Georgia Institute of Technology College of Sciences

It is a classical problem to study whether the h-principle holds for certain classes of maximally non-integrable distributions. The most studied case is that of contact structures, where there is a rich interplay between flexibility and rigidity, exemplified by the overtwisted vs tight dichotomy. For other types of maximally non-integrable distributions, no examples of rigidity are currently known.

In this talk I will discuss rigidity phenomena for fat distributions, which can be viewed as higher corank generalizations of contact structures. These admit natural symplectizations and contactizations. I will introduce a natural class of submanifolds in fat manifolds, called prelegendrians, which admit canonical Legendrian lifts to the contactization. The main result of the talk is that these submanifolds exhibit rigidity: in the “standard corank-2 fat manifold” there exists an infinite family of prelegendrian tori, all of them formally equivalent but pairwise not prelegendrian isotopic. In other words, the h-principle fails for prelegendrians. The talk is based on joint work with Álvaro del Pino and Wei Zhou.
 

Uncertainty is ubiquitous: both data and physical models inherently contain uncertainty. Therefore, it is crucial to identify the sources of uncertainty and control its propagation over time. In this talk, I will introduce two approaches to address this uncertainty propagation problem—one for the inverse problem and one for the forward problem. The main idea is to work directly with probability measures, treating the underlying PDE as a pushforward map. In the inverse setting, we will explore various variational formulations, focusing on the characterization of minimizers and their stability. In the forward setting, we aim to propose a new approach to tackle high-dimensional uncertainties.

This talk concerns the Benjamin-Ono (BO) equation of internal wave theory, and properties of the solution of the Cauchy initial-value problem in the situation that the initial data is fixed but the coefficient of the nonlocal dispersive term in the equation is allowed to tend to zero (i.e., the zero-dispersion limit). It is well-known that existence of a limit requires the weak topology because high-frequency oscillations appear even though they are not present in the initial data.  Physically, this phenomenon corresponds to the generation of a dispersive shock wave. In the setting of the Korteweg-de Vries (KdV) equation, it has been shown that dispersive shock waves exhibit a universal form independent of initial data near the two edges of the dispersive shock wave, and also near the gradient catastrophe point for the inviscid Burgers equation from which the shock wave forms. In this talk, we will present corresponding universality results for the BO equation. These have quite a different character than in the KdV case; while for KdV one has universal wave profiles expressed in terms of solutions of Painlevé-type equations, for BO one instead has expressions in terms of classical Airy functions and Pearcey integrals. These results are proved for general rational initial data using a new approach based on an explicit formula for the solution of the Cauchy problem for BO. This is joint work with Elliot Blackstone and Matthew Mitchell, based on other work with Blackstone, Louise Gassot, and Patrick Gérard.

The Ramsey number R(3,k) is the smallest n so that every triangle free graph on n vertices has an independent set of size k.  Upper bounds to R(3,k) consist of finding independent sets in triangle free graphs, while lower bounds consist of constructing triangle free graphs with no large independent set.  The previous best-known lower bound was independently due to works of Bohman-Keevash and Fiz Pontiveros-Griffiths-Morris, both of which analyzed the triangle-free process.  The analysis of the triangle-free process is a delicate dance of demonstrating self-correction and requires tracking a large family of graph statistics simultaneously.  We will discuss a new lower bound to R(3,k) and provide a gentle introduction to the concept of "the Rodl nibble," with an emphasis on which ideas simplify our analysis.  This is based on joint work with Marcelo Campos, Matthew Jenssen and Julian Sahasrabudhe.

Many real life problems rely on understanding the solutions to a system of polynomial equations. In this talk, I will outline some of these applications and how tools from algebraic geometry can provide answers to relevant engineering questions.

Kakeya sets are compact subsets of $\mathbb{R}^n$ that contain a unit line segment pointing in every direction and the Kakeya conjecture states that such sets must have Hausdorff dimension $n$. The property of stickiness was first discovered by by Katz-Laba-Tao in their 1999 breakthrough paper on the Kakeya problem. Then Wang-Zahl formalized the definition of a sticky Kakeya set as a subclass of general Kakeya sets in 2022. Sticky Kakeya sets played an important role as Wang and Zahl solved the Kakeya conjecture for  $\mathbb{R}^3$ in a major recent development.
The planebrush method is a geometric argument by Katz-Zahl which gives the current best bound of 3.059 for Hausdorff dimension of Kakeya sets in $\mathbb{R}^4$. Our new result shows that sticky Kakeya sets in $\mathbb{R}^4$ have dimension 3.25. The planebrush argument when combined with the sticky hypothesis gives us this better bound. 

One of the first results on concordance was a condition on the Alexander polynomials of slice knots, now known as the Fox-Milnor condition. In this talk, we discuss a generalization of the Fox-Milnor condition to links and their multivariable Alexander polynomials. The main tool is an interpretation of the Alexander polynomials in terms of “Reidemeister torsion”, a notion defined for general manifolds. We will see that the Fox-Milnor condition is a reflection of a certain duality theorem for Reidemeister torsion.

The Walsh-quantized baker's maps are models for quantum chaos on the torus. We show that for all baker's map scaling factors D\ge2 except for D=4, typically (in the sense of Haar measure on the eigenspaces, which are degenerate) the distribution of the matrix element fluctuations for a randomly chosen eigenbasis looks Gaussian in the semiclassical limit N\to\infty, with variance given in terms of classical baker's map correlations. This determines the precise rate of convergence in the quantum ergodic theorem for these eigenbases. The presence of the classical correlations highlights that these eigenstates, while random, have microscopic correlations that differentiate them from Haar random vectors. For the single value D=4, the Gaussianity of the matrix element fluctuations depends on the values of the classical observable on a fractal subset of the torus.

This series will tie together algebraic, complex analytic, symplectic, and contact geometries together in one coherent story. This will be done via the study of a series of couplets from different fields of geometry:

Algebraic manifolds:
Affine and quasi-projective varieties (non-compact models)
Projective varieties (compact models)

Complex manifolds:
Stein manifolds
Stein compactifications

Symplectic manifolds:
Liouville/ Weinstein geometry
Compact Kahler manifolds 

Depending on how long it takes to discuss these items, I will also attempt to include discussions on:

• Biran-Giroux decompositions of symplectic manifolds • Boothby-Wang bundles and contact plumbings of these • Milnor's fibration theorem for isolated singularities and connections to open book decompositions and Lefschetz fibrations • Open questions and interesting avenues of research

Most of our discussion will, as a side effect, outline the topological structure behind Type IIA String theory (the "topological A-model") which requires a 6-dimensional Calabi-Yau (Kahler) background.

I’ll describe some recent work (joint with Jeff Kahn and Jinyoung Park) on the "Second" Kahn--Kalai Conjecture (KKC2), the original conjecture on graph containment in $G_{n,p}$ that motivated what is now the Park--Pham Theorem (PPT). KKC2 says that $p_c(H)$, the threshold for containing a graph $H$ in $G_{n,p}$, satisfies $p_c(H) < O(p_E(H) log n)$, where $p_E(H)$ is the smallest p such that the expected number of copies of any subgraph of $H$ is at least one. Thus, according to KKC2, the simplest expectation considerations predict $p_c(H)$ up to a log factor. This serves as a refinement of PPT in the restricted case of graph containment in $G_{n,p}$. Our main result is that $p_c(H) < O(p_E(H) log^3(n))$. This last statement follows (via PPT) from our results on a completely deterministic graph theory problem about maximizing subgraph counts under sparsity constraints.