Date/Time : 13 Apr 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Xavier Fernández-Real, Swiss Federal Institute of Technology Lausanne (EPFL)

In this talk we present an introduction to integro-differential operators and the obstacle problem, and how they interact with each other in what is known as the fractional obstacle problem.

In $\R^n$, the fractional obstacle problem with a given obstacle $\varphi\in C_c^\infty(\R^n)$ can be written as

\[\min\{(-\Delta)^s u, u – \varphi\} = 0\quad\text{in}\quad \R^n.\]

The set $\{u = \varphi\}$ is called the contact set, and its boundary $\partial\{u = \varphi\}$ is called the free boundary, which are unknowns of the problem. The free boundary can be divided into two sets: regular points (where it is regular) and degenerate points. In general, degenerate points can exist in any dimension.

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Date/Time : 13 Apr 2023 16:00
Venue : S16 #03-06
Speaker : Karthyek Murthy, Singapore University of Technology and Design

Several problems in data-driven decision-making and risk management suffer from data imbalance, a term referring to settings where a small fraction of data has an outsized impact on estimating one or more decision-making criteria. Due to the paucity of relevant samples, such problems are usually approached with the “estimate, then optimize” workflow involving a model estimation from data in the first step before plugging in the trained model to solve various downstream optimization tasks. As biases due to model selection, misspecification, and overfitting to in-sample data are especially difficult to avoid in the first-step estimated model in settings affected by data imbalance, we construct novel locally robust optimization formulations in which the first-step estimation has no effect, locally, on the optimal solutions obtained.

We show that this local insensitivity translates to improved out-of-sample performance freed from the first-order impact of model errors introduced in the first-step estimation. A key ingredient in achieving this local robustness is a novel debiasing procedure that adds a non-parametric bias correction term to the objective. The debiased formulation retains convexity, and the imputation of the correction term relies only on a non-restrictive large deviations behavior conducive for transferring knowledge from representative data-rich regions to the data-scarce tail regions suffering from imbalance. The bias correction gets determined by the extent of model error in the estimation step and the specifics of the stochastic program in the optimization step, thereby serving as a scalable “smart-correction” step bridging the disparate goals in estimation and optimization.

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Date/Time : 13 Apr 2023 11:00
Venue : S17 #05-12 (SR4)
Speaker : Lucas Mason-Brown, University of Oxford

Let G and H be real reductive groups. To any L-homomorphism e: H^L \to G^L one can associate a map e_* from virtual representations of H to virtual representations of G. This map was predicted by Langlands and defined (in the real case) by Adams, Barbasch, and Vogan. Without further restrictions on e, this map can be very poorly behaved. A special case in which e_* exhibits especially nice behavior is the case when H is an endoscopic group. In this talk, I will introduce a more general class of L-homomorphisms which exhibit similar behavior to the endoscopic case. I will explain how this more general notion of endoscopic lifting relates to the theory of cohomological induction. I will also explain how this generalized notion of endoscopic lifting can be used to prove the unitarity of many Arthur packets. This is based on joint work with Jeffrey Adams and David Vogan.

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Date/Time : 13 Apr 2023 10:00
Venue : via Zoom
Speaker : Daniel Sanz-Alonso, University of Chicago

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https://nus-sg.zoom.us/j/81261210349?pwd=M2VlVjUvR1o4dGVodDY4MnBtRzMzUT09

Meeting ID: 812 6121 0349
Passcode: 557052

Semi-supervised learning refers to the problem of recovering an input-output map using many unlabeled examples and a few labeled ones. In this talk I will survey several mathematical questions arising from the Bayesian formulation of graph-based semi-supervised learning. These questions include the modeling of prior distributions for functions on graphs, the derivation of continuum limits for the posterior, the design of scalable posterior sampling algorithms, and the contraction of the posterior in the large data limit.

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Date/Time : 12 Apr 2023 16:00
Venue : via Zoom
Speaker : Tran Chieu-Minh, National University of Singapore

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https://nus-sg.zoom.us/j/81046462775?pwd=aDhaYnhQOTgybzdYS21yajlQdnBmZz09

Meeting ID: 810 4646 2775
Passcode: 23571113

The following question was described in Ben Green’s list of open problems, but also attributed to discussions with Emmanuel Breuillard: When A \subseteq SO(3, R) has sufficiently small normalized Haar measure, do we have \mu(AA) > 3.99 \mu(A)?
This problem is often considered the first serious obstacle in generalizing results of additive combinatorics to nonabelian settings when we also want sharp bounds.

In this talk, I will discuss a recent joint work with Jing Yifan and Zhang Ruixiang where we combine techniques from model theory, probability theory, and harmonic analysis to answer it positively.

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Date/Time : 12 Apr 2023 17:00
Venue : via Zoom
Speaker : Daniel Max Hoffmann, University of Warsaw

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Date/Time : 06 Apr 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Quan Zhao, University of Regensburg, Germany

In this talk, I will consider the structure-preserving front-tracking methods for a sharp-interface model of two-phase flow in both the unfitted and fitted mesh approaches, meaning that the volume preservation and energy decay are satisfied on the discrete level. In the unfitted approach, the constructed method is based on an Eulerian weak formulation, while in the fitted approach, the method is devised in the arbitrary Lagrangian-Eulerian (ALE) framework. We compare the numerical results of the two approaches by using the example of a rising bubble.

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Date/Time : 06 Apr 2023 11:00
Venue : S17 #05-12 (SR4)
Speaker : Nadya Gurevich, Ben Gurion University of the Negev

The classical Schur-Weyl duality, relating representations of the symmetric group S(r) and of sl(n), admits a quantum affine generalization. We explain its relation to the Whittaker functor on the category of Iwahori-spherical genuine representations of a central cover of GL(r) of degree n. 

In particular, we deduce that the space of Whittaker functionals on such representation admits an action of an affine quantum group.

This is a joint work with Fan Gao and Edmund Karasiewicz.

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Date/Time : 05 Apr 2023 17:00
Venue : S17 #05-11 (SR5)
Speaker : Frank Stephan, National University of Singapore

Languages over the unary alphabet have been studied in computer science for a long time. The present work investigates regular languages over the unary alphabet and one investigates the various forms of representing them by finite automata. Here three types of automata are studied:

(a) Deterministic finite automata where the finite automaton has exactly one run per word which is accepting iff the word is in the given language.

(b) Nondeterministic finite automata where there are choices in the states on how to procede with a run on various symbols, a word is in the language iff there is some run which is accepting.

(c) Unambiguous finite automata which are a special case of nondeterministic finite automata and which either reject a word or which have exactly one accepting run for a word.

The results presented are from the following two areas:

(1) This paper improves the upper bound of the equality problem of unary nondeterministic automata from an exponential in the second root to an exponential in the third root of the number of states. This almost matches a known lower bound based on the exponential time hypothesis by Fernau and Krebs.

(2) It is established that the standard regular operations of union, intersection, complementation and Kleene star cause either only a polynomial or a quasipolynomial blow-up.
Concatenation of two n-state ufas, in worst case, causes a blow-up from n to a function with an exponent of sixth root of n. Decision problems of finite formulas using regular operations and comparing languages given by n-state unambiguous automata, in worst case, require an exponential-type of time under the Exponential Time hypothesis and this complexity goes down to quasipolynomial time in the case that the concatenation of languages is not used in the formula. Merely comparing two languages given by n-state ufas in Chrobak Normal Form is in LOGSPACE.

This talk is joint work with Gordon Hoi, Sanjay Jain and Christopher Tan.

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