Friday Jan 11 |
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| 4:30-5:20 Caltech Ilia Binder (U Toronto) Computability questions in planar conformal geometry I will give an overview of the recent research on the computational properties of conformal maps. I will discuss the computability and complexity of computing a conformal map onto a given domain, both for the interior and boundary points. I will also talk about the computability of harmonic measure. Based on joint projects with M. Braverman, C. Rojas, and M. Yampolsky. |
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| 5:30-6:20 Caltech Terence Tao (UCLA) Embedding the Heisenberg group into a Euclidean space with bounded distortion From the work of Pansu and Semmes it is known that the Heisenberg group (with the Carnot-Caratheodory metric) cannot embed into Euclidean space (or even Hilbert space) in a bilipschitz fashion. However if one "snowflakes" the metric then this becomes possible thanks to work of Assouad. There is a lower bound on the distortion in doing so due to Austin, Naor, and Tessera; we show that this bound can be attained while embedding into a bounded dimensional Euclidean space, answering a question of Naor and Neiman in the negative. Our argument uses an iteration inspired by the Nash-Moser iteration scheme. |
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Friday Jan 25 |
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| 4:00-4:50 UCLA Ben Krause (Caltech) Euclidean Ramsey Theory via the Uncertainty Principle -- an Argument of Bourgain In this talk, I will discuss an approach of Bourgain used to handle problems in Euclidean Ramsey theory, and survey some results in this direction. |
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| 5:00-5:50 UCLA Alexei Poltoratski (Texas A&M) Completeness of exponential functions: Beurling-Malliavin and Type problems My talk will focus on two old problems on completeness of trigonometric polynomials in $L^2$-spaces, the Beurling-Malliavin (BM) and Type problems. I will discuss the classical solution to the BM problem and a recent type formula, along with connections and applications. |
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Friday Feb 8 |
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| 4:30-5:20 Caltech Gennady Uraltsev (Cornell) Uniform Bounds for the bilinear Hilbert transform The bilinear Hilbert transform is a prototypical modulation invariant multi-linear singular operator \begin{equation*} \text{BHT}_{\alpha}(f_{1},f_{2})(x):= \int_{\mathbb{R}} \int_{\mathbb{R}}\widehat{f_1}(\xi_{1})\widehat{f_2}(\xi_{2}) \mathbf{1}_{[0,+\infty)}(\xi_{1}-\alpha\xi_{2}) e^{2\pi i (\xi_{1}+\xi_{2})x}d \xi_{1}d \xi_{2}. \end{equation*} It arises in many contexts including Cauchy integrals along Lipschitz curves and in the study of Calderón commutators. BHT satisfies the bounds \begin{equation*} \| \text{BHT}_{\alpha}(f_{1},f_{2})(x) \|_{L^{p_{3}'}(\mathbb{R})}\leq C_{p_{1},p_{2},\alpha} \| f_{1} \|_{L^{p_{1}}(\mathbb{R})}\| f_{2} \|_{L^{p_{2}}(\mathbb{R})} \end{equation*} for any $p_{1,2,3}\in(1,+\infty)$ with $\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}=1$. A longstanding open problem is that of bounds for the bilinear Hilbert transform uniform in the parameter $\alpha$. The dyadic analog of this problem has been solved by Oberlin and Thiele (2010). We will give a general overview of the topic and of the main ideas involved in the proof of the above bound with a constant $C_{p_{1},p_{2}}$ independent of $\alpha$. We will particularly emphasize the role of the framework of outer measure $L^{p}$ spaces of Do and Thiele in proving the result. |
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| 5:30-6:20 Caltech Asgar Jamneshan (UCLA) An introduction to Boolean-valued analysis The aim of this talk is to provide an introduction to Boolean-valued analysis which refers to the application of Boolean-valued models to analysis. We focus on Boolean-valued models that are built from a sigma-finite measure space. We will discuss how basic Boolean-valued structures such as real numbers, vector spaces, and compact spaces are interpreted in standard analysis, and how certain objects in standard analysis and probability such as function spaces, Hilbert bundles, and conditional expectations can be translated to a Boolean-valued context. We build a bridge between classical functional analysis and Boolean-valued functional analysis via vector duality. As one major application, we present the use of Boolean-valued models to systematically preserve measurability and conditioning. No knowledge of mathematical logic is required. |
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Friday Feb 22 |
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| Linde Hall Inaugural Math Symposium No seminar. |
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Friday Mar 8 |
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| 4:00-4:50 UCLA David Beltran (BCAM) Local smoothing estimates for Fourier Integral Operators The sharp fixed-time Sobolev estimates for Fourier Integral Operators (and therefore solutions to wave equations in Euclidean space or compact manifolds) were established by Seeger, Sogge and Stein in the early 90s. Shortly after, Sogge observed that a local average in time leads to a regularity improvement with respect to the sharp fixed-time estimates. Establishing variable-coefficient counterparts of the Bourgain-Demeter decoupling inequalities, we improve the previous best known local smoothing estimates for FIOs. Moreover, we show that our results are sharp in both the Lebesgue and regularity exponent (up to the endpoint) in odd dimensions. This is joint work with Jonathan Hickman and Christopher D. Sogge. |
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| 5:00-5:50 UCLA Ciprian Demeter (IU) Refinements of Vinogradov's Mean Value Theorem We prove new exponential sum estimates on small balls, using refined Kakeya-type input and two-step decouplings. Joint with Larry Guth and Hong Wang. |
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Last update: Mar 6, 2019 | |