Date/Time : 01 Dec 2021 16:00
Venue : via Zoom
Speaker : Pierre-Francois Rodriguez, Imperial College, London

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https://nus-sg.zoom.us/j/85694125031?pwd=eGJuMWk5SXRCS2hWc1JITXpRWmdRUT09

Meeting ID: 856 9412 5031
Passcode: 918464

Abstract: I will report on recent progress regarding the near-critical behavior of certain statistical physics models in dimension 3. Our results deal with the critical scaling exponents associated to the second-order phase transition for two percolation problems involving the Gaussian free field in 3D. In one case, they determine a unique “fixed point” corresponding to the transition, which is proved to obey one of several scaling relations. Such laws are classically conjectured to hold by physicists on the grounds of a corresponding scaling ansatz.

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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 : 26 Nov 2021 10:00
Venue : via Zoom
Speaker : Wenyu Pan, University of Chicago

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https://nus-sg.zoom.us/j/82434099670?pwd=VlhicUs1SlZPd0IrTzZ5d0NrMm1oQT09

Meeting ID: 824 3409 9670
Passcode: 772471

Abstract: Let  Hⁿ be the hyperbolic n-space and Γ be a geometrically finite discrete subgroup in Isom + (Hⁿ) with parabolic elements. We investigate whether the geodesic flow (resp. the frame flow) over the unit tangent bundle T¹ (Γ\Hⁿ) (resp. the frame bundle F(Γ\Hⁿ)) mixes exponentially. This result has many applications, including spectral theory, orbit counting, equidistribution, prime geodesic theorems, etc. 

I will start with a survey of the past results and related problems on this topic. Then I will present the joint work with Jialun Li in which we show that the geodesic flow mixes exponentially. I will describe some ingredients in the proof. If there is time, I will also discuss the exponential mixing of the frame flow.

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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 : 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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