Date/Time : 26 Jul 2022 15:00
Venue : S17 #04-06 (SR1)
Speaker : Antonio De Rosa, University of Maryland

Abstract: We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3-dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard [Invent. Math.,1983] in dimension 3. Joint work with Guido De Philippis.

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Date/Time : 19 Jul 2022 16:00
Venue : via Zoom
Speaker : Piyush Srivastava, Tata Institute of Fundamental Research

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https://nus-sg.zoom.us/j/83199111482?pwd=L09Yd054dDNTbDBpRkNtc0tmQnk5Zz09

Meeting ID: 831 9911 1482
Passcode: 569991

This series of lectures will survey various applications to algorithms of a collection of techniques that are broadly referred to as “geometry of polynomials”. Specific examples include:

1) Real stable polynomials and constructions of Ramanujan expanders
2) Lee-Yang theory and deterministic “approximate counting” algorithms”
3) Lorentzian or “strongly log concave” polynomials and applications to sampling.

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Date/Time : 15 Jul 2022 16:00
Venue : S17 #04-04 (SR3)
Speaker : Defeng Sun, The Hong Kong Polytechnic University

Abstract: It has been widely recognized that the structured sparsity of the optimal solutions is an intrinsic property for large-scale optimization problems arising from modern applications in the big data era. In this talk, we shall first illustrate the structured sparsity of the solutions via some popular machine learning models. In particular, we shall show that the solution of the convex clustering model can be highly structurally sparse even if the solution itself is fully dense. We shall then introduce a dual semismooth Newton based proximal point algorithm (PPDNA) and explain why it can be much more efficient than the first-order methods for solving a class of large-scale optimization problems arising from machine learning. The key point is to adaptively make use of the second-order sparsity of the solutions in addition to the data sparsity so that, at each iteration, the computational costs of the second-order methods can be comparable or even lower than those of the first-order methods. Equipped with the PPDNA, we shall then introduce some adaptive sieving methodologies to generate solution paths for large-scale optimization problems with structured sparsity of particular importance in applications. In the last part of the talk, we shall illustrate the high efficiency of our approach with extensive numerical results on several important models including convex clustering, lasso, and exclusive lasso.

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Date/Time : 12 Jul 2022 16:00
Venue : via Zoom
Speaker : Piyush Srivastava, Tata Institute of Fundamental Research

Join Zoom Meeting
https://nus-sg.zoom.us/j/83199111482?pwd=L09Yd054dDNTbDBpRkNtc0tmQnk5Zz09

Meeting ID: 831 9911 1482
Passcode: 569991

This series of lectures will survey various applications to algorithms of a collection of techniques that are broadly referred to as “geometry of polynomials”. Specific examples include:

1) Real stable polynomials and constructions of Ramanujan expanders
2) Lee-Yang theory and deterministic “approximate counting” algorithms”
3) Lorentzian or “strongly log concave” polynomials and applications to sampling.

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Date/Time : 04 Jul 2022 16:00
Venue : S17 #06-11 (SR6)
Speaker : Sang-hyun Kim, Korea Institute of Advanced Study

Abstract:  We prove that for each finite index subgroup H of the mapping class group of a closed hyperbolic surface, and for each real number r>1 there does not exist a faithful C^r-action (in Hölder’s sense) of H on a circle. For this, we partially determine the optimal regularity of faithful actions by right-angled Artin groups on a circle. (Joint with Thomas Koberda and Cristobal Rivas).

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Date/Time : 30 Jun 2022 16:00
Venue : S17 #04-06 (SR1)
Speaker : Shirshendu Ganguly, University of California, Berkeley

Abstract: The Kardar-Parisi-Zhang (KPZ) universality class refers to a broad family of models of random growth which are believed to exhibit certain common features, such as universal scaling exponents and limiting distributions.

Planar first passage percolation models (FPP), where one considers a distortion of the planar Euclidean metric by random noise are canonical examples of random planar metrics expected to lie in this class. A key observable of interest is the geodesic, the (random) shortest path between specified endpoints.

These models can also be viewed as examples of disordered systems admitting complex energy landscapes. In this formulation, the geodesic is the ground state, the configuration with the lowest energy, lying at the base of the deepest valley.

In this talk we will report recent progress in understanding various geometric features of such energy landscapes for certain variants of FPP known as last passage percolation (LPP) admitting certain key algebraic features, and their manifestations in intricate fractal behavior as well as sensitivity to dynamical perturbations of the underlying noise (noise-sensitivity and chaos).

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Date/Time : 20 Jun 2022 14:00
Venue : S17 #04-06 (SR1)
Speaker : Yap Jit Wu, Harvard University

Abstract: For a rational function on P^1, it is known that the set of periodic points of period n converges weakly to a measure supported on its Julia set, known as the equilibrium measure, as n tends to infinity. In the 1980s, Szpiro-Ullmo-Zhang proved an arithmetic version of this for abelian varieties, where one is allowed to take just the Galois orbit of a torsion point, instead of all torsion points of order n. Since then, much more general arithmetic equidistribution theorems have been proven, along with the discovery of various applications to number theory.

In this talk, we will explain how having a stronger notion called logarithmic equidistribution will allow us to prove finiteness of S-integral points in certain situations and state some results in this direction.

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Date/Time : 17 Jun 2022 16:30
Venue : S17 #04-06 (SR1)
Speaker : John Harlim , Penn State University

Abstract:  I will discuss recent efforts in using the Diffusion Maps (DM) algorithm to solve elliptic PDEs on unknown manifolds using point clouds data. The key idea rests on the fact that away from the boundary, the second-order elliptic differential operators can be approximated by integral operators defined with appropriate Gaussian kernels. On manifolds with boundary, however, such an approximation is only valid for functions that satisfy the Neumann boundary condition. Motivated by the classical ghost-point correction in the finite-difference method for solving Neumann problems, we extend the diffusion maps algorithm with ghost points such that it is a consistent estimator in the pointwise sense even near the boundary. Applying the proposed algorithm, which we called the Ghost Points Diffusion Maps (GPDM), to solve the well-posed elliptic PDEs with Dirichlet, Neumann, or Robin boundary conditions, we establish the convergence of the approximate solution under appropriate smoothness assumptions. I will also discuss a neural-network regression solution on an algebraic equation induced by the DM/GPDM discretization. In addition to theoretical convergence for a wide enough neural network model, we numerically found that the proposed solver avoids the classical large matrix inversion problem. I will also discuss the application of DM/GPDM on Bayesian elliptic inverse problems. If time permits, I will also present symmetric Graph-Laplacian matrices that spectrally converge to the Laplace-Beltrami operator defined on compact manifolds with homogeneous Neumann and Dirichlet boundary conditions.

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Date/Time : 16 Jun 2022 11:00
Venue : S17 #04-06 (SR1)
Speaker : Dinesh Thakur, University of Rochester

Abstract: We will explain some basic congruences of number theory, questions, conjectures and results about them and connections with Zeta values.

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