Date/Time : 12 Feb 2025 16:00
Venue : S17 #04-04 (SR3)
Speaker : Fukushi Kenji, Kyoto University

The extension of Morse theory to S^1 valued Morse functions, more generally, closed 1-form is called Morse-Novikov theory. Morse-Novikov theory gives a new invariants of manifolds, the Novikov homology. As in Morse homology, the essence is the trajectory of the gradient flow connecting the critical points.

Based on this, Michael Farber extended Lusternik-Schnirelmann category to closed 1-form.

Unlike the ordinary result of Lusternik-Schnirelmann category about the number of critical points of functions, this extended category no longer gives the lower bound of the number of zeros of closed 1-forms. However, if the number of zeros of closed 1-form is less than this category, the dynamics of the gradient flow of the closed 1-form must have a Homoclinic cycle.

In this talk, I will explain the Lusternik-Schnirelmann category of closed 1-form, and I will introduce the extension of the Farber’s result to non compact manifolds.

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Date/Time : 12 Feb 2025 15:00
Venue : S17 #04-04 (SR3)
Speaker : Ni Lei, University of California, San Diego

In this talk, I will review some applications of Lie theory in geometry such as Cartan’s work on symmetric spaces, Hamilton’s observation and Boehm-Wilking’s constructions utilizing Lie brackets and some highly useful considerations from Lie theory. Then I will talk about some more recent applications in my joint/solo works.

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Date/Time : 11 Feb 2025 16:00
Venue : S17 #05-11 (SR5)
Speaker : Travis Scrimshaw, Hokkaido University

Key polynomials originally arose as characters of natural representations of upper triangular matrices coming from Schubert calculus. They have many algebraic properties similar to Schur functions, such as a Cauchy-type formula and form a basis for the polynomial ring. For this talk, we consider them to be special polynomials that are defined recursively by divided difference operators. We recall the Bump-Brubaker-Buciumas-Gustafsson construction using colored solvable lattice models to transform this into a combinatorial interpretation using semistandard tableaux. Using Kashiwara’s theory of crystal base, we recall another way to interpret the tableaux. We will conclude the talk with some recent generalizations of these constructions and open questions. This is based on joint work with Valentin Buciumas and Katherine Weber and with Eric Marberg.

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Date/Time : 11 Feb 2025 14:00
Venue : S17 #05-11 (SR5)
Speaker : Christian Johansson, University of Gothenburg

I will discussion joint work with James Newton and Carl Wang-Erickson, where we give an interpretation of the p-adic local Langlands correspondence for GL(2,Qp) as an embedding of the derived category of locally admissible representation into the category of ind-coherent sheaves on the moduli stack of Galois representations. Time permitting, I will also discuss local-global formulas.

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Date/Time : 10 Feb 2025 16:00
Venue : S16 #03-06
Speaker : Zhang Lei, Shanghai Jiao Tong University

Quantum computing has garnered considerable interest in the past few years owing to its potential to offer up to exponential acceleration compared to classical computational methods. In this talk, I will first introduce the Schrödingerisation method by Jin, Liu and Yu [1], followed by its concrete quantum circuit implementation [2], with examples of heat equation and Maxwell equation [3]. I will further demonstrate applications to linear multiscale PDEs [4] and multiscale Hamilton-Jacobi equations [5].

1. Shi Jin, Nana Liu and Yue Yu, Quantum Simulation of Partial Differential Equations via Schrödingerization,Phys. Rev. Lett. 133, 230602, 2024.
2. Junpeng Hu, Shi Jin, Nana Liu, Lei Zhang, Quantum Circuits for partial differential equations via Schrödingerisation, Quantum, 8, 1563, 2024.
3. Chuwen Ma, Shi Jin, Nana Liu, Kezhen Wang, Lei Zhang, Schrödingerization based Quantum Circuits for Maxwell’s Equation with time-dependent source terms, arXiv:2411.10999, 2024.
4. Junpeng Hu, Shi Jin, Lei Zhang, Quantum algorithms for multiscale partial differential equations, (SIAM) Multiscale Model. Simulation, 22(3), 1030-1067, 2024.
5. Kezhen Wang, Lei Zhang, Quantum Algorithms for Multiscale Hamilton-Jacobi Equation, preprint.

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Date/Time : 07 Feb 2025 11:00
Venue : S17 #04-04 (SR3)
Speaker : Wang Yutong, National University of Singapore

In the first course, we will briefly introduce Lean 4 and verifiable mathematics, then go through together some general usages of LLM assistants to speed up mathematics research, including techniques like in-context-learning and function calling. Then we will use examples to demonstrate how these techniques, combined with expert knowledge and symbolic tools, can contribute to both formal and informal mathematics. Specifically, we will try together some recent tools in Lean 4, to effectively drive formal proof-writing in subsequent courses.
This mini-course is designed for general audiences without a background in LEAN.

This mini-course will live stream. Here are the zoom ID and the link:
https://nus-sg.zoom.us/j/81322620719?pwd=Wg8FVTZbfX4QpZ7vbiYwaBN3n1MO7A.1

Meeting ID: 813 2262 0719
Passcode: 359631

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Date/Time : 06 Feb 2025 13:30
Venue : S17 #04-06 (SR1)
Speaker : Asbjørn Christian Nordentoft, Université Paris-Saclay

In the case of the modular surface Duke-Imamouglu-Toth defined a thin subgroup associated to each closed geodesic and studied the equidistribution of the projection of the associated hyperbolic orbifold with boundary (i.e. the partial coverings). In this talk, I will give a purely geometric generalization to any hyperbolic surface. Furthermore, I will explain that the equidistribution of closed geodesics in the unit tangent bundle implies equidistribution of the associated partial coverings. This yields a new proof of the result of Duke-Imammouglu-Toth as well as many other new equidistribution results.

This is joint work with Ser Peow Tan.

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Date/Time : 05 Feb 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Grigorios Pavliotis, Imperial College London

In this talk we present recent results on the quantitative study of stochastic interacting particle systems and of their mean field limit. We will start by presenting links between uniform propagation of chaos, the absence of phase transitions and Gaussian fluctuations around the mean field limit. We will present different methodologies for inferring parameters in the mean field (McKean) stochastic differential equation from observations of particle paths. Finally, we show how we can identify stationary states of the mean field PDE and how to steer the dynamics towards a chosen steady state using optimal control methodologies.

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Date/Time : 05 Feb 2025 14:00
Venue : S17 #04-06 (SR1)
Speaker : JongHae Keum, Korea Institute for Advanced Study

The surfaces in the title are complex algebraic surfaces with the Betti numbers of the complex projective plane, and are called (rational) homology projective planes. If such a surface has only quotient singularities (a kind of mild singularities), then its minimal resolution is a 2-dimensional complex manifold with zero geometric genus and irregularity. They are 2-dimensional generalizations of the complex projective line (the Riemann sphere), and are closely related to the theory of 4-manifolds. Fake projective planes and the complex projective plane are smooth examples of a rational homology projective plane.

My presentation will begin with basic definitions and examples, and then describe recent progress in the study of such surfaces, singular ones and fake projective planes.

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