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 : 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 : 16 Sep 2026 16:30
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Jackie Lok, National University of Singapore

Recently, a class of algorithms based on combining classical fixed-point iterations with repeated random sparsification has been successfully applied to very large-scale linear algebra problems, where storing the full solution vector itself may be prohibitive. However, a general mathematical understanding of when this approach can achieve dimension-independent computational and storage requirements remains limited.

In this talk, we study a randomly sparsified power method for computing the leading eigenvector of a column-stochastic matrix. We prove a dimension-independent error bound that depends on the approximate sparsity of the solution vector and the mixing time of the Markov chain associated with the matrix, establishing the possibility for high-quality approximation in very high dimensions. On the other hand, we show that a deterministically sparsified counterpart can fail, demonstrating a provable separation between deterministic and random sparsification.

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Date/Time : 09 Sep 2026 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Miri Son , National University of Singapore

Recently, Fisher and Melnick classified SL(n,R)-actions on n-dimensional manifolds for n≥3. In this talk, we generalize their result by classifying smooth and real-analytic SLnR-actions on m-dimensional manifolds for 3≤n≤m≤2n-3. This problem is related to the Zimmer program, since any action of a Lie group restricts to an action of its lattices.

This classification relies on linearization of SL(n,R)-actions near a fixed point. The analytic case was proved by Guillemin-Sternberg and Kushnirenko. In the smooth case, the key is the linear representation of SL(n,R). We discuss the smooth case, which is ongoing joint work with Insung Park.

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Date/Time : 09 Sep 2026 13:30
Venue : S16 #05-18
Speaker : Toshihisa Kubo, Ryukoku University

Let X be a smooth manifold and Y a smooth submanifold of X. Let G’ ⊂ G be Lie groups, with G acting on X and G’ acting on Y. A differential operator D from the space of smooth sections of a G-equivariant vector bundle over X and to that of a G’-equivariant vector bundle over Y is called a differential symmetry breaking operator (DSBO for short) if D is G’-intertwining.

In general, the classification and construction of DSBOs are challenging problems. Around 2010, Toshiyuki Kobayashi introduced a fascinating idea for addressing these problems when both X and Y are flag manifolds. We refer to the resulting machinery as the F-method. The F-method reduces the classification and construction of DSBOs to solving a certain system of partial differential equations via the algebraic Fourier transform of generalized Verma modules. In this talk, we shall first discuss recent progress on the F-method. We shall then give explicit formulas for the DSBOs for (G,G’)=(GL(3,R), GL(2,R)) with minimal parabolic subgroups. This talk is based on joint work with Jonathan Ditlevsen and Víctor Pérez-Valdés.

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Date/Time : 09 Sep 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Beomjun Choi, Korea Advanced Institute of Science & Technology (KAIST)

We will discuss how collective behavior in interacting particle systems can lead to qualitative changes in equilibria, and how these equilibria behave dynamically. The first part concerns long-time convergence for Keller–Segel equation, and more generally McKean–Vlasov equations, in relation to phase transitions and bifurcation. The second part concerns the local stability of mixed Nash equilibria for mean-field Langevin descent–ascent, where exponential convergence can be established despite the non-gradient and non-self-adjoint structure. We will also discuss a corresponding finite-particle stability result.

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Date/Time : 07 Sep 2026 15:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : David Hansen, National University of Singapore

I will formulate a collection of vanishing conjectures in the Fargues-Scholze setting, which describe the maps on suitable Grothendieck groups induced by Eisenstein functors, constant term functors, and the gluing functors between strata of Bun_G. I will also discuss the relationships among these conjectures, and explain how they imply a very general form of the Bertoloni Meli-Harris-Viehmann conjecture (which I will also recall in detail). I will then sketch a proof of the vanishing conjectures (and hence a proof of the generalized BMHV conjecture). Joint work in progress with Linus Hamann and Peter Scholze.

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