Date/Time : 21 Jan 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Yusuf Baris Kartal, University of Edinburgh

Morse theory provides a method for studying the topology of smooth manifolds through the critical points of a function and its gradient flow. In the 1980s, Andreas Floer applied ideas from Morse theory to the infinite dimensional setting, producing an invariant in symplectic topology that has since become a central tool in this field, and later understood to be closely related to enumerative geometry and mirror symmetry. Inspired by Floer’s insight, Cohen-Jones-Segal proposed a framework to obtain finer information from the data arising in Morse and Floer theory. However, this setup is not a priori compatible with the compact group actions, which limits their usefulness in the study of equivariant topology and geometry. In this talk, I will explain how, in joint work with Laurent Cote, we extend the setup and the construction of Cohen-Jones-Segal to study equivariant invariants. I will then discuss applications to topology and symplectic geometry, and explain how tools from homotopy theory can be used to relate symplectic invariants to the topology of the underlying manifold.

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Date/Time : 20 Jan 2025 09:00
Venue : Zoom link TBA
Speaker : Yangyang Li, The University of Chicago

Minimal surfaces, which generalize geodesics to higher dimensions, naturally model various physical phenomena and are fundamental tools in geometric analysis. In this talk, I will survey recent advances on the existence, regularity, and Morse-theoretic study of minimal (hyper)surfaces. Highlights include our recent results on the enumeration of minimal surfaces, the generic regularity of minimal hypersurfaces, and their anisotropic counterparts. I will also discuss how these findings illuminate the interplay between the space of minimal hypersurfaces and the ambient Riemannian manifold.

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Date/Time : 15 Jan 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Yash Lodha, The University of Hawaii at Manoa

The groups of homeomorphisms of the real line and the circle exhibit a remarkable subgroup structure, and consequently, a plethora of topological, algebraic and combinatorial phenomena. In this talk I shall provide some historical motivation to study such groups, and describe some concrete examples that illustrate the theory. I will also present some recent advances in the area.

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Date/Time : 14 Jan 2025 14:00
Venue : S17 #05-11 (SR5)
Speaker : Yu Xuejun, Rafael Cabral, Huawei Singapore Research Center

This talk explores the applications of Large Language Models (LLMs) in the domain of mathematical reasoning. We introduce the technical advancements that enable LLMs to perform tasks ranging from basic arithmetic to advanced theorem proving. Specifically, we present two pioneering research projects being developed at Huawei’s Singapore Research Center: the Formal Proof Agent and the Geometry Problem Solving Agent. The Formal Proof Agent is designed to solve mathematical proof questions without logical errors. It leverages LLMs to automate the generation of formal proofs by interacting with interactive theorem provers like Lean 4, ensuring reliable and rigorous logical reasoning. Meanwhile, in Geometry Problem Solving, there is the additional challenge of interpreting geometry diagrams, which current multimodal LLMs struggle with. Our work leverages computer vision tools to interpret geometry diagrams and, as before, solves geometry problems step-by-step, by combining formal symbolic languages with the creative capabilities of LLMs. These projects highlight the versatility of LLMs in addressing diverse and multimodal mathematical challenges, paving the way for future innovations in AI-driven reasoning.

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Date/Time : 13 Jan 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Yap Jitwu, Harvard University

Arithmetic dynamics is an emerging field that translates problems from arithmetic geometry into a dynamical framework. In this talk, I will provide an overview of arithmetic dynamics and demonstrate how the equidistribution of Galois orbits, due to Szpiro-Ullmo-Zhang, and Yuan, can be applied within our context. I will then discuss a quantitative version of Yuan’s theorem and along with some implications.

