Friday Jan 10 |
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| 4:00-4:50 UCLA, MS 6627 Ioan Bejenaru (UCSD) The multilinear restriction estimates for hypersurfaces with curvature We will present an overview of the multilinear restriction theory, with an emphasis on the case when the hypersurfaces have some curvature. We will discuss a new result: the case of $n-1$ hypersurfaces in $n$ dimensions where a fairly general theory is developed. |
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| 5:00-5:50 UCLA, MS 6627 Jose Ramon Madrid Padilla (UCLA) Improving estimates for discrete polynomial averages For a polynomial $P$ mapping the integers into the integers, define an averaging operator $A_{N} f(x):=\frac{1}{N}\sum_{k=1}^N f(x+P(k))$ acting on functions on the integers. We prove sufficient conditions for the $\ell^{p}$-improving inequality \begin{equation*} \|A_N f\|_{\ell^q(\mathbb{Z})} \lesssim_{P,p,q} N^{-d(\frac{1}{p}-\frac{1}{q})} \|f\|_{\ell^p(\mathbb{Z})}, \qquad N \in\mathbb{N}, \end{equation*} where $1\leq p \leq q \leq \infty$. For a range of quadratic polynomials, the inequalities established are sharp, up to the boundary of the allowed pairs of $(p,q)$. For degree three and higher, the inequalities are close to being sharp. In the quadratic case, we appeal to discrete fractional integrals as studied by Stein and Wainger. In the higher degree case, we appeal to the Vinogradov Mean Value Theorem, recently established by Bourgain, Demeter, and Guth. |
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Friday Jan 24 |
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| 4:30-5:20 Caltech, Linde 310 David Conlon (Caltech) Graph homomorphism inequalities The graph homomorphism density $t_H(G)$ is a measure of the number of copies of a fixed graph $H$ in another graph $G$. In this talk, we survey recent work concerning inequalities between $t_H(G)$ and $t_K(G)$ for different graphs $H$ and $K$, touching upon Sidorenko's conjecture, graph norms and related topics. Much of the material discussed is joint work with Joonkyung Lee. |
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| 5:30-6:20 Caltech, Linde 310 Matt Jacobs (UCLA) The $H^1$ projection problem and the Navier-Stokes equations Given a map $S$, the $H^1$ projection problem seeks a measure preserving bijection $Z$ that is as close as possible to $S$ in the $H^1$ norm. This problem is closely related to Arnold’s geometric interpretation of fluid mechanics. Indeed, the Euler-Lagrange equation associated to the H^1 projection problem can be viewed as a discrete-in-time approximation to the Lagrangian formulation of the Navier-Stokes equations. If the $H^1$ norm is replaced by the $L^2$ norm, the existence and uniqueness of minimizers follows from Brenier’s celebrated polar factorization theorem, however the corresponding results for the $H^1$ projection problem in 3-dimensions have long remained open. In this talk, I will present an argument establishing existence and uniqueness of minimizers for the $H^1$ projection problem under some regularity assumptions on $S$. This talk is joint work with Wilfrid Gangbo and Inwon Kim. |
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Friday Feb 7 |
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| 4:00-4:50 UCLA, MS 6627 Ruixiang Zhang (U. Wisconsin-Madison) Local smoothing for the wave equation in $2+1$ dimensions Sogge's local smoothing conjecture for the wave equation predicts that the local $L^p$ space-time estimate gains a fractional derivative of order almost $\frac{1}{p}$ compared to the fixed time $L^p$ estimates, when $p\geq \frac{2n}{n-1}$. Jointly with Larry Guth and Hong Wang, we recently proved the conjecture in $\mathbb{R}^{2+1}$. I will talk about a sharp square function estimate we proved which implies the local smoothing conjecture in dimensions $2+1$. A key ingredient in the proof is an incidence type theorem. |
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| 5:00-5:50 UCLA, MS 6627 Jing-Jing Huang (U. of Nevada, Reno) Counting rational points close to a hypersurface We will mainly talk about problems related to counting rational points in a thin neighborhood of a manifold. Using Fourier analytic methods, we have recently solved the case of hypersurfaces. A key step of the proof utilizes a duality relation between the counting functions for the hypersurface and its dual surface to bootstrap the counting bound. There are also significant applications of our results to diophantine inequalities and metric diophantine approximation on manifolds. |
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Friday Feb 21 |
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| 4:30-5:20 Caltech, Linde 310 Theresa Anderson (Purdue U.) Discrete Maximal functions over curved subvarieties Starting with a motivating distributional question about prime vectors on spheres, we study variants of the spherical maximal functions that arise in such a context. Maximal functions involving primes, bilinear variants involving two spheres and higher codimensional analogues involving triangles appear on the menu in today's talk. |
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| 5:30-6:20 Caltech, Linde 310 Zane Li (IU Bloomington) A bilinear proof of decoupling for the moment curve We give a proof of decoupling for the moment curve that is inspired from nested efficient congruencing. We also discuss the relationship between Wooley's nested efficient congruencing and Bourgain-Demeter-Guth's decoupling proofs of Vinogradov's Mean Value Theorem. This talk is based on joint work with Shaoming Guo, Po-Lam Yung, and Pavel Zorin-Kranich. |
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Friday Mar 6 |
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| 4:00-4:50 UCLA, MS 6627 Simone Steinbruechel (University of Zürich) Plateau's problem and geometric measure theory In the 19th century, the problem of finding the surface of least area among those spanning a given boundary was introduced as Plateau's problem. In this talk, we will present two different ways how to tackle the problem and then focus on the methods of Geometric Measure Theory in order to discuss existence, regularity and uniqueness of minimal surfaces. |
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| 5:00-5:50 UCLA, MS 6627 Michael Christ (UC Berkeley) On trilinear oscillatory integrals Integrals with rapidly oscillating factors arise in many contexts, and an extensive theory provides upper bounds and asymptotics in various circumstances. The theory of multilinear oscillatory integrals of degree three and higher is, however, relatively undeveloped. Several results for the trilinear case will be presented, the underlying method of analysis will be outlined, and an application to weak continuity of certain trilinear products will be discussed. |
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Last update: March 2, 2020 | |