Date/Time : 25 Nov 2022 15:00
Venue : S17 #04-06 (SR1)
Speaker : Slawomir Solecki, Cornell University

One of the areas of interest of Descriptive Set Theory is dynamics of Polish groups, that is, groups carrying a group topology that is separable and completely metrizable. Such groups, like for example, the unitary group of the infinite dimensional Hilbert space or the homeomorphism group of the unit interval, are not, in general, locally compact. Therefore, in studying their dynamics, classical methods relying on Haar measure are not available. They can often be replaced by descriptive set theoretic tools.

I will describe the descriptive set theoretic background and indicate how this point of view lead to a recent answer to an old question in Ergodic Theory. The question lies within a long-established theme, going back to the work of Halmos and Rokhlin, of investigating generic measure preserving transformations.

To be more specific: The behavior of a measure preserving transformation, even a generic one, is highly non-uniform. In contrast to this observation, a different picture of a very uniform behavior of the closed group generated by a generic measure preserving transformation has emerged. This picture included substantial evidence that pointed to these groups being all topologically isomorphic to a single group, namely, $L^0$—the group of all Lebesgue measurable functions from $[0,1]$ to the circle. Glasner and Weiss asked if this was the case. I will sketch a negative answer to this question that rests on an analysis of unitary representations of the non-locally compact Polish group $L^0$.

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Date/Time : 09 Nov 2022 17:15
Venue : S17 #04-05 (SR2)
Speaker : Benjamin T Castle, Ben Gurion University of the Negev

Abstract: Motivated by recent progress toward Zilber’s Restricted Trichotomy Conjecture, we study reducts of the complex field up to interdefinability over parameters. Precisely, we will consider structures of the form (C, P_1,…,P_n), where C is set of complex numbers and the P_i are polynomial maps of potentially different arities.   Somewhat surprisingly, our main result states that almost all such structures (in a precise sense) are interdefinable. The proof uses a mix of tools from geometric stability theory, combinatorics, and algebraic geometry.

This is joint work with Chieu-Minh Tran.

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Date/Time : 11 Oct 2022 16:00
Venue : S17 #05-11 (SR5)
Speaker : Mahan Maharaj, Tata Institute of Fundamental Research

Abstract: We develop a new orbit equivalence framework for mating Fuchsian groups with polynomial maps, adding to the Sullivan dictionary between Kleinian groups and rational dynamics. A surprising fallout is that topological orbit equivalence rigidity fails for Fuchsian groups. This is joint work with Sabyasachi Mukherjee.

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Date/Time : 02 Sep 2022 16:00
Venue : S17 #04-06 (SR1)
Speaker : Balint Virag, University of Toronto

Abstract : In Euclidean geometry, bisectors are perpendicular lines. What happens in a random geometry? Surprisingly, many features are not sensitive to which model we use. There is a universal random geometry and its bisectors help answer some open questions in particle systems.

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

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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).

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Date/Time : 08 Apr 2022 09:00
Venue : via Zoom
Speaker : Konstantin Tikhomirov, Georgia Institute of Technology

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https://nus-sg.zoom.us/j/6124909748?pwd=TDFJYitzZ3ZSeUZUcTBEWnhTZWtSUT09

Meeting ID: 612 490 9748
Passcode: 007151

Abstract: The Gaussian Elimination with Partial Pivoting (GEPP) is a classical algorithm for solving systems of linear equations. Although the worst-case loss of precision in GEPP due to roundoff errors is known to be very significant, empirical evidence suggests that for a typical coefficient matrix, GEPP is numerically stable. Until recently, this phenomenon has received little theoretical justification. We make a progress on this problem by showing that, given the random $n\times n$ standard Gaussian coefficient matrix, the {\it growth factor} of the Gaussian Elimination with Partial Pivoting is at most polynomially large in $n$ with probability close to one. This implies that with high probability the GEPP results only in logarithmic (in $n$) loss of bits of precision, which improves an earlier estimate of Sankar of order $O(\log^2 n)$. This is joint work with Han Huang.

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Date/Time : 11 Mar 2022 09:00
Venue : via Zoom
Speaker : Laura DeMarco, Harvard University

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https://nus-sg.zoom.us/j/86723583031?pwd=WDBVNVpqbU8xWWI3amhKMTdPdk45Zz09

Meeting ID: 867 2358 3031
Passcode: 232023

Abstract: The periodic orbits and their structure are a fundamental feature of a dynamical system. In an algebraic setting, where the system is defined by polynomials, we can use tools from algebraic or arithmetic geometry to study these orbits. As the past year has seen major breakthroughs in complex and arithmetic geometry — for example in the uniform versions of the Mordell or Manin-Mumford Conjectures proved recently by Dimitrov-Gao-Habegger, Kuhne, Yuan and others, studying families of abelian varieties — I will take this opportunity to describe some parallel questions and progress in complex and arithmetic dynamical systems.

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