Date/Time : 10 Feb 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Håkan Hedenmalm, KTH Royal Institute of Technology

Fourier series is a classical method to expand periodic functions on the line, with roots in the theory of heat conduction. If we want to expand non-periodic functions on the line, we are told to use the Fourier transform instead and accept that the result is a distribution. In 2011, Hedenmalm and Montes-Rodriguez suggested an alternative expansion. It now turns out that this expansion exists uniquely and that it connects with the Fourier interpolation of Radchenko and Viazovska (2019). We will explain how the hyperbolic Fourier series is very natural and corresponds to a “low temperature” expansion where automorphic forms and theta functions are examples from a high temperature regime where nonuniqueness holds.

This reports on joint work (with Viazovska, Radchenko, Bakan, Montes-Rodriguez).

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Date/Time : 20 Jan 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Sergiu Klainerman, Princeton University

The question whether black holes are real can be approached mathematically by addressing basic issues concerning their rigidity, stability and how they form in the first place. I will review these and focus on recent results concerning the nonlinear stability of slowly rotating Kerr black holes.

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Date/Time : 09 Jan 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Jonathan Eckstein, Rutgers University

This talk describes the solution of convex optimization problems that include uncertainty modeled by a finite but potentially very large multi-stage scenario tree.

In 1991, Rockafellar and Wets proposed the progressive hedging (PH) algorithm to solve such problems. This method has some advantages over other standard methods such as Benders decomposition, especially for problems with large numbers of decision stages. The talk will open by showing that PH is an application of the Alternating Direction Method of Multipliers (ADMM), and thus of the “Douglas-Rachford” operator splitting method. The equivalence of PH to the ADMM has long been known but not explicitly published.

Next, the talk will apply a different, more recent operator splitting method called “projective splitting” to the same problem. The resulting method is called “asynchronous projective hedging” (APH). Unlike most decomposition methods, it does not need to solve every subproblem at every iteration; instead, each iteration may solve just a single subproblem or a small subset of the available subproblems.

Finally, the talk will describe work integrating the APH algorithm into mpi-sppy, a Python package for modeling and distributed parallel solution of stochastic programming problems. Mpi-sppy uses the Pyomo Python-based optimization modeling sytem. Our experience includes using up to 2,400 processor cores to solve 2-stage and 4-stage test problem instances with as many as 1,000,000 scenarios.

Portions of the work described in this talk are joint with Patrick Combettes (North Carolina State University), Jean-Paul Watson (Lawrence Livermore National Laboratory, USA), and David Woodruff (University of California, Davis).

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Date/Time : 06 Jan 2023 14:00
Venue : LT34 (S17 Level 3)
Speaker : George Lusztig, MIT

According to Chevalley, semisimple groups can be defined over any field. More recently they were defined over some structures called semifields (which include the set of strictly positive real numbers under the usual addition and multiplication). It turns out that the theory over semifields can be used to get a better understanding of the theory over fields.

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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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