Abstract: In this talk, I will describe a new, transformation group theoretic, approach to the classification of Darboux integrable partial differential equations of the type
uxy = f(x, y, u, ux, uy).
Such equations are commonly referred to in the literature as f-Gordon equations, generalized wave map equations, or equations of Liouville type.
The main result of this group theoretic approach is that a complete list of all f-Gordon equations which are Darboux integrable at order 3 can be determined from a complete list of rank 2 distributions in 5 dimensions which admit intransitive 5-dimensional symmetry groups. In this way, the study of Darboux integrable f-Gordon equations is tied, quite remarkably, to Cartan's celebrated 1910 paper. Through this correspondence, we uncovered a new class of Darboux integrable equations which leads to a complete classification of all Darboux integrable equations at order 3.
This talk is based on joint work with Brandon P. Ashley, Southern Oregon University.
Robert Bryant (Duke University) - Graduate Student Lecture
Abstract: A Riemannian manifold (M, g) is said to be curvature-homogeneous if, for any two points x and y in M, there is an isometry of tangent spaces TxM and TyM that identifies the curvature operators at the two points. In other words, the Riemannian metric g is “homogeneous up to second order”. When M is a surface, this is the same as having constant Gauss curvature, and, in this case, the surface is actually locally homogeneous. In higher dimensions, this is no longer true, but exactly how ‘flexible’ curvature homogeneous spaces are is not well-understood. In this talk, I will survey some classical and recent results about curvature-homogeneous metrics and describe some of the tools that can be used to study this problem.
Abstract: In a recent work with Florit and Ziller, we completed the classification of curvature-homogeneous hypersurfaces in spaces of constant curvature, treating the one remaining unsolved case, that of a hypersurface in a 4-dimensional space form. It was a surprise to discover that, in this case, there exists an ‘exotic’ family of solutions that are not homogeneous as hypersurfaces, and it turns out that a variety of techniques are needed to analyze them fully.
Abstract: Compact holomorphic manifolds and Riemannian manifolds invite you all to participate in their three epic stories. In the first tale, the main character is going to be a compact holomorphic manifold, and as in every story, there will be some action going on. More specifically, the group of invertible complex numbers, or even better, several copies of those, act on the manifold. The spirit of the late Andrzej Białynicki-Birula until this day helps us to comprehend what is going on.
The second story is a tale of holonomies, it begins with "a long time ago,..." and concludes with "... and the last missing piece of this mystery is undiscovered till this day". The protagonist of this part is a quaternion-Kähler manifold, while the legacy of Marcel Berger is in the background all the time.
In the third part, we meet legendary distributions, which are subbundles in the tangent bundle of one of our main characters. Among others, distributions can be foliations or contact distributions, which like yin and yang live on opposite sides of the world, yet they strongly interact with one another. Ferdinand Georg Frobenius is supervising this third part.
Finally, in the epilogue, all the threads and characters so far connect in an exquisite theorem on the classification of low-dimensional complex contact manifolds. In any dimension, the analogous classification is conjectured by Claude LeBrun and Simon Salamon, while in low dimensions, it is proved by Jarosław Wiśniewski, Andrzej Weber, in a joint work with the narrator.
I will discuss the theorem, some motivations, and some of the ideas in the proof.
Abstract: A topological model for a knot invariant is a realization of the invariant as graded intersection pairings on coverings of configuration spaces. In this talk, I will describe a topological model for the HOMFLY-PT polynomial. I plan to discuss the motivation from previous work by Lawrence and Bigelow giving topological models for the Jones and sln polynomials, and our construction, joint with Cristina Anghel, which uses a state sum formulation of the HOMFLY-PT polynomial to construct an intersection pairing on the configuration space of a Heegaard surface of the link.
Abstract: I will explain the construction of the Fukaya-Morse category of a Riemannian manifold X—an A-infinity category (a category where associativity of composition holds only "up-to-homotopy") where the higher composition maps are given in terms of numbers of embedded trees in X, with edges following the gradient trajectories of certain Morse functions. I will give simple examples and explain different approaches to understanding the structure and proving the quadratic relations on the structure maps—(1a) via homotopy transfer, (1b) effective field theory approach, (2) topological quantum mechanics approach. The talk is based on a joint work with O. Chekeres, A. Losev, and D. Youmans, arXiv:2112.12756.
Abstract: I will start by reviewing my recent work with Dan Grady on the geometric cobordism hypothesis and locality of fully extended nontopological functorial field theories. I will then apply these results to explicitly compute, in terms of homotopy coherent representations of Lie groups, the space of 2-dimensional fully extended functorial field theories with geometric structures like flat 2-dimensional Riemannian metrics, 2|1-dimensional super Euclidean structures, or conformal structures. If time permits, I will discuss further constructions of field theories (ongoing work with Dan Grady) that involve differential characteristic classes, index and eta-invariants, and quantization.
Abstract: Comparison geometry is a significant topic in metric geometry and geometric analysis. Gromov proposes to study comparison theorems related to scalar curvature and mean curvature for manifolds with boundary. In this talk, I will summarize recent developments on this topic, utilizing tools from index theory and geometric analysis. I will also talk about our recent work, joint with Zhichao Wang and Bo Zhu, where we prove a scalar-mean rigidity theorem using capillary mu-bubbles.
Abstract: Solutions of certain PDEs, such as Einstein metrics and soliton metrics, are easier to obtain if one assumes a certain amount of symmetry. We will discuss this in the case of polar actions and actions whose generic orbits are hypersurfaces. Smoothness conditions for solutions are an important part of the problem, which in general is known to be subtle and difficult to describe. We will discuss some recent progress by Tinyue Liu in the case of polar metrics and, in joint work with Luigi Verdiani, in the case of cohomogeneity one metrics.