A classic problem in topology is determining how many smooth structures a given manifold can admit. We say that a manifold admits exotic smooth structures if there are at least two distinct smooth structures on the manifold. Understanding these exotic phenomena has been a central topic in both geometric and algebraic topology. Ever since Milnor discovered exotic 7-spheres, there has been extensive research searching for exotic smooth structures on various manifolds. Furthermore, the work of Donaldson and Freedman opened a new world for exotic phenomena in dimension 4. In this talk, we will provide an overview of exotic smooth structures on topological manifolds, review their history, and explore cutting-edge developments in the field.

ML/DL method are widely used for prediction in biomedical research, but predictive performance alone often provides limited scientific insight. Explainable artificial intelligence (XAI) and visualization methods can help identify important variables, characterize complex relationships, and generate interpretable findings beyond prediction accuracy. In this seminar, I will introduce practical XAI and visualization approaches and demonstrate their applications using two biomedical examples: drug-related expression prediction in human and mouse liver data, and analysis of hypnotic medication timing and use patterns. These examples illustrate how predictive modeling can be extended toward interpretation, hypothesis generation, and data-driven scientific discovery.

In critical PDEs, finite-time blow-up may occur while the scaling-critical norm remains bounded, with singularity formation arising from concentration and loss of compactness. In this first talk, we consider the radial focusing energy-critical wave equation in three dimensions and discuss the role of concentration-compactness and profile decomposition in the study of Type II blow-up. We then introduce radiation for the radial free wave equation and the channel of energy method. These ideas will provide the starting point for the rigidity analysis in the second talk. [References] 1. H. Bahouri, P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999), 131–175. 2. C. E. Kenig, A. Lawrie, B. Liu, W. Schlag, Channels of energy for the linear radial wave equation, Adv. Math. 285 (2015), 877–936.

Zeta and L-functions have been classically associated, first to schemes of finite type over Z, and then to the l-adic cohomology of smooth projective varieties over a global field. In the latter definition, independence from l and actual existence remain partly conjectural in characteristic 0. In these lectures, I will explain how to associate unconditionally an L-function to objects in triangulated categories of motives. For the motive of a smooth projective variety this definition differs from the one above in general, but only up to a finite number of Euler factors. This might be useful to tackle the Beilinson conjectures.

We discuss lower bounds on Lyapunov exponents for linear PDEs driven by random velocity fields, including the advection–diffusion equation and the linearized stochastic Navier–Stokes equations. In particular, we show that the exponential rate of decay of the L^2-norm for solutions to the advection–diffusion equation is optimal. A key ingredient in the proof is high-frequency stochastic instability, arising from the non-degeneracy of the driving noise. This talk is based on joint work with Martin Hairer, Tommaso Rosati, and Sam Punshon-Smith.

We present some basics of L-functions usually (to be) covered in the undergraduate and graduate entry level number theory classes.

We present some basics of L-functions usually (to be) covered in the undergraduate and graduate entry level number theory classes.

The stability theory of entropy solutions to the compressible Euler system on a half-line with prescribed boundary conditions, such as inflow, outflow, and impermeable boundary conditions, remains largely open. It is worth emphasizing that establishing the stability of even a single reflected shock at the boundary has been a highly challenging problem. In this talk, I will briefly review the classical stability theory for small-BV entropy solutions in the whole-space setting and then turn to the half-line problem, where I will present recent progress on the stability of entropy solutions in the presence of a boundary.