Date/Time : 03 Apr 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Xuhua He, Chinese University of Hong Kong

An invertible matrix is called totally positive if all its minors are positive. In 1994, Lusztig developed the theory of total positivity for arbitrary split real reductive groups and their flag manifolds. He further generalized the theory to arbitrary Kac-Moody groups in 2019. The theory of total positivity has found important applications in different areas: cluster algebras, higher Teichmuller theory, the theory of amplituhedron in physics, etc.

In this talk, we will discuss some remarkable combinatorial, geometric and representation-theoretic aspects of total positivity. This talk is based on some recent works with Huanchen Bao.

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Date/Time : 21 Mar 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Jie Shen, Purdue University

Many complex nonlinear systems have intrinsic structures such as energy dissipation or conservation, and/or positivity/maximum principle preserving. It is desirable, sometimes necessary, to preserve these structures in a numerical scheme.

I will present some recent advances on using the scalar auxiliary variable (SAV) approach and Lagrange multiplier approach to develop highly efficient and accurate structure preserving schemes for a large class of complex nonlinear systems. These schemes can preserve energy dissipation/conservation as well as other global constraints and/or are positivity/bound preserving, only require solving decoupled linear equations with constant coefficients at each time step, and can achieve higher-order accuracy.

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Date/Time : 08 Mar 2023 16:00
Venue : S17 #04-05 (SR2)
Speaker : Jurgen Jost, Max Planck Institute for Mathematics in the Sciences

Hypergraphs naturally emerge as models in many fields, ranging from chemical reaction systems to collaboration networks. Their mathematical theory, however, was underdeveloped until recently. I shall introduce some new ideas and concepts that we have developed, describe some of the interesting new mathematical developments and show some novel phenomena in pattern formation.

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Date/Time : 03 Mar 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Jun-Muk Hwang, Institute for Basic Science, Korea

When an analytic equation has a formal power series solution, does it have a convergent power series solution?

It has been well-known that this question has an affirmative answer if we work in local setting. The question is much subtler if we work in global setting. A typical example is Grauert’s formal prinicple for compact complex submanifolds in complex manifolds. We give an introductory overview of the results on the formal principle for rational curves in complex manifolds.

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Date/Time : 17 Feb 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Gabor Domokos, Budapest University of Technology and Economics

If we approximate natural fragments by convex polyhedra and count the respective numbers for faces, vertices and edges then, in most cases, we find averages remarkably close to 6,8,12, the values corresponding to the cube. Not only can this observation be translated into a simple Lemma about hyperplane convex mosaics, we also find that general convex mosaics may serve as particularly fitting models for natural fragmentation patterns. In 2D, this approach offers a complete, global catalog of natural tilings, ranging from atomic lattices to the geometry of tectonic plates. In 3D we can also build a similar catalog which suggests that the majority of natural fragments should indeed produce cubic averages. To verify our geophysical claims, we vetted field data from over 4000 natural fragments against computer simulations and found not only good agreement for the aforementioned cuboid averages, but we could also reproduce full distributions for many geophysical shape descriptors with remarkable accuracy.

The appearance of the cube (albeit in an averaged sense) in this context may remind one of Plato’s theory of the Element Earth. I will briefly comment on some purely geometric aspects of this connection.

This is joint work with Doug Jerolmack (U. Pennsylvania), Ferenc Kun (U. Debrecen) and János Török (Budapest U. of Technology and Economics).

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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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