Friday Apr 5 |
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| 4:30-5:20 Caltech Shaoming Guo (UW Madison) Polynomial Roth theorems in Salem sets Let $P(t)$ be a polynomial of one real variable. I will report a result on searching for patterns of the form $(x, x+t, x+P(t))$ within Salem sets, whose Hausdorff dimension is sufficiently close to one. Joint work with Fraser and Pramanik. |
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| 5:30-6:20 Caltech Irina Holmes (Michigan State University) On an inequality for the dyadic square function In this talk we discuss the weak $(1, 1)$ sharp inequality for the dyadic square function. Specifically, we outline a new approach via Bellman functions, inspired by an older paper of Bollobas. The new approach involves two Bellman functions to tackle the same inequality, instead of the usual one. Many of the properties of the two Bellman functions are mirror images of one another, and their intertwined behavior yields the sharp inequality. Joint with Paata Ivanisvili and Alexander Volberg. |
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Friday Apr 19 |
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| 4:00-4:50 UCLA Hongwei Gao (UCLA) Head and tail speeds of mean curvature flow with forcing Let us consider a moving interface forced by its mean curvature and a positive periodic forcing term. We investigate the interplay between the mean curvature and the forcing term in the long run. It turns out that the large time behavior of the interface can be characterized by its head speed and tail speed, which depend continuously on its direction of propagation. If these two speeds are equal in all directions, homogenization appears. Otherwise, the interface exhibits long fingers in a certain direction. In the laminar case, these long fingers converge to traveling waves with different speeds. Moreover, there exists a stationary solution of the cylindrical type. |
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| 5:00-5:50 UCLA Paata Ivanisvili (UC Irvine) Weissler's conjecture on the Boolean cube The necessary and sufficient conditions will be obtained on $L^p$ to $L^q$ boundedness of the Hermite semigroup $e^{z\Delta}$ on the boolean cube of an arbitrary dimension equipped with uniform counting measure, where $z$ is a complex number and $1\leq p \leq q$. This solves an old open problem in complex hypercontractivity theory on the Hamming. Certain cases of the triples $(p,q,z)$ were characterized by Bonami (1970); Beckner (1975); and Weissler (1979). Several applications will be presented. Work in progress with Fedja Nazarov. |
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Friday May 3 |
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| 4:30-5:20 Caltech Jonathan Luk (Stanford) High frequency limits and Burnett's conjecture in general relativity It is known in the physics literature that "high-frequency weak limits" of solutions to the Einstein vacuum equations are not necessarily vacuum solutions, but may have a non-trivial stress-energy-momentum tensor, which can be viewed physically as ''effective matter fields'' arising from back-reaction of high frequency gravitational waves. Burnett conjectured nonetheless that any such a limit is isometric to a solution to the Einstein-massless Vlasov system. We prove that Burnett's conjecture is true under a symmetry assumption and a gauge condition. This is a joint work with Cécile Huneau. |
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| 5:30-6:20 Caltech Jameson Graber (Baylor University) The First-Order Planning Problem in Mean Field Games The planning problem, as introduced by P.-L. Lions, is a model for Nash games with a continuum of players in which the initial and final distributions of states are prescribed. In the first-order case, we have a clear analogy with optimal transport problems, which is reminiscent of the Benamou-Brenier formulation of the Monge-Kantorovitch problem. In this presentation we will give conditions under which existence of solutions is guaranteed for any absolutely continuous initial/final measures. Additionally, we will show how, unlike for classical optimal transport problems, the running costs imposed on the density variable result in extra regularity in both time and space. |
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Friday May 17 |
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| 4:00-4:50 UCLA Nets Katz (Caltech) On the Discretized sum-product theorem We discuss joint work with L. Guth and J. Zahl in which we adapt Garaev's approach to the finite field sum product problem to obtain explicit bounds in the discretized sum product theorem. |
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| 5:00-5:50 UCLA Palina Salanevich (UCLA) Random Vector Functional Link Neural Networks as Universal Approximators Single layer feedforward neural networks (SLFN) have been widely applied to solve problems such as classification and regression because of their universal approximation capability. At the same time, iterative methods usually used for training SLFN suffer from slow convergence, getting trapped in a local minimum and being sensitivity to the choice of parameters. Random Vector Functional Link Networks (RVFL) is a randomized version of SLFN. In RVFL, the weights from the input layer to hidden layer are selected at random from a suitable domain and kept fixed in the learning stage. This way, only output layers are optimized, which makes learning much easier and cheaper computationally. Igelnik and Pao proved that the RVFL network is a universal approximator for a continuous function on a bounded finite dimensional set. In this talk, we provide a non-asymptotic bound on the approximation error, depending on the number of nodes in the hidden layer, and discuss an extension of the Igelnik and Pao result to the case when data is assumed to lie on a lower dimensional manifold. |
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Friday May 31 |
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| 4:30-5:20 Caltech Rupert Frank (Caltech/LMU München) Functions of perturbed self-adjoint operators We consider the difference $f(H_1)−f(H_0)$ for self-adjoint operators $H_0$ and $H_1$ acting in a Hilbert space. We establish a new class of estimates for the operator norm and the Schatten class norms of this difference. Our estimates utilise ideas of scattering theory and involve conditions on $H_0$ and $H_1$ in terms of the Kato smoothness. They allow for a much wider class of functions $f$ (including some unbounded ones) than previously available results do. As an example we consider the case where $H_0=−\Delta$ and $H_1=−\Delta+V$ are the free and the perturbed Schrödinger operators in $L^2(\mathbb{R}^d)$, and $V$ is a real-valued short range potential. The talk is based on joint work with A. Pushnitski. |
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| 5:30-6:20 Caltech Sylvester Eriksson-Bique (UCLA) Constructing and uniformizing Loewner Carpets Carpets are metric spaces that are homeomorphic to the standard Sierpinski carpet. By a theorem of Whyburn, they have a natural topological characterization. Consequently, they arise in many contexts involving dynamics, self similarity or geometric group theory. In these contexts, the Loewner property would have many structural and geometric implications for the space. However, unfortunately, we do not know if the Loewner property would be satisfied, or even could be satisfied, in many of the applications of interest. I will discuss recent results on finding large families of Loewner carpets, and some properties they must enjoy, such as explicit and inexplicit uniformizations in the plane. |
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Last update: May 20, 2019 | |