Friday Oct 5 |
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| 4:00-4:50 UCLA, MS 6627 Sarah Peluse (Stanford) Bounds for sets lacking $x,x+y,x+y^2$ Let $P_1,\dots,P_m\in\mathbb{Z}[y]$ be polynomials with zero constant term. Bergelson and Leibman's generalization of Szemerédi's theorem to polynomial progressions states that any $A\subset [N]$ lacking nontrivial progressions of the form $x,x+P_1(y),\dots,x+P_m(y)$ satisfies $|A|=o(N)$. Proving quantitative bounds in the Bergelson-Leibman theorem is an interesting and difficult generalization of the problem of proving bounds in Szemerédi’s theorem, and bounds are known only in a very small number of special cases. In this talk, I'll discuss a bound for subsets of $[N]$ lacking the progression $x,x+y,x+y^2$, which is the first progression of length at least three involving polynomials of differing degree for which a bound is known. This is joint work with Sean Prendiville. |
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| 5:00-5:50 UCLA, MS 6627 Philip Isett (Caltech) Local Dissipation of Energy for Continuous Incompressible Euler Flows I will discuss the construction of continuous solutions to the incompressible Euler equations that exhibit local dissipation of energy and the surrounding motivations from the study of turbulence. A significant open question, which represents a strong form of the Onsager conjecture, is whether such solutions exist that locally dissipate energy while having the maximal possible regularity of being 1/3-Hölder continuous. |
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Friday Oct 19 |
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| 4:30-5:20 Caltech, Linde 310 Maksym Radziwill (Caltech) Gaps between consecutive eigenvalues of the Laplacian on generic flat tori A basic problem in mathematical physics is to understand the spacings between eigenvalues of the Laplacian defined on various manifolds. The simplest case, which is already difficult, is the case of generic flat tori. I will discuss some conjectures and recent results on small and large gaps between consecutive eigenvalues of the Laplacian on generic flat tori. This will describe joint works with Aistlaitner, Blomer, Bourgain and Rudnick. |
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| 5:30-6:20 Caltech, Linde 310 Hong Wang (MIT) A restriction estimate in $\mathbb{R}^3$ If $f$ is a function supported on a truncated paraboloid, what can we say about $Ef$, the Fourier transform of $f$? Stein conjectured in the 1960s that for any $p>3$, $$\|Ef\|_{L^p(\mathbb{R}^3)} \lesssim \|f\|_{L^{\infty}}.$$ We make a small progress toward this conjecture and show that $p> 3+3/13\approx 3.23$. In the proof, we combine polynomial partitioning techniques introduced by Guth and the two ends argument introduced by Wolff and Tao. |
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Friday Nov 2 |
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| 4:00-4:50 UCLA, MS 6627 Erin Compaan (MIT) Well-posedness for Some Dispersive PDEs on the Half-line We discuss a method for obtaining well-posedness for dispersive PDEs posed on a half line. It allows us to apply powerful Fourier-restriction norm methods to half-line problems. In this talk, we show well-posedness and smoothing results obtained for the "good" Boussinesq model. We also discuss global results for the Klein-Gordon-Schrodinger system and recent work on the gKdV system. Based on joint work with N. Tzirakis. |
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| 5:00-5:50 UCLA, MS 6627 Dominique Maldague (UC Berkeley) A constrained optimization problem for the Fourier transform I will present some recent work concerning extremization in harmonic analysis. We will focus on the following question: Among functions $f$ majorized by indicator functions $1_E$, which functions have maximal ratio $\|\widehat{f}\|_q/|E|^{1/p}$? I will give some context for this question, and describe what is known so far. Then for exponents $q\in(3,\infty)$ sufficiently close to even integers, we identify the maximizers and prove a quantitative stability theorem. |
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Friday Nov 16 |
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| 4:30-5:20 Caltech, Linde 310 Andrés Chirre (IMPA) Bounds for the Riemann zeta-function via Fourier analysis This talk will be a brief survey on the history of some objects related with the Riemann zeta-function and his study via Fourier analysis. The machinery that appears here will be bandlimited approximations that solve the Beurling-Selberg extremal problem and the classical resonance method. In the final part, we will show recent bounds of some objects that study the distribution of zeros of the Riemann zeta-function, using functions that have Fourier transform eventually non-positive. |
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| 5:30-6:20 Caltech, Linde 310 José Ramón Madrid Padilla (UCLA) On the regularity of maximal operators In this talk we will discuss the regularity properties (boundedness and continuity) of the classical and fractional maximal operators when these act on the Sobolev space $W^{1,p}(\mathbb{R}^{n})$. We will focus on the case $p=1$. We will talk about the recent results obtained and some of the current open problems. |
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Friday Nov 30 |
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| 4:00-4:50 UCLA, MS 6627 Ruixang Zhang (UW-Madison) The pointwise convergence problem for the free Schrödinger equation Carleson proposed a problem on a.e. convergence for free Schrödinger solutions as time goes to zero. Recently it got a sharp answer (up to the endpoint) in all dimensions. We will talk about the new result in dimensions $n+1$ for all $n>2$ and ideas behind it (joint work with Xiumin Du). |
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| 5:00-5:50 UCLA, MS 6627 Igor Kukavica (USC) The Euler equations with a free interface We address the local existence of solutions for the water wave problem, which is modeled by the incompressible Euler equations in a domain with a free boundary evolving with the flow. We are particularly interested in the local existence for the initial velocity, which is rotational and belongs to a low regularity Sobolev space. We will review the available existence and uniqueness results for the problem with surface tension. We will also briefly mention the compressible case. The results are joint with M. Disconzi and A. Tuffaha. |
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Last update: Nov 21, 2018 | |