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12:00pm - Zoom - Probability and Analysis Webinar Brett Wick - (Washington University in St. Louis) TBA |
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4:00pm - NS2 1201 - Differential Geometry Tin Yau Tsang - (UC Irvine) Scalar curvature and Riemannian polyhedra To characterize scalar curvature, Gromov proposed the dihedral rigidity conjecture which states that a positively curved polyhedron having dihedral angles less than those of a corresponding flat polyhedron should be isometric to a flat one. In this talk, we will discuss some recent progress on this conjecture and its connection with general relativity (ADM mass and quasilocal mass). |
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2:00pm to 3:00pm - 510R Rowland Hall - Combinatorics and Probability Mateo Diaz - (Caltech) Clustering a mixture of Gaussians with unknown covariance Clustering is a fundamental data scientific task with broad application. This talk investigates a simple clustering problem with data from a mixture of Gaussians that share a common but unknown, and potentially ill-conditioned, covariance matrix. We start by considering Gaussian mixtures with two equally-sized components and derive a Max-Cut integer program based on maximum likelihood estimation. We show its solutions achieve the optimal misclassification rate when the number of samples grows linearly in the dimension, up to a logarithmic factor. However, solving the Max-cut problem appears to be computationally intractable. To overcome this, we develop an efficient spectral algorithm that attains the optimal rate but requires a quadratic sample size. Although this sample complexity is worse than that of the Max-cut problem, we conjecture that no polynomial-time method can perform better. Furthermore, we present numerical and theoretical evidence that supports the existence of a statistical-computational gap. Finally, we generalize the Max-Cut program to a k-means program that handles multi-component mixtures with possibly unequal weights and has similar guarantees. |
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9:00am to 10:00am - Zoom - Inverse Problems Jorge Passamani Zubelli - (Khalifa University & IMPA) A Splitting Strategy for the Calibration of Jump-Diffusion Models |
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10:00am to 11:00am - https://uci.zoom.us/j/95053211230 - Number Theory Martin Cech - (Concordia University) Ratios conjecture and multiple Dirichlet series Conrey, Farmer and Zirnbauer formulated the ratios conjectures, which give asymptotic formulas for the ratios of products of shifted L-functions from some family. They have many corollaries to other problems in arithmetic statistics, such as the computation of various moments or the distribution of zeros in a family of L-functions. During the talk, we will show how to use multiple Dirichlet series to prove the conjectures in the family of real Dirichlet L-functions for some range of the shifts. The talk will be accessible even to those with little background in analytic number theory. |
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11:00am - zoom ID: 949 5980 5461. Password: the last four digits of the zoom ID in the reverse order - Harmonic Analysis Dmitriy Stolyarov - (St. Petersburg State University) On Maz’ya’s \Phi-inequalities. I will speak about an unusual way to correct the (invalid) endpoint case of the Hardy—Littlewood—Sobolev inequality. Usually the correction is done by imposing additional linear constraints on the function we apply the Riesz potential to. Being the gradient of another function is an example of such a constraint. The inequalities obtained this way are often called Bourgain—Brezis inequalities. In 2010, Maz’ya suggested another approach: instead of constraining the right hand side we should replace the L_p norm on the left with an expression \Phi, which alongside with having the same homogeneity properties as the L_p norm, possesses additional cancellations. He conjectured that if \Phi satisfies a natural necessary condition, then the modified Hardy—Littlewood—Sobolev inequality holds true. I will try to survey the proof of Maz’ya’s conjecture. Based on https://arxiv.org/abs/2109.08014 |
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3:00pm - RH 306 - Colloquium Gunther Uhlmann - (University of Washington) Seeing Through Space-Time The inverse problem we address is whether we can determine the structure of a region in space-time by measuring point light sources coming from the region. We can also observe gravitational waves since the LIGO detection in 2015. We will also consider inverse problems for nonlinear hyperbolic equations, including Einstein's equations, involving active measurements. |
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4:00pm - RH 306 - Colloquium Igor Krasovsky - (Imperial College) Quasiperiodicity and quasiintegrability We will discuss the critical almost Mathieu operator: Azbel/Hofstadter/Harper model of an electron on the square lattice in a magnetic field. When the commensurability parameter between the lattice and the magnetic field is irrational, the spectrum of the model is a zero-measure Cantor set and its Hausdorff dimension is not larger than 1/2. We will emphasize the significance of the two-dimensionality of the problem, which was used in recent work of the speaker with S. Jitomirskaya. We will also discuss some similarities with integrable two-dimensional statistical models: the Ising model and the dimer problem. |