Date/Time : 07 Dec 2020 09:00
Venue : via Zoom
Speaker : Kwan Matthew, Stanford University

Join Zoom Meeting https://nus-sg.zoom.us/j/86360904887?pwd=MEFqS1MwcHRTQ0djTHhXUHFFaG43Zz09 Meeting ID: 863 6090 4887 Password: 557853 Abstract: Anti-concentration inequalities provide limits on the extent to which random variables can be concentrated: for example, they commonly give uniform upper bounds on the probability that a random variable takes any particular value. In this talk I’ll discuss some of the many connections between anti-concentration and combinatorics, initially focusing on applications to Ramsey graphs but also touching on a few other topics such as the polynomial Littlewood-Offord problem and permanents of random matrices.

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Date/Time : 01 Dec 2020 09:00
Venue : via Zoom
Speaker : Sun Zheng, Ohio State University

Join Zoom Meeting https://nus-sg.zoom.us/j/87275619715?pwd=WVJrYXVyVVFZbk5meXhna3kycTJEdz09 Meeting ID: 872 7561 9715 Password: 617596 In scientific and engineering computing, we usually encounter numerical discretizations of time-dependent equation systems associated with conserved or decreasing energy. The strong stability of a time stepping method refers to its ability to preserve the monotone energy-decay law at the discrete level, which may provide improved robustness for simulations. In this talk, we present a unified study on strong stability of explicit Runge-Kutta methods for linear semi-negative problems. In particular, we prove a both necessary and sufficient condition for strong stability of arbitrary odd-order methods.

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Date/Time : 25 Nov 2020 09:00
Venue : via Zoom
Speaker : Tong Jiajun, University of California, Los Angeles

Join Zoom Meeting https://nus-sg.zoom.us/j/87962970385?pwd=QnRENCtzSW55VXArejBpWUtXcmlyUT09 Meeting ID: 879 6297 0385 Password: 808752 Abstract:  We study a two-phase tumor growth model in two space dimensions, where proliferation of the tumor cells leads to expansion of the tumor domain and migration of surrounding normal tissues into the exterior vacuum. Instead of considering evolution of cell densities, our focus will be on the dynamics of two moving interfaces in the model that separate the tumor, the normal tissue, and the exterior vacuum. We will discuss well-posedness results on the interface evolution starting from a nearly radial initial configuration, under the assumption that the tumor has lower mobility than the normal tissue, which is in line with the well-known Saffman-Taylor condition in viscous fingering. We will briefly discuss possible ill-posedness scenarios when these conditions are violated, as well as implications of our results on further analysis of the tumor growth models. This is joint work with Inwon Kim.

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Date/Time : 23 Nov 2020 09:00
Venue : via Zoom
Speaker : Yao Yao, Georgia Institute of Technology

Join Zoom Meeting https://nus-sg.zoom.us/j/81762064402?pwd=Z250ZmtKTFVhUDNqTzFlN0pvdldoQT09 Meeting ID: 817 6206 4402 Password: 845588 Abstract:  For any nonnegative density f and radially decreasing interaction potential W, the celebrated Riesz rearrangement inequality shows the interaction energy E[f] = \int f(x)f(y)W(x-y) dxdy satisfies E[f] <= E[f^*], where f^* is the radially decreasing rearrangement of f. It is a natural question to look for a quantitative version of this inequality: if its two sides almost agree, how close must f be to a translation of f^*? Previously the stability estimate was only known for characteristic functions. I will talk about a recent work with Xukai Yan, where we found a simple proof of stability estimates for general densities. I will also discuss its connection with some nonlocal PDEs such as the aggregation-diffusion equation.

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Date/Time : 19 Nov 2020 09:00
Venue : via Zoom
Speaker : Shen Junliang, MIT

Join Zoom Meeting https://nus-sg.zoom.us/j/88455891763?pwd=Sm5ISjNHSnpDVDMzYkFVd2ZBTit2QT09 Meeting ID: 884 5589 1763 Password: 869110 Abstract:  The P=W conjecture by de Cataldo, Hausel, and Migliorini suggests a surprising connection between the topology of Hitchin systems and Hodge theory of character varieties. In this talk, we will focus on interactions between topology of Lagrangian fibrations and Hodge theory in general hyper-Kaehler geometries. Such connections shed new light on both the P=W conjecture for Hitchin systems and the Lagrangian base conjecture for compact hyper-Kähler manifolds.

