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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Date/Time : 27 Nov 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Ruiyi Yang, Princeton University

Gaussian processes (GPs) are important random function models with desirable analytic properties that have found wide applications in inverse problems, spatial statistics, and machine learning. In this talk, we shall investigate a generalization of the popular Matérn GP to the manifold setting. In the first part, we formalize its definition and introduce a graph-based approximation that is computable given only a point cloud of samples from the manifold. The resulting graph Matérn GP enjoys a sparsity structure that facilitates computation and is widely applicable in many aspects of Bayesian methodologies. In the second part, we discuss in detail its application in a manifold optimization problem where gradients are intractable to obtain. Exploiting tools from Bayesian optimization, we propose an efficient algorithm with provable guarantees, whose effectiveness is further demonstrated by numerical examples.

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Date/Time : 28 Oct 2024 10:00
Venue : via Zoom
Speaker : Qijun Tan, Google DeepMind

Join Zoom Meeting
https://nus-sg.zoom.us/j/82351206827?pwd=L6NM7PEa9BfIlfHqeare8XkoY0IKnr.1

Meeting ID: 823 5120 6827
Passcode: 077539

In this talk, we will explain how Large Language Models (LLM) obtain their capability in solving difficult math problems. In particular, We will go over key techniques that unearth the reasoning potential of LLMs. We will also illustrate key challenges in LLM reasoning.

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Date/Time : 17 Oct 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Wang Yutong, National University of Singapore

The growing of verifiable formal languages, particularly interactive proof assistants like Lean 4, is shaping the future of mathematics and transforming how large language models (LLMs) can assist in automated reasoning. However, navigating the vast and complex Mathlib and composing reliable formal statements remains a challenge, with arcane syntax and scattered lemmas.

In this talk, I will present our recent developments: the LeanSearch platform V2 and the Herald translator, which features Lean-aware LLMs to streamline proof writing and auto-formalization of statements. We will explore specialized model training and data augmentation techniques that enhance LLM-accelerated formalization. I will demonstrate how these tools can assist both mathematicians and Lean developers.

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Date/Time : 14 Aug 2024 16:00
Venue : S17 #06-11 (SR6)
Speaker : Felix Schremmer, The University of Hong Kong

This talk presents a novel, interdisciplinary study that leverages a Machine Learning (ML) assisted framework to explore the geometry of affine Deligne-Lusztig varieties (ADLV). The primary objective is to investigate the nonemptiness pattern, dimension and enumeration of irreducible components of ADLV. Our proposed framework demonstrates a recursive pipeline of data generation, model training, pattern analysis, and human examination, presenting an intricate interplay between ML and pure mathematical research. Notably, our data-generation process is nuanced, emphasizing the selection of meaningful subsets and appropriate feature sets. We demonstrate that this framework has a potential to accelerate pure mathematical research, leading to the discovery of new conjectures and promising research directions that could otherwise take significant time to uncover. We rediscover the virtual dimension formula and provide a mathematical proof of a newly identified problem concerning a certain lower bound of dimension. This is joint work with Bin Dong, Xuhua He, Pengfei Jin and Qingchao Yu, and the corresponding paper can be found on arXiv and the Peking Mathematical Journal.

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