Date/Time : 14 Mar 2022 10:00
Venue : via Zoom
Speaker : Emanuel Indrei, Purdue University

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https://nus-sg.zoom.us/j/85357515356?pwd=ajE1TlB3RXJZMzBRVHB3VmFmYXhaZz09

Meeting ID: 853 5751 5356
Passcode: 934049

Abstract: In 1947, Polya proved that if $n=3,4$ the regular polygon $P_n$ minimizes the principal frequency of an n-gon with given area $\alpha>0$ and suggested that the same holds when $n \ge 5$. In $1951,$ Polya \& Szego discussed the possibility of counterexamples. Recently, I constructed explicit $2n-4$-dimensional polygonal manifolds $\mathcal{M}(n, \alpha)$ and proved the existence of a computable $N \ge 5$ such that for all $n \ge N$, the admissible $n$-gons are given via $\mathcal{M}(n, \alpha)$ and there exists an explicit non-empty set $ \mathcal{A}_{n} \subset \mathcal{M}(n,\alpha)$ such that $P_n$ has the smallest principal frequency among $n$-gons in $\mathcal{A}_{n}$. Inter-alia a formula is proved for the principal frequency of $P \in \mathcal{M}(n,\alpha)$ when $n \ge 3$ in terms of an equilateral $n$-gon $P_E$ with the same area. The techniques involve a partial symmetrization, tensor calculus, the spectral theory of circulant matrices, and $W^{2,p}$ estimates. Last, an application is given in the context of electron bubbles.

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Date/Time : 11 Mar 2022 10:00
Venue : via Zoom
Speaker : Jean Honorio, Purdue University

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https://nus-sg.zoom.us/j/9775606531?pwd=cVZ6eTRYS3ZzVUlFNXRuSlhMVjJSUT09

Meeting ID: 977 560 6531
Passcode: 123321

Several problems in machine learning are combinatorial (e.g., clustering, learning sparse representations.) Convex relaxations have been used in the past, in order to circumvent the NP-hardness of the combinatorial problems, albeit being approximate solutions. My long-term research goal is to uncover large classes of continuous relaxations (beyond convexity) of worst-case NP-hard problems, that admit exact solutions.

In this talk, I will introduce a unifying framework, which uses the power of continuous relaxations (beyond convexity), Karush-Kuhn-Tucker conditions, primal-dual certificates and concentration inequalities. This framework has produced novel algorithms not only for classical worst-case NP-hard problems, e.g., Bayes nets, structured prediction, clustering, but also for aspects of fairness and meta learning.

As a first example, we study the problem of fair sparse regression on a biased dataset where bias depends upon a hidden binary attribute. The presence of a hidden attribute adds an extra layer of complexity to the problem by combining sparse regression and clustering with unknown binary labels. The corresponding optimization problem is combinatorial but we propose an invex relaxation, a subclass of nonconvex relaxations. We then provide exact recovery guarantees.

As a second example, we study the problem of exact partitioning of high-order models. Exact partitioning can be formulated as a tensor optimization problem. We relax this high-order combinatorial problem to a convex conic form problem. To this end, we carefully define the Caratheodory symmetric tensor cone, and show its convexity, and the convexity of its dual cone. We also provide exact recovery guarantees.

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Date/Time : 11 Mar 2022 09:00
Venue : via Zoom
Speaker : Laura DeMarco, Harvard University

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https://nus-sg.zoom.us/j/86723583031?pwd=WDBVNVpqbU8xWWI3amhKMTdPdk45Zz09

Meeting ID: 867 2358 3031
Passcode: 232023

Abstract: The periodic orbits and their structure are a fundamental feature of a dynamical system. In an algebraic setting, where the system is defined by polynomials, we can use tools from algebraic or arithmetic geometry to study these orbits. As the past year has seen major breakthroughs in complex and arithmetic geometry — for example in the uniform versions of the Mordell or Manin-Mumford Conjectures proved recently by Dimitrov-Gao-Habegger, Kuhne, Yuan and others, studying families of abelian varieties — I will take this opportunity to describe some parallel questions and progress in complex and arithmetic dynamical systems.

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Date/Time : 08 Mar 2022 09:00
Venue : via Zoom
Speaker : Weinan Zhang, University of Virginia

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https://nus-sg.zoom.us/j/84683629566?pwd=SXBHQzdmcUoxLzhrQ3pqU29ERS9KZz09

Meeting ID: 846 8362 9566
Passcode: 746935

Abstract: Introduced by Lusztig in the early 1990s, the braid group symmetries constitute an essential part in the theory of quantum groups. The i-quantum groups are coideal subalgebras of quantum groups arising from quantum symmetric pairs, which are now viewed as natural generalizations of quantum groups. In this talk, I will present our construction of relative braid group symmetries (associated to the relative Weyl group of a symmetric pair) on i-quantum groups of arbitrary finite types. These new symmetries inherit most properties of Lusztig’s symmetries, including compatible relative braid group actions on modules. This is joint work with Weiqiang Wang.

