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 : 24 Nov 2021 15:00
Venue : via Zoom
Speaker : Jonathan Gruber, École Polytechnique Fédérale de Lausanne

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https://nus-sg.zoom.us/j/85892358313?pwd=L2ZESjRtQ0xyV244T2RkZEt4UWdXUT09

Meeting ID: 858 9235 8313
Passcode: 873255

Abstract: Let U be a quantum group at a primitive l-th root of unity, obtained by extension of scalars from Lusztig’s integral form of the quantized enveloping algebra of a complex simple Lie algebra. A key role in the representation theory of U is played by the so-called tilting modules. The category Tilt(U) of tilting U-modules is closed under tensor products, and in many respects, its monoidal structure is better understood than that of the category Rep(U) of all finite-dimensional U-modules.

In this talk, I will first explain how a U-module can be represented by a unique minimal complex of tilting modules. Then I will discuss how minimal complexes can be used to transfer information about the structure of tensor products from Tilt(U) to Rep(U).

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Date/Time : 24 Nov 2021 09:00
Venue : via Zoom
Speaker : Yair Shenfeld, MIT

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https://nus-sg.zoom.us/j/87967891890?pwd=R1k4L2FLUGZKVURjd0ZiU0h5L013dz09

Meeting ID: 879 6789 1890
Passcode: 176920

Abstract: What does a ball and a half space have in common? The former is the unique shape which minimizes surface area among all shapes of equal Euclidean volume, while the latter solves the analogous optimization problem for Gaussian volume. These shapes are examples of extremal structures and they play a prominent role in geometric and
probabilistic phenomena: from the shape of water droplets to concentration of measure.

Once we go deeper into the world of extremal structures, we find that they seem hopelessly complex, while permeating mathematics, ranging from probability to combinations to algebraic geometry. In fact, the
characterization of some of these structures is only conjectural, while for others, not even conjectures exist. I will talk about how we resolve several of these conjectures using stochastic, analytic, and geometric tools.

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Date/Time : 23 Nov 2021 09:00
Venue : via Zoom
Speaker : Rahul Singh, University of Pittsburgh

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https://nus-sg.zoom.us/j/84886649094?pwd=Mi9uUE1uREd5eVI3RG55c0JHZmlMQT09

Meeting ID: 848 8664 9094
Passcode: 337125

Abstract: A Higgs bundle over an algebraic curve is a vector bundle with a twisted endomorphism. An important question is to calculate the volume of the groupoid of Higgs bundles over finite fields. In 2014, Olivier Schiffmann succeeded in finding the corresponding generating function and together with Mozvogoy reduced the problem to counting pairs of a vector bundle and a nilpotent endomorphism. It was generalized recently by Anton Mellit to the case of Higgs bundles with regular singularities. An important step in Mellit’s calculations is the case of P^1 and 2 marked points, which allows him to relate the corresponding generating function with the Macdonald polynomials

It is a natural question to generalize Mellit’s calculations to arbitrary reductive groups. In my work, I consider the case of P^1 with 2 points for an arbitrary split connected reductive group G over F_q. Firstly, I give an explicit formula for the number of F_q-rational points of generalized Steinberg varieties of G. Secondly, for each principal G-bundle over P^1, I give an explicit formula counting the number of triples consisting of parabolic structures 0 and ∞ and compatible nilpotent sections of the associated adjoint bundle.

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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 : 19 Nov 2021 10:00
Venue : via Zoom
Speaker : Po-Shen Loh, Carnegie Mellon University

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https://nus-sg.zoom.us/j/84707855085?pwd=dVVUa0puWjlMT3NvN3hRWUV4N016QT09

Meeting ID: 847 0785 5085
Passcode: 687520

Abstract: We introduce a categorically new approach for pandemic control, which transforms selfish human behavior from a liability to an asset. Amidst a high-lethality pandemic, it could leverage individual survival instincts across the entire population, naturally incentivizing system-wide behavior that mitigates pandemic spread.

One of the central tools in pandemic control is contact tracing, which informs people after they have already been exposed to an infected person, and asks all of those people to remove themselves from society, to protect everyone else from themselves.

Informed by Game Theory, we reverse the incentives with a new infrastructure based on mobile devices which helps individuals directly reduce their own chance of infection (protecting themselves from everyone else). The core concept is to inform every participating user of their network-theoretic distance to each newly reported case, against the network of frequent close physical contacts.

This is fundamentally different from contact tracing, because it does not seek to control individuals after they are exposed, but rather empowers not-yet-exposed individuals to take selfish action to avoid exposure in the first place. It is analogous to arming every individual with their personal pandemic network radar. This opportunity was only recently unlocked by the mass proliferation of connected mobile devices, and opens a new category of intervention at the intersection of Game Theory, Technology, and Health.

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Date/Time : 18 Nov 2021 09:00
Venue : via Zoom
Speaker : Yusheng Lei, Boston College

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https://nus-sg.zoom.us/j/83129514590?pwd=U2VWNnhCUGtQR1NpUGo5Zlh3TUhxdz09

Meeting ID: 831 2951 4590
Passcode: 415969

Abstract:  In 2003, Bump-Friedberg-Ginzburg constructed the generalized global theta representation on a metaplectic double cover of an odd special orthogonal group. which was used later to construct the non-minimal theta liftings between double covers of orthogonal groups. This can be viewed as a generalization of the classical theta correspondence. In particular, it enjoys the tower property similar to the Rallis tower in the classical setting. This raises the question of when the first non-zero lifting will occur for a fixed theta tower. Bump-Friedberg-Ginzburg analyzed this problem when the automorphic representations are generic. In this talk, we will show the way to construct such theta liftings and talk about some progress towards understanding the non-generic cases.

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Date/Time : 12 Nov 2021 16:00
Venue : via Zoom
Speaker : Noga Alon, Princeton and Tel Aviv

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https://nus-sg.zoom.us/j/84886693609?pwd=ODArd3JMUDBlVjB2RTBTZlBjVnViUT09

Meeting ID: 848 8669 3609
Passcode: 821993

Abstract:  I will show how basic properties of polynomials can be used in the study of problems in Combinatorics, Additive Number Theory, Combinatorial Geometry and Graph Theory, describing recent and less recent results, problems and algorithmic challenges.

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Date/Time : 12 Nov 2021 10:00
Venue : via Zoom
Speaker : Clement Canonne, University of Sydney

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

Meeting ID: 977 560 6531
Passcode: 123321

Abstract: In this talk, I will introduce the primitive of “domain compression” over the probability simplex, which enables us to reduce the domain size from k to L while contracting (with high probability) the pairwise total variation distances by a factor at most √(L/k). This can equivalently be seen as some type of dimensionality reduction for probability distributions, preserving (in a one-sided way) the L1 distances.

Starting with the corresponding statement for L=2, I will show how to perform this domain compression in a randomness-efficient fashion, and discuss its optimality. As an application, I will describe how this yields sample-optimal algorithms for goodness-of-fit under various constraints, such as communication (bandwidth) constraints, local privacy, and shuffle privacy.

Based on joint works with Jayadev Acharya, Cody Freitag, Yanjun Han, Ziteng Sun, and Himanshu Tyagi (in particular, (https://arxiv.org/abs/1907.08743; COLT 2020)); and some insights from a paper by Amin, Joseph, and Mao (https://arxiv.org/abs/1911.01452; COLT 2020).

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