Date/Time : 02 Dec 2021 13:00
Venue : via Zoom
Speaker : Yang Lei, Sichuan University

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https://nus-sg.zoom.us/j/89266647804?pwd=K2p0K2V3R1VMcEQxdjlKMFFFYlJEQT09

Meeting ID: 892 6664 7804
Passcode: 445251

Abstract: In this talk, we will prove the convergence part of Khintchine’s theorem on non-degenerate manifolds. This confirms a conjecture of Kleinbock and Margulis in 1998. Our approach uses geometric and dynamical ideas together with a new technique of `major and minor arcs’. In particular, we establish sharp upper bounds for the number of rational points of bounded height lying near `major arcs’ and give explicit exponentially small bounds for the measure of `minor arcs’. This is joint work with Victor Beresnevich.

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Date/Time : 30 Nov 2021 09:00
Venue : via Zoom
Speaker : Goh Jun Le, University of Wisconsin-Madison

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https://nus-sg.zoom.us/j/83885192152?pwd=a2RDZTFvalVMM3VFT3lGazhyV2doQT09

Meeting ID: 838 8519 2152
Passcode: 769950

Abstract: Effective mathematics seeks to measure the algorithmic content of mathematical objects. One framework for doing so is the Turing degrees (Turing 1939), which allows us to measure the complexity of a variety of objects such as real numbers, sets of words in groups, and countable graphs.

However, the Turing degrees are inadequate for measuring the complexity of continuous real-valued functions (Miller 2004). Such objects (and more) can be measured in the broader structure of the enumeration degrees.

We shall discuss two lines of research in the enumeration degrees: (1) relationships between its subclasses, of which there are many, arising from various areas of mathematics; and (2) the enumeration degrees as a partial order, with the goal of algorithmically deciding all order-theoretic questions with a particular syntactic form.

The new results in this talk are joint work with Kalimullin, Miller, Soskova and Lempp, Ng, Soskova.

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Date/Time : 25 Nov 2021 09:00
Venue : via Zoom
Speaker : Minh Chieu Tran, University of Notre Dame du lac

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https://nus-sg.zoom.us/j/84777353294?pwd=cklXS3NTN2lIbUJxOC9ob245d0JHQT09

Meeting ID: 847 7735 3294
Passcode: 821366

Abstract: Let $G$ be a connected locally compact group with a left Haar measure $\mu$, and let $A,B \subseteq G$ be nonempty and compact. Assume further that $G$ is unimodular, i.e., $\mu$ is also the right Haar measure; this holds, e.g., when $G$ is compact, a nilpotent Lie group, or a semisimple Lie group. In 1964, Kemperman showed that

$$ \mu(AB) \geq \min \{\mu(A)+\mu(B), \mu(G)\} .$$

The inverse Kemperman problem (proposed by Griesmer, Kemperman, and Tao) asks when the equality happens or nearly happens. I will discuss the recent solution of this problem, highlighting the roles played by model theory and descriptive set theory. (Joint with Jinpeng An, Yifan Jing, and Ruixiang Zhang)

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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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