Date/Time : 18 Nov 2026 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Kenji Nakanishi, Kyoto University

Since the pioneering work of Kenig and Merle, there have been numerous results on global behavior of large general solutions to nonlinear dispersive equations that vary depending on the initial data, from asymptotically linear or scattering solutions to solitons and blow-up. The main question is to distinguish the types of solutions or to predict them from the initial data, without any asymptotic solution formula such as those provided by the inverse scattering theory in the special case of completely integrable systems. Despite of some spectacular progress such as the soliton resolution without size restriction, the range of applicable equations has been rather limited, such as the nonlinear wave equation or Schrodinger with simple nonlinearity, because certain monotonicity or invariance properties play crucial roles in the analysis without small factor. The Zakharov system is a model of the Langmuir turbulence in plasma, coupling the Schrodinger and the wave equations, and may be regarded also as a mathematical model to extend the methods from the simple scalar equations. It has been quite successful in the last decade for space dimensions 3 and 4, in spite of a few difficulties such as the derivative loss, the lower degree of nonlinear interactions, and the loss of invariance for scaling and for momentum or coordinate velocity. This talk will review the state of art, together with some detail on most recent results and open problems.

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Date/Time : 07 Oct 2026 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Kyu-Hwan Lee, University of Connecticut

The remarkable advances in AI are having a profound impact on mathematical research and are beginning to bring about substantial changes in both the practice and the possibilities of doing mathematics. In this talk, I will discuss these changes through the example of the recent resolution of the inverse Galois problem for $M_{23}$. In particular, I will illustrate how AI can contribute to mathematical exploration and computation, the discovery of new patterns, and the process of solving challenging problems. I will also offer some perspectives on how continued advances in AI may reshape the way mathematical research is conducted in the years ahead.

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Date/Time : 07 Oct 2026 16:30
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Shirley Ho, Flatiron Institute

Foundation models like GPT-4 have dramatically altered the modern work landscape for many industries reliant on language tasks, but no equivalent model exists yet for scientific applications. Incorporating foundation models into research workflows could enable unprecedented discoveries that connect traditionally distinct scientific sub-disciplines. However, mainstream foundation models trained on human-scale datasets will be insufficient for analyzing most scientific phenomena — a foundation model for science will require special consideration for the requirements of scientific datasets, especially those with wide dynamic ranges.

In this talk , I will introduce the Polymathic AI initiative: our goal is to accelerate the development of versatile foundation models tailored for numerical datasets and scientific machine learning tasks. The challenge we are undertaking is to build AI models which leverage information from heterogeneous datasets and across different scientific fields, which, contrary to domains like natural language processing, do not share a unifying representation (i.e., text). Such models can then be used as strong baselines or be further fine-tuned by scientists for specific applications. This approach has the potential to democratize AI in science by providing off-the-shelf models that have stronger priors (i.e., background knowledge) for shared general concepts such as causality, measurement, signal processing, and even more specialized shared concepts like wave-like behavior, which otherwise would need to be learned from scratch.

I will present our initial papers and projects, including large scientific datasets designed for large scale training “MultiModal Universe” and “The Well”.

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Date/Time : 02 Oct 2026 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Enrique Zuazua, Friedrich-Alexander-Universität Erlangen-Nürnberg

Machine Learning is creating a new universe of systems that compute, learn, interact, and act. Understanding their behavior raises fundamental mathematical questions about representation, inference, dynamics, optimization, and control.

This lecture introduces a mathematical framework for studying high-dimensional, adaptive, and interconnected intelligent systems. I will discuss examples from applied mathematics: neural networks viewed as dynamical systems, diffusion models and time reversal, neural methods for scientific computing, and hybrid modelling that combines physical knowledge with data-driven components.

The aim is to show how classical mathematics is contributing to the foundations of Artificial Intelligence, while intelligent learning systems themselves open new questions for mathematics.

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Date/Time : 30 Sep 2026 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Jack Thorne, University of Cambridge

Although computing the set of solutions to a given Diophantine equation is hard (!), the solution sets of families of equations often display striking statistical regularity – this is the case, for example, for the family of Weierstrass elliptic curves over Q given by the equations y^2 = x^3+ Ax + B. For a randomly chosen curve in this family, one expects to see either no solutions or infinitely many solutions, each case occurring with probability 1/2. Striking results in this direction have been proved for these curves and for the hyperelliptic curves y2 = x^{2n+1} + … + c_{2n+1} in the Fields medal-winning work of Bhargava (with co-authors including Shankar and Gross).

One of the main ingredients in these works is a bridge between rational points and the space of lattices (i.e. discrete, cocompact subgroups in Euclidean space) — a Riemannian manifold of finite volume that plays a recurring role in number theory, and is familiar (in the rank 2 case) as the quotient of the upper half plane by the group SL(2, Z). I will survey this picture and describe recent work with Jef Laga, which shows that we can use this formalism to understand not only the number of rational points but their physical properties (e.g. their heights).

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Date/Time : 23 Sep 2026 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Frank den Hollander, Leiden University

In this talk I present an overview of what is known and is not known about random processes on dynamic random graphs in co-evolution, i.e., with mutual feedback. The literature offers plenty of heuristics, simulations and conjectures, but so far mathematical results are extremely scarce. I describe some of these results, with a focus on opinion dynamics. In particular, I consider random graphs in which vertices can have one of two possible opinions. Pairs of vertices connected by an edge share their opinions according to a rate that may depend on the size of the graph. Each edge turns on or off according to a rate that depends on whether the vertices at its two endpoints have the same opinion or not. I exhibit joint evolution equations and describe the occurrence of phase transitions between consensus and polarisation.

Based on joint work with Siva Athreya (Bangalore) and Adrian Rollin (Singapore).

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Date/Time : 12 Aug 2026 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Boaz Klartag, Tel Aviv University and Weizmann Institute of Science

We prove there exists a lattice sphere packing in R^n whose density is at least c n^2 / 2^n, where c > 0 is a universal constant. Previously known constructions of sphere packings in R^n had densities of the order of magnitude of n / 2^n, up to logarithmic factors. Our proof utilizes a stochastically evolving ellipsoid that accumulates at least c n^2 lattice points on its boundary, while containing no lattice points in its interior except for the origin.

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Date/Time : 29 Jul 2026 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Zhongkai Tao, Institut des Hautes Études Scientifiques

The fractal uncertainty principle, introduced by Dyatlov and Zahl, has become an important tool in the study of spectral gaps for systems with trapping. It quantifies how a function and its Fourier transform can both concentrate near fractal sets. I will talk about some recent progress in the fractal uncertainty principle and its applications to spectral problems. This talk is partly based on joint works with Aidan Backus, James Leng, Xiaoqi Huang, Christopher Sogge, Zhexing Zhang, Alex Cohen and Zhenhao Li.

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Date/Time : 29 Apr 2026 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Juncheng Wei, The Chinese University of Hong Kong

In 2023, one year before his death, H. Brezis posted 28 open questions which he described as “challenging and difficult”. In this talk I will give complete answer to the first open question—Problem 1.1 on Brezis-Nirenberg problem on three dimensional unit ball, and the seventh open question—Problem 3.1 on harmonic map equation on disk.

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