Date/Time : 10 Mar 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Jie Ma, University of Science and Technology of China

A fundamental theorem of Rademacher from 1941 led to the study of supersaturation problems of graphs, which aim to count the minimum number of copies of a given graph F among all graphs with n vertices and m edges. This is closely related to a central concept in Extremal Graph Theory — the Turán number of F, which denotes the maximum number of edges in an n-vertex graph which does not contain F as a subgraph. Famous results of Erdős, and Lovász and Simonovits determine the minimum number of cliques K_r in graphs whose number of edges exceed the Turán number of K_r. Subsequent works of Mubayi as well as Pikhurko and Yilma extend these classical results from cliques to color-critical graphs, a rich family playing important roles in extremal problems. In this talk, we will discuss supersaturation problems beyond color-critical graphs and investigate natural enumerative parameters.

Our results go beyond the previous results and show that supersaturation problems for general graphs can be rather complicated. Among others, we disprove a conjecture of Mubayi. Joint work with Long-Tu Yuan.

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Date/Time : 09 Mar 2023 11:00
Venue : S17 #05-12 (SR4)
Speaker : David Hansen, National University of Singapore

In recent work, Fargues-Scholze constructed a natural candidate for the local Langlands correspondence (LLC) for all reductive groups over local fields, and showed that it matches the Harris-Taylor-Henniart LLC for GL_n. It is a natural problem to ask whether their construction agrees with previously known constructions of LLC for more general groups. After giving some motivation for this problem and discussing previously known results, I will sketch a proof that for SO_2n+1 and its unique inner form, the Fargues-Scholze LLC agrees with the LLC constructed by Arthur and Moeglin.

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Date/Time : 08 Mar 2023 16:00
Venue : S17 #04-05 (SR2)
Speaker : Jurgen Jost, Max Planck Institute for Mathematics in the Sciences

Hypergraphs naturally emerge as models in many fields, ranging from chemical reaction systems to collaboration networks. Their mathematical theory, however, was underdeveloped until recently. I shall introduce some new ideas and concepts that we have developed, describe some of the interesting new mathematical developments and show some novel phenomena in pattern formation.

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Date/Time : 08 Mar 2023 17:00
Venue : S17 #05-11 (SR5)
Speaker : Chong Chitat, National University of Singapore

Let T be the full infinite binary tree (i.e. the Cantor space). An infinite subtree S of T is a strong subtree if (i) S is isomorphic to T, (ii) if sigma is a node in S, and sigma_0, sigma_1 are its immediate successors in S, then sigma_0 and sigma_1 have the same length in T and furthermore (iii) the intersection of sigma_0 and sigma_1 is sigma.

The Halpern-Lauchli Theorem is a theorem in combinatorial mathematics which states that for any finite number of infinite full binary trees T_1,T_2,…, T_d, and any finite coloring of the d-row vectors in T_1 x
T_2 x … x T_d, there exist strong subtrees S_1 subseteq T_1, S_2 subseteq T_2, …, S_d subseteq T_d for which all d-row vectors in S_1 x S_2 x … x S_d have the same color.

This Ramsey type theorem is yet another striking demonstration of the existence of order within chaotic disorder. There is no known simple proof of this theorem in the published literature.

In this talk we discuss the proof-theoretic strength of the Halpern-Lauchli theorem from the reverse mathematics point of view. In particular, we give a characterisation of the inductive strength of this theorem as well as its conservation strength over a base theory weaker than Sigma_2-induction.

This is joint work with Li Wei and Yang Yue.

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Date/Time : 03 Mar 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Jun-Muk Hwang, Institute for Basic Science, Korea

When an analytic equation has a formal power series solution, does it have a convergent power series solution?

It has been well-known that this question has an affirmative answer if we work in local setting. The question is much subtler if we work in global setting. A typical example is Grauert’s formal prinicple for compact complex submanifolds in complex manifolds. We give an introductory overview of the results on the formal principle for rational curves in complex manifolds.

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Date/Time : 02 Mar 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Wu Yifei, Tianjin University

In recent years, more and more attention has been paid to the low regularity numerical study based on the practical needs. In this talk, some Fourier integrators are proposed for solving the KdV equation and the nonlinear Schrodinger equation. The designation of the scheme is based on the exponential-type integration, Splitting methods and the Phase-Space analysis of the nonlinear dynamics. By the rigorous analysis, the new schemes provide the first-order or second-order accuracy in Sobolev spaces for rough data, and reduce the regularity requirement of existing methods so far for optimal convergence. Moreover, the conservation laws of the numerical solutions are considered.

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Date/Time : 02 Mar 2023 11:00
Venue : S17 #05-12 (SR4)
Speaker : Luo Yi, National University of Singapore

In a recent work by Tasho Kaletha, local langlands conjectures were extended to a certain class of disconnected groups. Moreover, a local Langlands correspondence for disconnected tori was established, where the main ingredient is Tate-Nakayama duality in Galois hypercohomology. In this talk, we review and reformulate Kaletha’s construction. Using these local data, we show an automorphic multiplicity formula for disconnected tori over a number field.

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Date/Time : 01 Mar 2023 15:00
Venue : via Zoom
Speaker : Shukun Wu, California Institute of Technology

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

Meeting ID: 810 4646 2775
Passcode: 23571113

The Kakeya conjecture suggests a fundamental phenomenon in Euclidean space: (thin) tubes with different directions are essentially disjoint. The problem also arises naturally in other contexts, (e.g. Fourier analysis, geometric measure theory), and it often serves as a main enemy—tool. In this talk, I will briefly survey the conjecture and some techniques people developed to attack it. Along the way, I will also discuss some recent progress and applications.

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Date/Time : 01 Mar 2023 17:00
Venue : via Zoom
Speaker : Linus Richter, Victoria University of Wellington

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

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

A recent “point-to-set principle” of Jack Lutz and Neil Lutz characterises the Hausdorff dimension of any subset of Euclidean space in terms of the Kolmogorov complexity of its individual points.
Sets with particular fractal properties can now be constructed point-by-point, by coding “enough” information into each point, bit-by-bit.

After introducing the point-to-set principle, I will present a new result in fractal geometry: under V=L (the axiom of constructiblity), I will outline the construction of co-analytic sets of Euclidean space which fail Marstrand’s Projection Theorem, a classical result in fractal geometry concerning the dimension of orthogonal projections of analytic plane sets onto lines.

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