Date/Time : 15 Oct 2021 10:00
Venue : via Zoom
Speaker : Ju Sun, University of Minnesota

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

Meeting ID: 977 560 6531
Passcode: 123321

Abstract:  Phase retrieval (PR) concerns the recovery of a signal (1D, 2D, or 3D) from its Fourier magnitudes, and is central to numerous areas of scientific imaging. Over the past decade, there have been numerous breakthroughs in understanding simplified or modified versions of the problem, but the existence of effective and robust methods for the original PR problem remains elusive and so theoretical insights are out of the question. In this talk, I will discuss why PR is difficult, the strength and limitation of conventional formulations, the promise and perils of deep learning approaches, as well as our recent methods that solve practical PR problems in an unprecedented regime.

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Date/Time : 13 Oct 2021 16:00
Venue : via Zoom
Speaker : Jonas Beyrer, University of Bonn

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

Meeting ID: 875 1147 6637
Passcode: 468399

Abstract:  The study of higher rank Teichmüller spaces has attracted a lot of attention in the last 20 years. Those spaces are connected components of the character variety Hom(H,G)/G that consist entirely of discrete and faithful representations; here G is a Lie group and H the fundamental group of a closed surface. Those components generalize classical Teichmüller space, the case G=PSL(2,R), which also identifies with the moduli space of hyperbolic structures on the surface. Representations in higher rank Teichmüller spaces, so called „positive representations“, are not only discrete and faithful, but have many other striking properties and similarities to hyperbolizations (e.g. constant entropy, collar lemmas, etc). However surprisingly there are many representations that are not in higher rank Teichmüller spaces and still share many of this striking properties: representations that satisfy a „partial positivity“. In this talk on joint work with Beatrice Pozzetti I want to discuss partially positive representations and their main difference to representations in higher rank Teichmüller spaces: degenerations.

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Date/Time : 13 Oct 2021 14:00
Venue : via Zoom
Speaker : Hankyung Ko, Uppsala University

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https://nus-sg.zoom.us/j/86403704548?pwd=ZWFSVHpqMVd0TkNNVTM0ZEg0MzhSdz09

Meeting ID: 864 0370 4548
Passcode: 234932

Abstract:  A Coxeter system is a presentation of a group by generators and a specific form of relations, namely the braid relations and the reflection relations. The Coxeter presentation leads to, among others, a similar presentation of the (Iwahori-)Hecke algebras and the Kazhdan-Lusztig theory, which provides combinatorial answers to major problems in Lie theoretic representation theory and geometry. Extending such applications to the `singular land’ requires the singular version of the Hecke algebra. Underlying combinatorics of the singular Hecke algebra/category comes from the parabolic double cosets and is the first step in understanding the singular Hecke category. In this talk, I will present a Coxeter theory of the parabolic double cosets developed in a joint work with Ben Elias. In particular, I will explain a generalization of the reduced expressions and describe the braid and non-braid relations.

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Date/Time : 13 Oct 2021 16:00
Venue : via Zoom
Speaker : Liao Yuke,

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

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

Abstract: Hindman’s theorem is a Ramsey type theorem related to finite sum while its proof-theoretic strength still has a huge gap. One form of the question is that if any computable coloring function would have an arithmetic solution (homogeneous set). We will construct a computable coloring functions that no Pi-3 set can be homogeneous.

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Date/Time : 08 Oct 2021 16:00
Venue : via Zoom
Speaker : Siddhartha Mishra, ETH Zürich

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https://nus-sg.zoom.us/j/88995092115?pwd=UmxidFJqTzZyQ2JRLyt5SGNnaHhnUT09

Meeting ID: 889 9509 2115
Passcode: 309203

Abstract:  We will describe PINNs and illustrate several examples for using PINNs for solving PDEs. Our aim would be to elucidate mechanisms that underpin the success of PINNs in approximating classical solutions to PDEs by deriving rigorous bounds on the resulting error. Examples of a variety of linear and nonlinear PDEs will be provided including PDEs in high dimensions and inverse problems for PDEs.