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Date/Time : 09 Jan 2025 15:00
Venue : S17 #04-05 (SR2)
Speaker : Gianluca Ceruti, University of Innsbruck

The curse of dimensionality presents a formidable challenge in many fields, from molecular quantum dynamics to computer vision and kinetic equations. This phenomenon, characterized by an exponential growth in parameters with increasing system complexity, demands innovative approaches to enable efficient computation and data handling. In this talk, we delve into the promise of Dynamical Low-Rank Approximation (DLRA) as a model order reduction technique for addressing large-scale, time-dependent problems. By dynamically projecting the dynamics of high-dimensional systems onto a manifold of reduced complexity, DLRA reduces computational costs while preserving essential structural properties. Key to this approach are advanced tensor decompositions, such as the tensor train and hierarchical Tucker formats, which provide a foundation for efficient representation and computation. We will highlight recent advances, including the Basis-Update and Galerkin (BUG) integrator and its extensions, which enhance DLRA’s capabilities for preserving system dynamics and reducing memory requirements. Applications will illustrate DLRA’s versatility, from solving time-dependent partial differential equations to enhancing machine learning inference efficiency and reducing training costs in computer vision.

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Date/Time : 09 Jan 2025 09:15
Venue : S17 Level 3, IMS Executive Seminar Room
Speaker : Various,

Organizing Committee

Co-chairs

• Jun Le Goh (National University of Singapore)

• Yue Yang (National University of Singapore)

To honor Professor Chi Tat Chong on the occasion of his retirement after an extraordinary 50 years of service at the National University of Singapore (NUS), a conference will be held on 9–10 January 2025 at the Institute for Mathematical Sciences (IMS), NUS.

Professor Chong has made profound contributions to the fields of recursion theory, reverse mathematics, and mathematical logic. This conference will highlight his impactful work in computability, definability, and provability, areas where his scholarly influence has been deeply felt.

Beyond his academic achievements, Professor Chong has played a pivotal role in nurturing the mathematical community in Singapore, China, and across Asia. Notably, in 2000, he collaborated with colleagues at NUS to establish the Institute for Mathematical Sciences. In addition, since the Reform and Opening-up in China, he has mentored and supported a new generation of logicians, leaving a lasting legacy in the region.

The two-day conference is jointly organized by the Department of Mathematics, the Faculty of Science, and the Institute for Mathematical Sciences. It will feature a diverse program of talks by distinguished scientists from around the world. The event will bring together Professor Chong’s colleagues and friends, many of whom have cherished his friendship and mentorship over the past five decades.

Activities

Venue

IMS Executive Seminar Room

Block S17, Level 3
10 Lower Kent Ridge Road
Singapore 119076

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Date/Time : 05 Dec 2024 15:30
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Igor Balla, Masaryk University

Despite being elementary objects in geometry, large sets of equiangular lines appear naturally in a wide variety of fields including elliptic geometry, frame theory, Banach space theory, spectral graph theory, and quantum information theory with connections to algebraic number theory and Hilbert’s twelfth problem. In 1973, Lemmens and Seidel posed the problem of determining the maximum number of equiangular lines in an r-dimensional Euclidean space with a given angle. In this talk, we will present new upper bounds on this problem that unify and significantly extend or improve essentially all previously known results, as well as being best possible either exactly or up to a factor of 2. These recent results were obtained independently or together with Matija Bucic.

Roughly speaking, our approach relies on new lower bounds on the second eigenvalue of a graph as well as new upper bounds on its multiplicity. In particular, we obtain the first extension of the Alon–Boppana theorem to dense graphs, with equality for strongly regular graphs corresponding to families of r+1 choose 2 equiangular lines in r dimensions. Time permitting, we will also mention some results in the complex setting.

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Date/Time : 04 Dec 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Thomas Karam, University of Oxford

This talk shall be essentially composed of four parts, each consisting of a short introduction to a research area followed by some contributions. We shall begin by discussing a central conjecture in Ramsey theory unifying several generalisations of the van der Waerden theorem. Thereafter, our focus shall move to the extension of basic properties of the rank of matrices to the different notions of rank of tensors, followed by the extension of basic properties of polynomials over finite fields to restrictions of these polynomials with Boolean variables. Finally, we shall study the Brunn-Minkowski inequality in more arbitrary ambient spaces than R^n, in particular in the torus (R/Z)^n for subsets satisfying various levels of constraints.

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