Date/Time : 17 May 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Susanne Brenner, Lousiana State University

Elliptic optimal control problems are infinite dimensional least-squares data fitting problems where the models are defined by second order elliptic boundary value problems. In the presence of pointwise constraints on the state variable, the traditional approach becomes more challenging due to the reduced regularity of the adjoint state. In this talk I will explain how to adopt finite element methods originally designed for fourth order elliptic boundary value problems to this class of optimal control problems and present a unified framework for their analysis.

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Date/Time : 15 May 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Jeong Han Kim, Korea Institute for Advanced Study

Majority dynamics on a graph G is a deterministic process such that every vertex updates its +/- 1-assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, O’Donnell, Tamuz and Tan conjectured that, in the Erdos-Renyi random graph G(n,p), the random initial +/- 1-assignment converges to a 99%-agreement with high probability whenever p = \omega(1/n). This conjecture was first confirmed for p > \lambda n^{-1/2} for a large constant \lambda by Fountoulakis, Kang and Makai. Although this result has been reproved recently by Tran and Vu and by Berkowitz and Devlin, it was unknown whether the conjecture holds for p < \lambda n^{-1/2} . We break this \Omega(n^{-1/2})-barrier by proving the conjecture for sparser random graphs G(n,p), where \lambda’ n^{-3/5} \log n < p < \lambda n^{-1/2} with a large constant \lambda’ > 0.

(Joint work with D. Chakraborti, J. Lee, and T. Tran)

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Date/Time : 12 May 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Chan Hon Fu Raymond, City University of Hong Kong

With the advances in sensor technology, data has become ubiquitous. To make sense of data we have to solve higher and higher dimensional problems that may seem intractable. However, many high-dimensional problems have solutions that live in low-dimensional spaces. Sparsity is a way to exploit the low-dimensional structure of the solutions to obtain feasible solutions for high-dimensional problems. In this talk, we will see how sparsity can be applied to facilitate the solutions of imaging problems, including magnetic resonance imaging, single-molecule localisation microscopy, and ground-based astronomy.

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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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