Georgia Institute of Technology College of Sciences

Brill--Noether theory studies linear systems on an algebraic curve. K3 surfaces have been central in Brill--Noether theory since Lazarfeld's proof of the Brill--Noether theorem, which describes the linear series on a general curve of genus $g$. In this talk, we'll survey some recent results concerning the Brill--Noether theory of curves on K3 surfaces. Specifically, we'll discuss a refined Brill--Noether theory, which aims to understand the linear series on curves with a given Brill--Noether special linear series, and related unstable vector bundles on K3 surfaces with applications to the dimensions of components of Giesker--Petri loci in $\mathcal{M}_g$, parameterizing curves admitting a linear series with non-injective Petri map.

TBA

Wave turbulence theory provides a unifying framework for analyzing the nonlinear interactions of dispersive waves across diverse physical regimes. These interactions manifest over an extensive range of scales and media, encompassing phenomena such as gravitational waves in astrophysical contexts, surface waves in fluid dynamics, coherent structures in quantum systems, and even pattern formation in everyday fluid flows.

Despite their widespread occurrence, the mathematical analysis of such systems remains exceptionally challenging due to the interplay between nonlinearity, dispersion, and randomness. These difficulties have catalyzed the development of sophisticated analytical and probabilistic tools, drawing from partial differential equations, statistical mechanics, and dynamical systems theory.

In this lecture, I will present a short survey of recent advances in the rigorous treatment of wave turbulence, emphasizing key techniques, foundational results, and persistent open problems. Some attention will be given to the derivation and analysis of kinetic equations and the role of resonant interactions in the long-time dynamics of weakly nonlinear wave systems.

Local invariant manifolds provide a natural way to separate center,
stable, and unstable dynamics near an invariant object and, at the
same time, to construct lower-dimensional models of the dynamics. In
this talk, I will present a unified numerical procedure for computing
high-order local invariant manifolds attached to fixed points and
reducible quasi-periodic invariant tori of Poincaré maps. The
fixed-point setting appears naturally as the zero-dimensional case of
the quasi-periodic formulation.

For a quasi-periodic skew-product, the normal variational dynamics
along the torus defines a linear cocycle over a rigid
rotation. Assuming reducibility, a Floquet transformation converts
this cocycle into a constant matrix, so that a spectral splitting can
be introduced as in the fixed-point case. The desired invariant
manifold is then represented as a graph over a selected spectral
subspace. Expanding this graph in Fourier–Taylor series leads, order
by order, to cohomological equations whose solvability is governed by
the interaction between the base rotation and the Floquet
spectrum. The same construction also provides the nonlinear dynamics
induced on the selected variables, yielding an explicit reduced model
together with a lifting to the full phase space.  

I will illustrate the method first on a (quasi-)periodically forced
pendulum, which provides a simple setting in which to visualize the
response torus, its center reduction, and the dynamics generated by
the reduced map. Time permitting, I will also discuss how the same
ideas arise in (quasi-)periodically forced models from celestial
mechanics.


Joint works with M. Barcelona, L. Domingo, and M. Jorba-Cusco.

TBA