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 : 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 : 26 Aug 2026 17:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Rod Downey, Victoria University of Wellington

A computable Banach space X = (X, || · ||, +, ·) is a Banach space with a computable dense sequence of points which makes X a computable metric space using the norm, for which the operations are computable functions.  Most, but not all, familiar spaces of analysis and physics commonly met are such spaces. In the same way as classical vector space theory revolves around (Hamel) bases, Banach spaces are best understood using what is called a Schauder basis. There are many, many questions around this concept, both classically and in the effective content.

In this talk I will overview many of the questions and results we have so far, and particularly report on some recent work seeking to understand the effective content of Kadets’ Theorem, which states (more or less) that all infinite dimensional Banach spaces look the same if we forget the norm: any two infinite-dimensional, separable Banach spaces are homeomorphic as topological spaces. Surprisingly, this result can be proven for computable Banach spaces using only the halting problem, which seems rare in effective topology.

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Date/Time : 26 Aug 2026 14:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Zach Turner, Duke University

Chemotaxis plays a crucial role in the facilitation of biological and ecological processes, often to significantly improve reaction rates between two or more interacting species. One important example is signaling in the immune system; immune cells release chemokines when they sense invaders, which mediates the chemotaxis of lymphocytes towards the site of infection. Another example is reproduction, in which spermatozoa follow the chemical gradient of a chemoattractant released by oocytes. In this talk reporting on a work in progress, we consider a system of two PDEs which describe the flux-limited chemotaxis of two reacting densities. We build on the results of An et al. J. Math. Biol. 92, 57 (2026)., by treating the three-dimensional case. We argue that whether significant reaction enhancement occurs is largely governed by the spatial scale of the initial concentration of the attracted density, distinguishing it from two-dimensional chemotaxis. In particular, we show that there are reasonable scenarios where three-dimensional chemotaxis could possibly fail to practically improve biological reaction rates.

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Date/Time : 26 Aug 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Shi-Zhuo Looi, California Institute of Technology

Dyadic shell models are infinite systems of ODEs that retain the quadratic nonlinearity, energy identity, and scaling of the 3D incompressible Navier-Stokes equations. Such models can be realized as averaged Navier-Stokes equations at the PDE level, and some of them blow up in finite time despite satisfying the energy identity, showing that these features do not preclude singularity formation. Among dyadic models with these features, the Obukhov model is distinguished by its regulated cascade mechanism: the inviscid version is globally regular. However, the inviscid proof relies on exact energy conservation, which viscosity destroys. We prove that the viscous Obukhov model is globally regular for all initial data in the critical Sobolev space and above, obtaining a sharp threshold exponent result. The proof introduces a critical rescaling that shows a viscous activation threshold for each shell, and establishes regularity by showing that two competing scales, viscous damping and nonlinear growth, become incompatible at high frequencies. Time permitting, we discuss a proof of global regularity for the Katz-Pavlovic dyadic model close to the sharp threshold exponent (above which blowup is known).

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Date/Time : 20 Aug 2026 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Yongnam Lee, Institute for Basic Science

Motivated by DeVleming’s work on moduli of surfaces in P^3 and Chen-Hu-Jiang’s work on moduli of threefolds with volume 2 and geometric genus 4, we study the deformation of pairs of P^3 and hypersurfaces using the classification of Q-Gorenstein degenerations of P^3 with canonical singularities. We prove that if a degenerating threefold has canonical singularities, then the moduli space is smooth at the corresponding pair. Consequently, we find some boundary divisors of the moduli of smooth hypersurfaces. Finally, using the double cover method, we derive some information on the moduli space of threefolds X with canonical singularities with the same volume and geometric genus as a double cover of P^3 branched over a hypersurface. This is joint work with Jungkai Chen and Phin-Sing Soo.

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Date/Time : 20 Aug 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Minsung Kim, Pohang University of Science and Technology

In chaotic systems, the mixing property is known for the fast decay of correlation. It is called rapid mixing if the correlation function decays super-polynomially. The mixing mechanism for hyperbolic systems and their compact group extensions was studied by Dolgopyat in a series of his papers in the late 90s’.

In this talk, we prove rapid mixing for almost all random walks generated by m larger equal than 2 translations on an arbitrary nilmanifold. For several classical classes of nilmanifolds, we show m=2 suffices. This provides a partial answer to the question raised in the work of Dolgopyat (’02) about the prevalence of rapid mixing for random walks on homogeneous spaces.

This is joint work with Dmitry Dolgopyat and Spencer Durham.

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Date/Time : 19 Aug 2026 16:30
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Liu Lei, Chongqing University

Nonlinear waves on nonzero asymptotic backgrounds, such as breathers and rogue waves, have been hot research topics in recent years. Starting with the three-wave resonant interactions system, we identify a specific rogue waves-building mechanism based on modulational instability (MI) which includes zero wavenumber in the gain band. An essential finding is that this mechanism works solely under a linear relation between the MI gain and a vanishingly small wavenumber of the modulational perturbation. We also report the emergence of a previously unexplored species of solitary waves-time-localized solitons, which coincides with the condition for the occurrence of the zero-wavenumber-gain MI. These results can be extended to the Massive Thirring model

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Date/Time : 19 Aug 2026 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Silu Yin, Hangzhou Normal University

In this talk, I will introduce a Morawetz type estimate for the wave equations in $\R^2$ and develop a non-negative Morawetz type energy. With this estimate in hand, an alternative proof of global existence for the Cauchy problem of quasilinear wave equation with small and compactly supported data is given. What is more, such weighted Morawetz type energy estimate also works for certain wave system with multiple speeds, which can be used to prove global existence of some admissible harmonic elastic wave system in $\R^2$.

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