Date/Time : 02 Apr 2025 10:00
Venue : S17 #05-11 (SR5)
Speaker : Edmund Karasiewicz, National University of Singapore

In this mini course we will study two topics. The first is the PGL2xF4 exceptional theta correspondence over a p-adic field. The second is the dual group of a spherical variety. Over the course of four lectures we will:

1. Introduce exceptional groups, focusing on constructions and tools relevant to the exceptional theta correspondence.
2. Describe the PGL2xF4 exceptional theta correspondence over a p-adic field.
3. Present the combinatorial construction of the dual group of a spherical variety due to Knop-Schalke.
4. Connect the exceptional theta correspondence to the F4/Spin9 spherical variety.

Join Zoom Meeting
https://nus-sg.zoom.us/j/84884850855?pwd=bvokLMShAf5wFeAHh2zemmel1dvb9f.1

Meeting ID: 848 8485 0855
Passcode: 226224

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Date/Time : 02 Apr 2025 17:00
Venue : S17 #04-04 (SR3)
Speaker : Jia Zekun,

Post’s problem is one of the most fundamental questions in the early development of recursion theory. In 1944, Post proved the existence of a non-recursive m-incomplete r.e. set and generalized his result to tt-reducibility and Q-reducibility. However, it seemed hopeless to further extend Post’s technique to T-reducibility, and this problem remained open for a decade until it was solved in the affirmative by Friedberg and Muchnik independently. Surprisingly, Post’s method was revived from a series of works by Malcev, Ershov, Marchenkov and Degtev in the 1970s. By introducing an underlying r.e. equivalence relation eta over the natural numbers, they showed that Post’s problem could indeed be solved in Post’s way. A key concept in the proof is the so-called eta-maximal set, resembling Post’s idea of a maximal set. Although Chong and Yang proved that the existence of a maximal set is equivalent to ISigma_2 over PA- + BSigma_2, Downey conjectured that the existence of an eta-maximal set may be simpler despite their similarities in definition. In this talk, we will improve Degtev’s construction of an eta-maximal set and show that Downey is right: PA- + ISigma_1 is enough to carry out and verify the construction.

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Date/Time : 01 Apr 2025 14:00
Venue : S17 #05-11 (SR5)
Speaker : Li Anlun, National University of Singapore

In the study of the spherical variety X=H\G, it’s crucial to study the period integral P_H of some (cuspidal) automorphic forms of G. In this talk we will study the case G=G_2 and H=SL_3. Under assumptions about the dimension 1 condition of some Hom spaces, we can show that the global period integral can be related to the local H-invariant linear functional that comes from the Plancherel theorem of Mp_2 and the theta correspondence from Mp_2 to G_2. Moreover, since this Hom space is (assumed to be) 1 dimensional, we can determine the coefficient by showing an analogue of the Siegel-Weil formula.

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Date/Time : 01 Apr 2025 14:50
Venue : S17 #05-11 (SR5)
Speaker : Colin Loh, National University of Singapore

Recent work of Mao, Wan and Zhang have provided a complete list of strongly tempered hyperspherical varieties and they proposed some new Rankin-Selberg integrals. In this talk, I will present 3 new Rankin-Selberg integrals associated to dual of strongly tempered distinguished polarised hyperspherical varieties and discuss the L-functions these integrals represent, as examples of the Relative Langlands Duality.

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Date/Time : 27 Mar 2025 16:00
Venue : S17 #06-11 (SR6)
Speaker : Jiuzu Hong, University of North Carolina at Chapel Hill

Global Schubert varieties of parahoric Bruhat-Tits group schemes were introduced by Xinwen Zhu in his proof of the coherence conjecture of Pappas-Rapoport. These global Schubert varieties provide flat families relating spherical Schubert varieties in affine Grassmannians and other Schubert varieties or their unions in partial affine flag varieties. In a series of talks, I will give an introduction to this topic and discuss several applications based on my own research.

