Date/Time : 01 Nov 2023 16:30
Venue : S17 #05-11 (SR5)
Speaker : Lin Ting, Peking University

In this talk, I will introduce the construction of the $C^r$ finite element space in any dimension. We show that $C^r$ finite element in $\mathbb R^d$ can be constructed, with the local shape function space $\mathcal P_{k}$ ($k\ge 2^d r+1$). Moreover, we show that the assumptions in our construction are sharp. This is based on joint work with Jun Hu (PKU), Qingyu Wu (PKU), and Beihui Yuan (YMSC).

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Date/Time : 31 Oct 2023 17:00
Venue : S17 #04-06 (SR1)
Speaker : Samuel Alfaro Tanuwijaya, National University of Singapore

The talk will deal with relativised versions of the Posner Robinson theorem and explore how far one can go and how to prove these relatived versions.

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Date/Time : 30 Oct 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : David Chew, NUS, Statistics & Data Science

In this talk, I will describe the effort at the Department of Statistics and Data Science to offer more courses in the Blended Learning (BL) format.

Here are some aspects of the general strategy adopted by the department:
1. Replace one of the two weekly lecture meetings by pre-recorded videos that give high level overviews of topics covered.
2. Hold in-person lectures once a week; dive deeper into the materials by going through proofs, computational applications and/or in-depth examples.
3. In all our gateway classes, senior students are hired as student helpers (SHs) to provide extra support to students. SHs are each assigned 12 ~ 15 students, to answer queries, provide consultation/coding help during (and outside of) lecture and tutorial hours.

As for the tool used for making videos, the department recommends the use of Descript, an all-in-one audio and video editing software. It allows one to make videos using AI generated voices, so that videos are made with a “neutral” voice. This make the videos more “transferrable” — they can be (easily adapted and) used by another instructor, or in other courses.

We will also discuss other tools used by some colleagues (TeX, ImageMagick, etc) and other practical considerations when offering a course in BL format: logistics, manpower etc.

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Date/Time : 25 Oct 2023 16:00
Venue : S16 #03-06
Speaker : Yin Jianyuan, National University of Singapore

Rotational Bose-Einstein condensates can exhibit quantized vortices as topological excitations. In this study, the ground and excited states of the two-dimensional rotational Bose-Einstein condensates are systematically studied by calculating the stationary points of the Gross-Pitaevskii energy functional with the constraint. A solution landscape method is introduced with the constrained high-index saddle dynamics. Various excited states and their connections at different rotational frequencies are revealed with solution landscapes. Four excitation mechanisms are identified: vortex addition, rearrangement, merging, and splitting. We demonstrate changes in the ground state with increasing rotational frequencies and decipher the stability evolution of ground states.

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Date/Time : 25 Oct 2023 14:00
Venue : via Zoom
Speaker : Simon Machado, ETH Zurich

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

Meeting ID: 810 4646 2775
Passcode: 23571113

Abstract: The doubling constant of a subset $A$ of a group is defined as the ratio of the size of $A^2$ (the set of products of two elements of $A$) by the size of $A$. It is now well-established that the doubling constant $K$ of $A$ captures a great deal of information about the additive structure of $A$. For instance, if $K < 3/2$, then $A$ itself must be a coset of a group. The generalised Breuillard—Green conjecture asserts that, conversely, if one knows that $A$ is a subset of a compact simple Lie group, then $K$ must be bounded below by an explicit dimensional constant, and provides examples saturating this inequality.

I will discuss a proof of that conjecture assuming that $A$ is sufficiently small to begin with. This will take us through the theory of additive combinatorics and, perhaps surprisingly, some optimal transport. If time permits I will also discuss what we know about subsets whose doubling constant is near the minimal value.

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Date/Time : 24 Oct 2023 17:00
Venue : S17 #04-06 (SR1)
Speaker : Meng-Che Ho, California State University Northridge

Classically, the word problem of a group is the set of words equal to the identity of the group, and we analyze them using Turing reductions. In this talk, we consider the word problem of a group as a computably enumerable equivalence relation (ceer), namely, two words are equivalent if and only if they are equal in the group. We compare ceers using the computable reduction: $E$ is reducible to $F$ if there is a computable function $f$ so that $iEj$ if and only if $f(i)Ff(j)$.

We will discuss some recent results and see that the landscape of word problems as ceers is very different from the classical theory. For instance, in the classical setting, any Turing degree can be realized as a word problem by first constructing a countable group and then embedding it into a finitely presented group via the Higman embedding theorem. However, we prove that in the ceer setting, there is a group $G$ whose word problem is not universal, but for any nontrivial $H$, the free product $G*H$ has a universal word problem.

This is a joint work with Uri Andrews and Luca San Mauro.

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Date/Time : 20 Oct 2023 15:00
Venue : S17 #05-11 (SR5)
Speaker : Apoorva Khare, Indian Institute of Science, Bangalore

We explain recent results involving Schur polynomials – viewed as functions on the positive orthant – which have led to progress along several classical directions of study:

(1) Understanding functions that preserve positive semidefiniteness when applied entrywise to matrices. This journey takes us from work of Schur, Schoenberg, Rudin, and Loewner (1900s) to modern-day results.

(2) The proofs of these recent results lead to characterizing (weak) majorization of real tuples using Schur polynomials; this has since been generalized to all Weyl groups. Majorization inequalities have been studied since Maclaurin and Newton (1700s).

(3) We also explain how all Schur polynomials are lurking within each smooth function, by completing a 1960s determinantal computation of Loewner. This helps extend determinantal identities by Cauchy and Frobenius (1800s) involving one or two geometric series, to all power series, over all unital commutative rings.

(Partly based on joint works with Alexander Belton, Dominique Guillot, Mihai Putinar, and with Terence Tao.)

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Date/Time : 19 Oct 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Ruobing Zhang, Princeton University

The studies of Poincaré-Einstein manifolds originate from the AdS/CFT correspondence in mathematical physics, which relates quantum gravity theory and conformal field theory. On the mathematics side, numerous conformal invariants were successfully constructed via the microlocal analysis on Poincaré-Einstein spaces, so that it has become a central area in conformal geometry.

A primary object in this talk is the so-called fractional GJMS operators, which provide a uniform treatment of conformally covariant operators with different orders. From the analytic point of view, the degenerate and nonlocal equations arising in this setting are particularly interesting. We will mainly exhibit a series of sharp dimension bounds and Hausdorff measure estimates for the singular set of those solutions. The proof is based on the quantitative differentiation theory for degenerate elliptic operators and involves subtle regularity issues on the Poincaré-Einstein metrics.

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Date/Time : 18 Oct 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Ruobing Zhang, Princeton University

This lecture concerns the metric Riemannian geometry of Einstein manifolds, which is a central theme in modern differential geometry and is deeply connected to a large variety of fundamental problems in algebraic geometry, geometric topology, analysis of nonlinear PDEs, and mathematical physics. We will exhibit the rich geometric/topological structures of Einstein manifolds and specifically focus on the structure theory of moduli spaces of Einstein metrics.

My recent works center around the intriguing problems regarding the compactification of the moduli space of Einstein metrics, which tells us how Einstein manifolds can degenerate. Such problems constitute the most challenging part in the metric geometry of Einstein manifolds. We will introduce recent major progress and propose several open questions in the field.

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