Date/Time : 24 Aug 2023 14:00
Venue : S17 #04-06 (SR1) (Colloquium (joint with IMS))
Speaker : Yujiro Kawamata, University of Tokyo

We will describe infinitesimal deformations of complex manifolds to the direction of something having possibly non-commutative structure sheaves by using Hochschild cohomology. We will also describe global non-commutative deformations of some surfaces.

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Date/Time : 08 Aug 2023 15:00
Venue : S17 #05-12 (SR4)
Speaker : Barbara Wohlmuth, Technical University of Munich

In this talk, we discuss several aspects of time‐fractional differential operators. The non‐local nature of the operator is challenging from a theoretical and algorithmic point of view but allows us to account for memory effects in the mathematical model. We consider a wide class of time‐fractional PDE models, including gradient flow, viscoelasticity, random field generators, and polymeric fluids. Of special interest is the equivalence of the time fractional gradient flow model with an integer order system in a higher dimension. Exploiting this equivalence allows us to construct efficient algorithms for time fractional PDEs. Replacing the kernel by a finite sum of exponentials transforms the non‐local PDE into an integer‐order system for the modes. To balance the discretization error in time and space with the kernel approximation error, the number of modes has to grow logarithmically. However, we observe numerically that the number of modes is very moderate if the finite sum of exponentials is based on a rational approximation of the kernel function. In this talk, we present theoretical existence results, algorithmic aspects and show several applications in which fractional operators play an important role. Numerical examples illustrate the large flexibility of the proposed techniques.

Joint work with Marvin Fritz (RICAM, Austria), Brendan Keith (Brown University, USA), Ustim Khristenko (Safran, France) and Endre Süli (University of Oxford, UK).

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Date/Time : 27 Jul 2023 15:00
Venue : S17-04-06 (SR 1)
Speaker : Weinan E, AISI and Peking University

For many years, the lack of good algorithms has severely limited our ability to conduct scientific research. At its heart, the difficulty comes from the notorious “curse of dimensionality” problem. Deep learning is exactly the kind of tool needed to address this problem. In the last few years, we have seen a tremendous amount of scientific progress made as a result of the AI revolution, both in our ability to make use of the fundamental principles of physics and in our ability to make use of experimental data.

In this talk, I will analyze the origin of the AI for Science revolution, review some of the major progresses made so far, and discuss how it will impact the way we do scientific research.

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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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