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Date/Time : 08 Oct 2021 10:00
Venue : via Zoom
Speaker : Peter Mueller, University of Texas at Austin

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

Meeting ID: 977 560 6531
Passcode: 123321

Abstract: Nonparametric Bayesian methods allow for principled model-based inference on unknown functions, distributions and structures without restriction to specific parametric models. The flexible prediction rules and the more realistic and complete description of related uncertainties are useful in many examples of clinical trial design and related data analysis. In this talk we review four related examples. The first example involves the comparison of two event time distributions for the time until resolution of air leaks after lung surgery. The second example is about inference for dynamic treatment regimes. Another example is a Bayesian basket trial design with a nonparametric Bayesian survival regression to find benefiting subpopulations. A last example is about the construction of a synthetic control arm to supplement a single-arm trial using real world data.

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Date/Time : 07 Oct 2021 16:00
Venue : via Zoom
Speaker : Xavier Fernández-Real, EPFL

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https://nus-sg.zoom.us/j/86013696968?pwd=VHJFQkVhVmg5N0Urdk0yV3A2MEJQQT09

Meeting ID: 860 1369 6968
Passcode: 328086

Abstract:  The fractional or thin one-phase problem is the study of minimizers of the functional
J_s(u) = [u]_{H^s(B_1)}^2 + |{u > 0} \cap B_1|.
The aim of this talk is to study homogeneous stable solutions to the fractional one-phase free boundary problem. The problem of classifying stable (or minimal) homogeneous solutions in dimensions n ≥ 3 is completely open. In this context, axially symmetric solutions are expected to play the same role as Simons’ cone in the classical theory of minimal surfaces, but even in this simpler case the problem is open.

The goal of this talk is twofold and is based on a recent work with X. Ros-Oton. On the one hand, we present our first main contribution: the stability condition for the thin one-phase problem. Quite surprisingly, this requires the use of “large solutions” for the fractional Laplacian, which blow up on the free boundary. On the other hand, using our new stability condition, we show that any axially symmetric homogeneous stable solution in dimensions n ≤ 5 is one-dimensional, independently of the parameter s∈(0,1).

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Date/Time : 06 Oct 2021 15:00
Venue : via Zoom
Speaker : Jan Frahm, Aarhus University

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https://nus-sg.zoom.us/j/81603760900?pwd=d0lGRktVY0hJSTM5ODBCWmZmTVRrUT09

Meeting ID: 816 0376 0900
Passcode: 883638

Abstract:  ranching problems ask for the behaviour of the restriction of an irreducible representation of a group $G$ to a subgroup $H$. In the context of smooth representations of real reductive groups, this typically leads to the study of multiplicities with which an irreducible representation of $H$ occurs as a quotient of an irreducible representation of $G$. Here, both quantitative results such as multiplicity-one theorems and qualitative results such as the Gan-Gross-Prasad conjectures are of interest. In the context of unitary representations of real reductive groups, one can go a step further and explicitly decompose in irreducible representation of $G$ into a direct integral of irreducible representations of $H$. I will explain how branching laws for unitary representations are related to those in the smooth category, and how one can use an analytic continuation procedure along a principal series parameter to obtain explicit branching laws from certain Plancherel formulas for homogeneous spaces.

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Date/Time : 06 Oct 2021 16:00
Venue : via Zoom
Speaker : Chong Chi Tat, 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:   In this talk we discuss the proof-theoretic strength, from the reverse mathematics perspective, of combinatorial principles concerning the coloring of binary trees and finite products of binary trees.  Beginning with the principle TT^1, which states that every finite coloring of the full binary tree has an isomorphic monochromatic subtree, we will cover its strengthening to the existence of a strong monochromatic subtree, and to the full generalization of the latter known as the Halpern-Laeuchli Theorem.

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