Georgia Institute of Technology College of Sciences

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.

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.
 

Cislunar space—the region between Earth and the Moon—has reemerged as a critical area for space exploration. From a mathematical perspective, this region is governed by multi-body dynamics that give rise to rich structures, including invariant manifolds, resonant orbits, and homoclinic chaos. This talk will introduce classical and modern tools from celestial mechanics to analyze motion in the Earth–Moon system, with an emphasis on restricted 3- and 4-body problems. We will discuss how perturbative methods (normal forms) and invariant manifold theory (parameterization method) reveal the underlying organization of the phase space. Particular attention will be placed on connecting the perturbative regime, where classical methods apply, with the realistic system, which often lies far outside that regime, using computer-assisted techniques. Our ultimate goal is to establish rigorous results for the real solar system while enhancing the engineering capabilities needed to design and fly missions, highlighting how mathematics contributes both to theory and to the practical challenges of contemporary space exploration.

No prior knowledge is needed; the talk will be self-contained and accessible. Undergraduates are encouraged to attend. 

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.