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Date/Time : 18 Nov 2020 16:00
Venue : via Zoom
Speaker : Yu Yue Tony, Universit Paris-Saclay (Paris-Sud)

Join Zoom Meeting https://nus-sg.zoom.us/j/81669932092?pwd=WSt6V01ESFFpY0dVRnhVdEt3cUIrQT09 Meeting ID: 816 6993 2092 Password: 287641 Abstract:  Mirror symmetry is one of the most mysterious dualities in mathematics. My research explores a new approach to mirror symmetry, via non-archimedean geometry. I will give an overview of the recent progress in this direction. I will talk about non-archimedean enumerative geometry, the Frobenius structure conjecture, and derived non-archimedean geometry. I will also describe an application towards cluster algebras, and another application towards the moduli space of Calabi-Yau pairs.

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Date/Time : 17 Nov 2020 09:00
Venue : via Zoom
Speaker : Huang Hao, Emory University

Join Zoom Meeting https://nus-sg.zoom.us/j/86270578593?pwd=cTYreG1hQlExWHJERStZN0hUWjNWQT09 Meeting ID: 862 7057 8593 Password: 313348 Abstract:  The biclique partition number bp(G) is the minimum number of complete bipartite graphs needed to partition the edges of a graph G. Its study was initiated in 1972 by Graham and Pollak, who showed that the complete graph with n vertices has biclique partition number exactly n-1. Since then, this important graph parameter has found many connections with problems arising from discrete geometry, communication complexity and network addressing. In this talk we will explore some of these connections and discuss counterexamples to two decade-old conjectures: one of Alon, Saks, and Seymour in 1991 saying the biclique partition number is strictly greater than the chromatic number; and another one of Erdos from the ’80s on random graphs. At the end of this talk, we will also outline the significance of studying its generalization for hypergraphs.

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Date/Time : 11 Nov 2020 09:00
Venue : via Zoom
Speaker : Yu Hui, Columbia University

Join Zoom Meeting https://nus-sg.zoom.us/j/85408305509?pwd=OHpHQkdtUSt3YzdLN1BTVkxOeTRXQT09 Meeting ID: 854 0830 5509 Password: 843914 Abstract:  The N-membrane problem is the study of shapes of elastic membranes being pushed against each other. The main questions are the regularity of the functions modeling the membranes, and the regularity of the contact regions between consecutive membranes. These are classical questions in free boundary problems. However, very little is known when N is larger than 2. In this case, there are multiple free boundaries that cross each other, and most known techniques fail to apply. In this talk, we discuss, for general N, the optimal regularity of the solutions in arbitrary dimensions, and a classification of blow-up solutions in 2D. Then we focus on the regularity of the free boundaries when N=3. We also discuss how the techniques developed here can be applied to other problems involving multiple free boundaries. This talk is based on two recent joint works with Ovidiu Savin (Columbia University).

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Date/Time : 09 Nov 2020 09:00
Venue : via Zoom
Speaker : Yang Yunan, New York University

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https://nus-sg.zoom.us/j/83496316224?pwd=STdTeDhKQk9pTnlLVFREaWs5UzFkZz09 Meeting ID: 834 9631 6224 Password: 665418

Optimal transport has been one interesting topic of mathematical analysis since Monge (1781). The problem’s close connections with differential geometry and kinetic descriptions were discovered within the past century, and the seminal work of Kantorovich (1942) showed its power to solve real-world problems. Recently, we proposed the quadratic Wasserstein distance from optimal transport theory for inverse problems, tackling the classical least-squares method’s longstanding difficulties such as nonconvexity and noise sensitivity. The work was soon adopted in the oil industry. As we advance, we discover that the advantage of changing the data misfit is more general in a broader class of data-fitting problems by examing the preconditioning and “implicit” regularization effects of different mathematical metrics as the objective function in optimization, as the likelihood function in Bayesian inference, and as the measure of residual in numerical solution to PDEs.

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