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.

Ptychographic phase retrieval is an ill-posed nonlinear inverse problem typically formulated as large-scale nonconvex optimization. I will present stochastic multigrid methods, beginning with the known-probe case as a foundation for blind ptychography. The approach uses quadratic surrogate models that majorize the data-misfit objective, together with compatible coarse-grid models that accelerate surrogate minimization. We extend this construction to blind ptychography, where the object and probe must be recovered jointly. Geometric-mean updates coordinate changes to the object and probe within each scan, while intensity-weighted minibatch synthesis combines the resulting local updates across scan positions. I will discuss the mathematical analysis and present results using synthetic data and experimental X-ray measurements.

A contact 3-manifold is planar if it is compatible with an open book whose page genus is 0. Wendl's theorem that any symplectic filling of a planar contact 3-manifold comes as a Lefschetz fibration filling a given planar open book translates questions about symplectic fillings of planar contact 3-manifolds into group theoretical questions about Dehn twist factorizations in the planar mapping class groups. I will outline work with Matt Day studying the nilpotent levels and truncations of the planar mapping class groups. We have calculated the ranks of the first three levels in the lower central series. I will explain these calculations and will outline a procedure to determine whether a mapping class element has a positive factorization. This work is joint with Matt Day.

The Conway knot C is an 11-crossing knot that bounds a locally flat disk in B^4. After many decades of unknown status, C was proven to not bound a smoothly embedded disk by Piccirillo. In this talk I will show that in fact, no sum of Conway knots bounds a smoothly embedded disk. This is joint work with Marco Golla (and independent of work of Piccirillo and of Donatone, Kegel, Lewark, and Truöl.)

I will discuss various aspects of steady solutions to the incompressible Euler equations, specifically their flexibility, rigidity, and stability properties.

First of two talks.

In Bernoulli percolation, we place edges independently with probability p between vertices of a lattice, and study the resulting random geometry. In the first talk, I will give a friendly introduction to the model and discuss some of the motivating questions and the history of methods in the high dimensional setting. In the second talk, I will discuss our recently developed tool, the incipient infinite cluster, and describe how we use it to derive the exact limiting distribution of distances within clusters, as well as prove sharp asymptotics for k-point connectivity functions. These results together establish several aspects of the conjecture that the scaling limit of high dimensional critical clusters is the Integrated Super-Brownian Excursion. Based on joint work with Shirshendu Chatterjee, Jack Hanson, and Philippe Sosoe.

We give a polynomial-time algorithm for finding a maximum weight independent set in a graph embedded in a surface of bounded genus and with a bounded number of vertex-disjoint odd faces. In particular, we obtain an algorithm for graphs with an even-faced embedding in the torus and for planar graphs with a bounded number of odd faces, answering a question of Thomassé (2018) and a question of Choi, Gorsky, Kim, McFarland, and Wiederrecht (2025) respectively. We reduce the original problem to finding a minimum cost non-negative integer circulation with homology constraints in a bidirected dual of the original graph. Joint work with Samuel Fiorini and Stefan Kober.

Murmurations were first observed as surprising correlations between Frobenius traces and root numbers of elliptic curves. They were then rigorously established for the family of modular forms in the level aspect by Zubrillina, and have since been studied in many families of L-functions in the number field setting. As pointed out by Sarnak, murmurations can also be interpreted as a phase transition in the one-level density of the low-lying zeros of the family. The goal of this talk is to investigate the function field analogue of this phenomenon. We will compute the murmuration density for the family of quadratic characters, relate the mean of the murmuration density to the stable homology of the moduli space of hyperelliptic curves, and clarify the relationship this has to one-level density. As a corollary, we will also obtain non-vanishing results for the L-functions of these curves. This is joint work with Lalín, Lee, and Oliver. 

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.

The success of modern generative models (LLMs) is a technological breakthrough, but foundations on which it is built has remained largely a mystery. A foundational understanding of the distributions that they sample from would vastly improve our ability to both diagnose potential issues (e.g. from a safety perspective) and also ability to push the applications of these models (e.g. in challenging settings such as scientific discovery). In this talk, I will broadly survey some aspects of work in this line while zooming in on a particular aspect: understanding sampling at inference time. In particular, we ask the question of how we can elicit, understand and control new behavior from language models particularly aimed towards long horizon tasks such as reasoning. Typical algorithmic interventions here suffer the issue of having errors compound with the time horizon. In this talk, we will see how these questions here can be connected to classical algorithmic questions from the theory of sampling and Markov chain Monte Carlo.  Using this new perspective, we will present an algorithm VGB, that circumvents compounding of error with the horizon. Time permitting, we will survey the broader algorithmic landscape, with further connections to particle filtering and stochastic control. 

This talk is based on joint work with Dhruv Rohatgi, Yuchen Li, Donya Salless, Andrej Risteski, Ankur Moitra and Dylan Foster.

A unit fraction representation of a positive rational number writes it as a sum of finitely many unit fractions $\frac{1}{n}$ with distinct positive integers $n$. In 1202, Fibonacci showed that every positive rational number has such a representation. Since then, people have been interested in representations with special properties. In this spirit, Bloom asked: which sets $A \subseteq \mathbb{N}$ can represent every positive rational number, subject to some local conditions, using only denominators from $A$? Croot proved in 2003 that for any finite colouring of $\mathbb{N}$, $1$ can be written as sum of unit fractions with denominators in a single-color class. Bloom showed that any $A$ with positive upper logarithmic density works, and Liu and Sawhney improved this to sets with upper logarithmic density $(\log N)^{-1/5 + o(1)}$. Bloom also showed that the shifted primes $p-1$ have this property.

To characterize all sets $A$ with this special property, we need to understand which arithmetic properties make this possible. One way to do that is to study thinner sets that still have this special property. In this talk we show that the shifted Golomb primes have this property. The Golomb primes are a subset of primes defined by $q_1 = 3$ and, once $q_1, \dots, q_k$ are known, $q_{k+1}$ is the smallest prime $p$ with $p \not\equiv 1 \pmod{q_i}$ for all $i \le k$. They are a natural generalization of the primes, and their relative density in the primes is about $1/(loglogx)$, which goes to $0$.

This talk is postponed to a later date.

What is the maximum number of rainbow triangles in an edge-colored graph with m edges and r colors? Using entropic techniques, we prove an upper bound of C(r)*m^{3/2} rainbow triangles — the explicit constant C(r) we derive is best possible whenever there is an affine plane of order r-1. We also show that constructions attaining at least (C(r)−eps)*m^{3/2} rainbow triangles must exhibit an affine plane structure for sufficiently small epsilon, which further improves the upper bound if no such affine plane exists. Lastly, we consider the problem of counting properly edge-colored copies of K_d with d≥4. Surprisingly the optimal construction is the same for any d≥4, but differs from the affine plane construction that was optimal for d=3.