Date/Time : 22 Nov 2021 09:00
Venue : via Zoom
Speaker : Chan Swee Hong, University of California, Los Angeles

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https://nus-sg.zoom.us/j/82339078564?pwd=b1pEQ0xyVlBhT0ZGOExkYkJvSXNGdz09

Meeting ID: 823 3907 8564
Passcode: 214318

Abstract:  The study of log-concave inequalities for combinatorial objects have seen much progress in recent years. One such progress is the solution to the strongest form of Mason’s conjecture (independently by Anari et. al. and Brándën-Huh). In the case of graphs, this says that the sequence $f_k$ of the number of forests with $k$ edges, form an ultra log-concave sequence.

In this talk, we discuss an improved version of these results, proved by using a new tool called the combinatorial atlas method. This is a joint work with Igor Pak.

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Date/Time : 21 Apr 2021 08:00
Venue : via Zoom
Speaker : Yudong Chen, Cornell University

Join Zoom Meeting https://nus-sg.zoom.us/j/83859343651?pwd=Yy9pTWRiZ1N1SGs5dVVNbHpaRTBaQT09 Meeting ID: 838 5934 3651 Password: 685719 Estimation of mixture models is a fundamental problems in machine learning and statistics. This problem is inherently nonconvex, and may possess spurious local minimizers that are not globally optimal, both in theory and in practice. In this talk, we will present three positive results for mixture problems in the lens of nonconvex optimization. For a mixture of two components, we show that all local minima are globally optimal, and EM converges to a global optimum from random initialization. This result applies to a broad class of log-concave mixture problems including mixtures of Gaussians and linear regressions. For more than two components, spurious local minima provably exist. One such local minimizer puts multiple centers at one true component, and another center in the middle of multiple true components. We prove that this is essentially the only type of spurious local minima under certain conditions. Consequently, standard algorithms such as gradient descent and EM are guaranteed to converge to a solution that partially identifies the true model. Leveraging the above structural results, we will discuss several algorithmic ideas for avoiding local minima and finding the global optimum. Over-parametrization, in which one over-specifies the number of components, stands out as a promising approach. We characterize the benefit and price of over-parametrization in terms of its computational and statistical efficiency.

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Date/Time : 18 Jan 2021 09:00
Venue : via Zoom
Speaker : Sourav Sarkar, University of Toronto

Join Zoom Meeting https://nus-sg.zoom.us/j/85233351827?pwd=U2dESEhORVN4eHJwZlhNZEt0MEE3Zz09 Meeting ID: 852 3335 1827 Password: 149940 Abstract: Universality in disordered systems has always played a central role in the direction of research in Probability and Mathematical Physics, a classical example being the Gaussian universality class (the central limit theorem). In this talk, I will describe a different universality class for random growth models, called the KPZ universality class. Since Kardar, Parisi and Zhang introduced the KPZ equation in their seminal paper in 1986, the equation has made appearances everywhere from bacterial growth, fire front, coffee stain to the top edge of a randomized game of Tetris; and this field has become a subject of intense research interest in Mathematics and Physics for the last 15 to 20 years. The random growth processes that are expected to have the same scaling and asymptotic fluctuations as the KPZ equation and converge to the universal limiting object called the KPZ fixed point, are said to lie in the KPZ universality class, though this KPZ universality conjecture has been rigorously proved for only a handful of models till now. Here, I will talk about some universal geometric properties of the KPZ fixed point and show that the KPZ equation and exclusion processes converge to the KPZ fixed point under the 1:2:3 scaling, establishing the KPZ universality conjecture for these models, which were long-standing open problems in this field.

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Date/Time : 08 Dec 2020 16:00
Venue : via Zoom
Speaker : Zhang Shuangjian Kelvin, Ecole Normale Superieure Paris

Join Zoom Meeting https://nus-sg.zoom.us/j/88422472817?pwd=UmFmckRDR2drSVdCZzExR0tGQVVkZz09 Meeting ID: 884 2247 2817 Password: 591367 Abstract:  A monopolist wishes to maximize her profits by finding an optimal price policy. After she announces a menu of products and prices, each agent will choose to buy that product which maximizes his own utility, if positive. The monopolist’s profits are the sum of the net earnings produced by each product sold. These are determined by the costs of production and the distribution of products sold, which in turn are based on the distribution of anonymous agents and the choices they make in response to the monopolist’s price menu. In this talk, I will first review the history of this bi-level maximization problem in microeconomics, including the celebrated work of two Nobel Laureates and one Fields Medalist as well as recent progress on existence, uniqueness, concavity, regularity, and numerical results. When present, concavity makes the problem more amenable to computational and theoretical analysis; it is key to obtaining uniqueness and stability results for the monopolist’s strategy in particular. The beautiful connection to the regularity theory in Optimal Transport will also be addressed. Then I will list a few open problems and discuss some current progress. Part of this talk represents joint work with Professor Robert J. McCann and Professor Guillaume Carlier.

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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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