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Date/Time : 08 Mar 2022 16:00
Venue : via Zoom
Speaker : Yang Pengyu, ETH Zurich

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https://nus-sg.zoom.us/j/87511476637?pwd=VURaUDVzaHpMYmduN2Fod0h0Y29sQT09

Meeting ID: 875 1147 6637
Passcode: 468399

Abstract: We study translates of natural measures on curves/submanifolds under diagonal flows in homogeneous spaces, and describe geometric/arithmetic obstructions to equidistribution. Our basic strategy is to use Ratner’s unipotent rigidity theorem, and I will discuss the new ingredients from geometric invariant theory.

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Date/Time : 04 Mar 2022 10:00
Venue : via Zoom
Speaker : Yunlong Hou, National University of Singapore

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https://nus-sg.zoom.us/j/9775606531?pwd=cVZ6eTRYS3ZzVUlFNXRuSlhMVjJSUT09

Meeting ID: 977 560 6531
Passcode: 123321

Abstract: We design and analyze VA-LUCB, a parameter-free algorithm, for identifying the best arm under the fixed-confidence setup and under a stringent constraint that the variance of the chosen arm is strictly smaller than a given threshold. An upper bound on VA-LUCB’s sample complexity is shown to be characterized by a fundamental variance-aware hardness quantity H_VA. By proving a lower bound, we show that sample complexity of VA-LUCB is optimal up to a factor logarithmic in H_VA. Extensive experiments corroborate the dependence of the sample complexity on the various terms in H_VA. By comparing VA-LUCB’s empirical performance to a close competitor RiskAverse-UCB-BAI by David et al. (2018), our experiments suggest that VA-LUCB has the lowest sample complexity for this class of risk-constrained best arm identification problems, especially for the riskiest instances.

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Date/Time : 02 Mar 2022 16:00
Venue : via Zoom
Speaker : Federico Franceschini, ETH Zurich

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https://nus-sg.zoom.us/j/81974497199?pwd=S0h6QXU0TmFpbkxWeW4wSmIzMTNEdz09

Meeting ID: 819 7449 7199
Passcode: 422551

Abstract: The thin obstacle problem is a classical free boundary problem: it consists of an elliptic PDE which characterise minimisers of a certain variational problem. The boundary of the zero set of solutions of this PDE is called the “free boundary” and understanding its features is an active research field for people in PDE. In this talk I will first explain some known basic results (examples, existence and smoothness of solutions). Then I will present a recent joint work with Prof. Serra concerning the analysis of the free boundary.

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Date/Time : 02 Mar 2022 16:00
Venue : via Zoom
Speaker : Lavinia Picollo, National University of Singapore

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https://nus-sg.zoom.us/j/83049258042?pwd=UWViaWNvTFUrdFdhOHJCdEVydnVkdz09

Meeting ID: 830 4925 8042
Passcode: 1729=x3+y3

Abstract: Truth predicates are widely believed to be capable of serving a certain logical or quasi-logical function. There is little consensus, however, on the exact nature of this function. We offer a series of formal results in support of the thesis that disquotational truth is a device to simulate higher-order resources in a first-order setting. More specifically, we show that any theory formulated in a higher-order language can be naturally and conservatively interpreted in a first-order theory with a disquotational truth or truth-of predicate. In the first part of the talk we focus on the relation between truth and full impredicative sentential quantification. The second part is devoted to the relation between truth-of and full impredicative predicate quantification.

This is joint work with Thomas Schindler

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Date/Time : 02 Mar 2022 16:30
Venue : via Zoom
Speaker : Subhajit Goswami, Tata Institute of Fundamental Research

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https://nus-sg.zoom.us/j/86418786459?pwd=cVE2Wk5OSWthOUJtSEF5L0RzNURIdz09

Meeting ID: 864 1878 6459
Passcode: 815262

Abstract: In this talk I will give a brief introduction to Liouville first-passage percolation (LFPP) which is a model for random metric on the plane or a finite planar grid graph. It is studied primarily as a way to understand the random metric associated with Liouville quantum gravity (LQG). In short, the Liouville quantum gravity is a one parameter family of “canonical” random metrics on the plane. I will discuss a result on the estimates of the typical distance between two points and its implication for the Hausdorff dimension of the LQG distance. I will highlight how this estimate refutes a conjecture made in the physics literature about the LQG distance.

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