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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Date/Time : 05 Feb 2025 17:00
Venue : S17 #04-04 (SR3)
Speaker : Frank Stephan, National University of Singapore

The talk presents joint work with David Belanger (Singapore), C.T.~Chong (Singapore), Rupert Hoelzl (Munich), Gordon Hoi (Singapore), Sanjay Jain (Singapore), Dilip Raghavan (Singapore), Frank Stephan (Singapore), Daniel Turetsky (Wellington) and Jing Zhang (Toronto) published in four papers. The first one dealt with comparing the growth behaviour of functions or families of functions partial-recursive relative to an oracle and classified oracles with respect to this measure. The second, third and fourth papers are about reverse mathematics and study weakly represented families, that are families of functions computed relative to some oracle in the second order part of the model (all functions in the family use the same oracle). These families are used as a tool to define notions like dominated (every weakly represented family is dominated by some function in the model), hyperimmune, biimmune and so on. Furthermore, one can such families also use as a hypothesis space for classes to be learnt – this extends the study to inductive inference. This is a survey talk with sample proofs.

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Date/Time : 04 Feb 2025 14:00
Venue : S17 #05-11 (SR5)
Speaker : David Hansen, National University of Singapore

The Bernstein center of a p-adic reductive group G is a beautiful and explicit commutative ring which acts on “everything” related to the representation theory of G. In recent years, the idea has emerged that this ring contains a canonical subring – the stable Bernstein center – which should be intimately related with the local Langlands correspondence. However, while it is easy to define the stable Bernstein center, it is very difficult to exhibit elements in this subring. On the other hand, recent work of Fargues-Scholze defines another totally canonical subring of the Bernstein center, whose construction uses V. Lafforgue’s theory of excursion operators adapted to the Fargues-Fontaine curve. After reviewing these stories, I’ll sketch a proof that the Fargues-Scholze subring is actually contained in the stable Bernstein center, for all G.

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Date/Time : 03 Feb 2025 16:30
Venue : S16 #03-04
Speaker : Lin Bo, Huairou Laboratory

In this talk, I will introduce a flexible and high-order hybridizable discontinuous Galerkin (HDG) scheme with harmonic averaging technique to solve the drift-diffusion model for semiconductor devices. The proposed scheme combines the robustness of the Scharfetter-Gummel method, which is a classical discretization for this model, with the high-order accuracy and hp-flexibility by the HDG scheme. Gradient-based indicators are introduced to utilize the hp-adaptivity. Numerical examples suggest that the proposed scheme does not exhibit oscillations even when applied to heavily doped and sharp PN-junctions.

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Date/Time : 27 Jan 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Wil Schilders, President of ICIAM, Eindhoven University of Technology & TU Munich-Institute for Advanced Study

Scientific machine learning (SciML) has been taking the academic world by storm as an interesting blend of traditional scientific modeling with machine learning (ML) methodologies like deep learning. While traditional machine learning methodologies have difficulties with scientific issues like interpretability, and enforcing physical constraints, the blend of ML with numerical analysis and differential equations has evolved into a novel field of research which overcome these problems while adding the data-driven automatic learning features of modern machine learning. Many successes have already been demonstrated, with tools like physics-informed neural networks, universal differential equations, deep backward stochastic differential equation solvers for high dimensional partial differential equations, and neural surrogates showcasing how deep learning can greatly improve scientific modeling practice. Consequently, SciML holds promise for versatile application across a wide spectrum of scientific disciplines, ranging from the investigation of subatomic particles to the comprehension of macroscopic systems like economies and climates.

However, despite notable strides in enhancing the speed and accuracy of these methodologies, their utility in practical and specifically industrial settings remain constrained. Many domains within the scientific community still lack comprehensive validation and robustness testing of SciML approaches. This limitation is particularly pronounced when confronted with complex, real-world datasets emanating from interactions between machinery and environmental sensors as usually addressed in industry. Still if appropriately addressed, SciML with its promise to accelerate innovations and scientific discoveries by orders of magnitudes, offers unique opportunities to address the insatiable desire for faster and more accurate predictions in many fields.

This presentation is dedicated to exploring recent advancements in the implementation of SciML techniques. We will discuss how methodologies can be refined to ensure their practical viability and scalability, particularly in industrial sectors where digital and physical components converge.

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Date/Time : 24 Jan 2025 15:00
Venue : S17 #04-04 (SR3)
Speaker : Zhihan Wang, University of Chicago

I’ll discuss an upcoming work joint with Otis Chodosh, Christos Mantoulidis and Felix Schulze, in which we prove generic regularity of area-minimizing hypersurfaces in ambient dimension 11, extending the previous result of my co-authors in dimension 9 & 10.

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