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).

Add to calendar :

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).

Add to calendar :

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.

Add to calendar :

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.

Add to calendar :

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.

Add to calendar :

Date/Time : 09 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 arithmetic derivatives.

Add to calendar :

Date/Time : 01 Jun 2022 14:00
Venue : S17 #04-06 (SR1)
Speaker : Edmund Karasiewicz, University of Utah

Abstract: The Fourier coefficients of automorphic forms are important objects of study due to their connection to L-functions. In the adelic framework, constructions of L-functions involving Fourier coefficients (e.g. Langland-Shahidi and Rankin-Selberg methods) naturally lead to spherical Whittaker functions on p-adic groups. Thus we would like to study these spherical Whittaker functions to better understand L-functions. Casselman-Shalika determined a formula for the spherical Whittaker functions, and basic algebraic manipulations reveal that their formula can be succinctly expressed in terms of characters of the Langlands dual group. We will describe a new approach to the Casselman-Shalika formula that provides a conceptual explanation of the appearance of characters. This is joint work with Nadya Gurevich.

Add to calendar :

Date/Time : 27 May 2022 09:00
Venue : via Zoom
Speaker : Alan Chang, Princeton University

Join Zoom Meeting
https://nus-sg.zoom.us/j/85408531758?pwd=VDVXRXVFQ09RNjQ3QWFNeC9wUDlLdz09

Meeting ID: 854 0853 1758
Passcode: 633851

Abstract: The Kakeya needle problem asks the following: What is the smallest area in the plane in which a unit line segment can be rotated 180 degrees? This question, which can be solved with elementary geometry, has surprising connections to many other topics (e.g., harmonic analysis, PDEs, analytic number theory, theoretical computer science). In this talk, we will give a survey of this classical problem, and then we will discuss a variant where the line segment has been replaced by a C^1 curve (or more generally, a 1-rectifiable set). This is joint work with Marianna Csörnyei.

Add to calendar :

Date/Time : 13 May 2022 16:00
Venue : via Zoom
Speaker : Jan S Hesthaven, EPFL

Join Zoom Meeting
https://nus-sg.zoom.us/j/82587724360?pwd=dGJ4RTZVb25rK1dGRFI5Y0FyTUJiUT09

Meeting ID: 825 8772 4360
Passcode: 015672

Abstract: The development of reduced order models for complex applications promises rapid and accurate evaluation of the output of complex models under parameterized variation with applications to problems which require many evaluations, such as in optimization, control, uncertainty quantification and applications where near real-time response is needed. However, many challenges remain to secure the flexibility, robustness, and efficiency needed for general large scale applications, in particular for nonlinear and/or time-dependent problems.

After a brief introduction to reduced order models, we discuss the development of methods which seek to conserve chosen invariants for nonlinear time-dependent problems. We develop structure-preserving reduced basis methods for a broad class of Hamiltonian dynamical systems, including canonical problems and port-Hamiltonian problems, before considering the more complex situation of Hamiltonian problems endowed with a general Poisson manifold structure which encodes the physical properties, symmetries and conservation laws of the dynamics.

Time permitting we also discuss the extension of structure preserving models within a framework for nonlinear reduced order models in which a local basis allows to maintain a small basis even for problems with a slowly decaying Kolmogorov n-width such a transport dominated problems. We shall demonstrate the efficiency of such techniques for nonlinear transport dominated problems, including problems in kinetic plasma physics,

This work is done in collaboration with B, Maboudi (DTU), C. Pagliantini (TU/e), N. Ripamonti (EPFL).

Add to calendar :