and regularity for the Navier-Stokes equations on two-dimensional Riemannian manifolds without boundary. Our techniques are inspired by an approach of Mattingly and Sinai which was developed in the context of periodic boundary conditions on a flat background, and which is based on a maximum principle for Fourier coefficients. The extension to general manifolds requires several new ideas, connected to the less favorable spectral localization properties in our setting. Our arguments make use of frequency projection operators, multilinear estimates that originated in the study of the non-linear Schrödinger equation, and ideas from microlocal analysis.
This is joint work with Aynur Bulut. |
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| 4:00-4:50pm Jared Speck (Vanderbildt) Stable big-bang formation in general relativity: the somplete sub-critical regime, slides The celebrated theorems of Hawking and Penrose show that under appropriate assumptions on the matter model, a large, open set of initial data for Einstein’s equations lead to geodesically incomplete solutions. However, these theorems are “soft” in that they do not yield any information about the nature of the incompleteness, leaving open the possibilities that
i) it is tied to the blowup of
some invariant quantity (such as curvature) or Despite the “general ambiguity,” in the mathematical physics literature, there are heuristic results, going back 50 years, suggesting that whenever a certain “sub-criticality” condition holds, the Hawking–Penrose incompleteness is caused by the formation of a Big Bang singularity, that is, curvature blowup along an entire spacelike hypersurface. In recent joint work with I. Rodnianski and G. Fournodavlos, we have given a rigorous proof of the heuristics. More precisely, our results apply to Sobolev-class perturbations – without symmetry – of generalized Kasner solutions whose exponents satisfy the sub-criticality condition. Our main theorem shows that – like the generalized Kasner solutions – the perturbed solutions develop Big Bang singularities.
In this talk, I will provide an overview of our result and explain
how it is tied to some of the main themes of investigation by the mathematical general relativity
community, including the remarkable work of Dafermos–Luk on the stability of Kerr Cauchy horizons.
I will also discuss the new gauge that we used in our work, as well as intriguing connections to
other problems concerning stable singularity formation.
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Tuesday November 3 |
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| 10:00-10:50am Thomas Bloom (Cambridge) Spectral structure and arithmetic progressions, slides How much additive structure can we guarantee in sets of integers, knowing only their density? The study of which density thresholds are sufficient to guarantee the existence of various kinds of additive structures is an old and fascinating subject with connections to analytic number theory, additive combinatorics, and harmonic analysis. In this talk we will discuss recent progress on perhaps the most well-known of these thresholds: how large do we need a set of integers to be to guarantee the existence of a three-term arithmetic progression? In recent joint work with Olof Sisask we broke through the logarithmic density barrier for this problem, establishing in particular that if a set is dense enough such that the sum of reciprocals diverges, then it must contain a three-term arithmetic progression, establishing the first case of an infamous conjecture of Erdos.
We will give an introduction to this problem and sketch some of the recent ideas that have made this progress
possible. We will pay particular attention to the ways we exploit 'spectral structure' - understanding
combinatorially sets of large Fourier coefficients, which we hope will have further applications in number theory
and harmonic analysis.
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| 11:00-11:50am Thomas Beck (Fordham) Two-phase free boundary problems and the Friedland-Hayman inequality, slides The Friedland-Hayman inequality provides a lower bound on the first Dirichlet eigenvalues of complementary subsets of the sphere. In this talk, we will describe a variant of this inequality to geodesically convex subsets of the sphere with mixed Dirichlet-Neumann boundary conditions. Using this inequality, we prove an almost-monotonicity formula and Lipschitz continuity up to the boundary for the minimizer of a two-phase free boundary problem. This is joint work with David Jerison and Sarah Raynor. |
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Tuesday November 17 |
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| 3:00-3:50pm Cristian González-Riquelme (IMPA) BV and Sobolev continuity for maximal operators, slides The regularity of maximal operators has been a topic of interest in harmonic analysis over the past decades. In this topic we are interested in what can be said about the variation of a maximal function Mf given some information about the original function f. In this talk we present some recent results about the continuity of the map f↦ ∇Mf for the uncentered Hardy-Littlewood maximal operator in both the BV(ℝ) and the W1,1rad(ℝd) settings.
