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 : 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

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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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Date/Time : 12 Aug 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Daniel Bartl, University of Vienna

A major challenge in modern optimization problems is their instability, where small data perturbations can have disproportionately significant effects. We address this challenge in two settings. First, we focus on a distributionally robust approach, a popular method to counteract instability generically using optimal transport. Specifically, we demonstrate how this approach can systematically quantify the (in-)stability of a given optimization problem. In the second part, we focus on statistical robustness, dealing with data subject to outliers, such as those caused by adversarial corruption. We design robust estimators for optimization problems and risk measures.

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Date/Time : 08 Jul 2024 16:00
Venue : S17 #04-06 (SR1)
Speaker : Panel Discussion: Various speakers,

The leap from higher education to industry is a pivotal stage for many graduate students. Emphasising the necessity of an interdisciplinary approach and the application of the robust analytical skills honed during graduate studies, this panel discussion offers a comprehensive exploration of the journey that awaits MSc and PhD graduates who are looking to bridge the gap between academic accomplishment and professional fulfilment in the industry. Panellists will provide candid reflections on their personal pathways, highlighting the obstacles they encountered and the successes they achieved within various industrial domains. Attendees of the session will be encouraged to engage with live inquiries and raise pertinent topics for discussion. Discussions will revolve around the symbiotic relationship between advanced academic preparation and industry expectations, strategic planning for entering the corporate sector, the diverse opportunities ranging from employment in small to large-scale organisations to the launch of deep tech ventures, as well as the innovative trends and fruitful careers found in today’s industrial research and development.

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