In the first talk, I will cover the basics of affine Grassmannian and Schubert varieties.
In the second talk, I will discuss parahoric group schemes and flag varieties of loop groups.
In the last talk, I will introduce the global Schubert varieties of parahoric Bruhat-Titis group schemes and discuss some of their applications.

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Date/Time : 26 Mar 2025 17:00
Venue : S17 #04-04 (SR3)
Speaker : Chong Chi Tat, National University of Singapore

Combinatorial principles is a prominent area of investigation in reverse mathematics.

The computational complexity of a solution of an instance of a principle began with Jockusch work (1972) on Ramsey’s theorem over the standard model of Peano arithmetic. In general, every instance of a combinatorial principle has a definable solution over the standard model.

For an arbitrary model of RCA_0, however, the situation is not clear. There exist weak models and instances of combinatorial principles (such as Ramsey’s theorem for pairs) for which no definable solution is known. In this talk we present an analysis of the problem of definable solutions and some nonexistence results.

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Date/Time : 26 Mar 2025 16:30
Venue : S17 #06-11 (SR6)
Speaker : Wang Chushan, National University of Singapore

The nonlinear Schrödinger equation (NLSE) arises from various applications in quantum physics and chemistry, nonlinear optics, plasma physics, Bose-Einstein Condensates, etc. In these applications, it is necessary to incorporate low-regularity or singular potential and nonlinearity into the NLSE. Typical examples of such potential and nonlinearity include the discontinuous square-well potential, the singular Coulomb potential, the non-integer power nonlinearity, and the logarithmic nonlinearity. Such low regularity and singularity pose significant challenges in the analysis of standard numerical methods and the development of accurate, efficient, and structure-preserving schemes.

In this talk, I will introduce several new analysis techniques to establish optimal error bounds for some widely used numerical methods under optimally weak regularity assumptions. Based on the analysis, we also propose novel temporal and spatial discretizations to handle the low regularity and singularity more effectively.

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Date/Time : 26 Mar 2025 10:00
Venue : S17 #05-11 (SR5)
Speaker : Edmund Karasiewicz, National University of Singapore

In this mini course we will study two topics. The first is the PGL2xF4 exceptional theta correspondence over a p-adic field. The second is the dual group of a spherical variety. Over the course of four lectures we will:

1. Introduce exceptional groups, focusing on constructions and tools relevant to the exceptional theta correspondence.
2. Describe the PGL2xF4 exceptional theta correspondence over a p-adic field.
3. Present the combinatorial construction of the dual group of a spherical variety due to Knop-Schalke.
4. Connect the exceptional theta correspondence to the F4/Spin9 spherical variety.

Join Zoom Meeting
https://nus-sg.zoom.us/j/84884850855?pwd=bvokLMShAf5wFeAHh2zemmel1dvb9f.1

Meeting ID: 848 8485 0855
Passcode: 226224

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Date/Time : 26 Mar 2025 15:00
Venue : S17 #04-04 (SR3)
Speaker : Kim Sunghan, Uppsala University

Constraint maps are critical points to the Dirichlet energy subject to an image constraint. Namely, the constraint restricts these maps to lie outside a given (smooth, open-connected) set in the target space, which can then be thought of as an obstacle. The target space is multi-dimensional, making this problem a natural extension of the obstacle problem into the vectorial setting. The presence of the obstacle naturally induces free boundary, which is the interface between the set where the image of the maps lies away from the obstacle and the set where their image lies on the obstacle. Yet, constraint maps also develop singularities, such as discontinuous points and branch points which have been the main subjects in the theory of harmonic maps and minimal surfaces, respectively. In this talk, I will discuss the recent breakthroughs on the tantalizing interplay between the free boundary and the mapping singularities for energy-minimizing constraint maps, based on a series of my collaboration with Alessio Figalli, André Guerra (ETH), and Henrik Shahgholian (KTH).

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