This is based on joint works with D. Kosz (BV case) and E. Carneiro and
J. Madrid (radial case)
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| 4:00-4:50pm Yao Yao (Georgia Tech) Two results on the interaction energy, slides For any nonnegative density f and radially decreasing interaction potential W, the celebrated Riesz rearrangement inequality shows the interaction energy E[f] = ∫ f(x)f(y)W(x-y) dxdy satisfies E[f] ≤ E[f*], where f* is the radially decreasing rearrangement of f. It is a natural question to look for a quantitative version of this inequality: if its two sides almost agree, how close must f be to a translation of f*? Previously the stability estimate was only known for characteristic functions. I will discuss a recent work with Xukai Yan, where we found a simple proof of stability estimates for general densities. I will also discuss another work with Matias Delgadino and Xukai Yan, where we constructed an interpolation curve between any two radially decreasing densities with the same mass, and show that the interaction energy is convex along this interpolation. As an application, this leads to uniqueness of steady states in aggregation-diffusion equations with any attractive interaction potential for diffusion power m ≥ 2, where the threshold is sharp. |
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Tuesday December 1 |
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| 3:00-3:50pm Paata Ivanisvili (NC State) Sharpening the triangle inequality in Lp spacesslides The classical triangle inequality in Lp estimates the norm of the sum of two functions in terms of the sums of the norms of these functions. Perhaps one drawback of this estimate is that it does not see how "orthogonal" these functions are. For example, if f and g are not identically zero and they have disjoint supports then the triangle inequality is pretty strict (say for p>1). Motivated by the L2 case, where one has a trivial inequality ||f+g||2 ≤ ||f||2 + ||g||2 + 2 |fg|1, one can think about the quantity |fg|1 as measuring the "overlap" between f and g. What is the correct analog of this estimate in Lp for p different than 2? My talk will be based on a joint work with Carlen, Frank and Lieb where we obtain one extension of this estimate in Lp, thereby proving and improving the suggested possible estimates by Carbery, and another work with Mooney where we further refine these estimates. The estimates will be provided for all real p's. |
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| 4:00-4:50pm David Damanik (Rice) Proving Positive Lyapunov Exponents: Beyond Independence, slides We discuss the problem of proving the positivity of the Lyapunov exponent for Schrödinger operators with potentials defined by a hyperbolic base transformation and a Hölder continuous sampling function. Prominent examples of such base transformations are given by the doubling map and the Arnold cat map. The talk is based on joint work with Artur Avila and Zhenghe Zhang. |
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Tuesday December 16 |
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| 10:00-10:50am Emanuel Carneiro (ICTP) Uncertain signs, slides We consider a generalized version of the sign uncertainty principle for the Fourier transform, first proposed by Bourgain, Clozel and Kahane in 2010 and revisited by Cohn and Gonçalves in 2019, in connection to the sphere packing problem. In our setup, the signs of a function and its Fourier transform resonate with a generic given function P outside of a ball. One essentially wants to know if and how soon this resonance can happen, when facing a suitable competing weighted integral condition. The original version of the problem corresponds to the case P=1. Surprisingly, even in such a rough setup, we are able to identify sharp constants in some cases. This is a joint work with Oscar Quesada-Herrera (IMPA - Rio de Janeiro). |
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| 11:00-11:50am Aleksandr Logunov (Princeton) Zero sets of Laplace eigenfunctions, slides In the beginning of 19th century Napoleon set a prize for the best mathematical explanation of Chladni’s resonance experiments. Nodal geometry studies the zeroes of solutions to elliptic differential equations such as the visible curves that appear in these physical experiments. We will discuss geometrical and analytic properties of zero sets of harmonic functions and eigenfunctions of the Laplace operator. For harmonic functions on the plane there is an interesting relation between local length of the zero set and the growth of harmonic functions. The larger the zero set is, the faster the growth of harmonic function should be and vice versa. Zero sets of Laplace eigenfunctions on surfaces are unions of smooth curves with equiangular intersections. Topology of the zero set could be quite complicated, but Yau conjectured that the total length of the zero set is comparable to the square root of the eigenvalue for all eigenfunctions. We will start with open questions about spherical harmonics and explain some methods to study nodal sets. |
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Last update: December 16, 2020 | |