UIUC-WUSTL   Joint Seminar in Symplectic and Poisson Geometry

Title: Lagrangian Intersections and the Shape Invariant

Abstract: Let  X be a subset of $\mathbb{R}^{2n}$, The shape invariant of X is a set-valued symplectic capacity that encodes the product Lagrangian tori that can be symplectically embedded into X. There are many open questions about this embedding problem, and hence the shape invariant, even when X is a four-dimensional toric domain. I will discuss some new flexibility and rigidity results in this direction which allow one to compute the shape invariant of large families of toric domains. This is joint work in progress with Richard Hind (Notre Dame).

Title: Fukaya categories of surfaces and pants decompositions

Abstract: A Riemann surface of negative Euler characteristic can be decomposed into pairs of pants. I will present joint work with Nicolò

Sibilla that reconstructs the Fukaya category of a surface from several copies of the Fukaya category of the pair of pants. 

For the UIUC locals: this talk will focus on some different aspects from my talk with the same title earlier this semester.

Title: Geometric structures and algebroids

Abstract: In geometry, it is common to study geometric structures constrained by partial differential equations (PDEs), focusing on their existence and classification.

In certain special cases, such as Bochner-Kähler metrics, Bryant observed that a Lie algebroid underlies the associated existence and classification problem. Building on this insight, Fernandes and Struchiner established a precise connection between the classification of complete solutions and the integrability of the corresponding Lie algebroid.

In this talk, I will show that algebroids are not limited to special cases but also play a fundamental role in generic existence and classification problems. I will introduce the concept of relative algebroids, a framework that generalizes both PDEs and traditional Lie algebroids. Through illustrative examples, I will demonstrate how algebroids arise naturally as quotients of PDEs by symmetries, providing a unified explanation for their role in classifying geometric structures up to symmetry. This is joint work with Rui Loja Fernandes.

Title: On simplicial sheaves on groupoids

Abstract: One common type of Morita invariant arises from the ``groupoid cohomology’’ of a Lie groupoid. These invariants can be thought of as models for algebra of functions, deRham cohomology and more. In this talk, we will discuss a way to generalize these results by viewing them in terms of sheaves on the nerve of a groupoid. This talk concerns joint work with Xiang Tang.

Title: Analytic Morse theory under Lie group actions

Abstract: In this talk, under the action of a compact connected Lie group G, we first define the G-invariant Thom-Smale chain complex associated to a special type of Morse-Bott functions. Then, we give an analytic interpretation by the G-invariant Witten instanton chain complex. When G is a torus, we find a chain isomorphism. When G is non-abelian, we find a one-to-one correspondence between chains. Finally, we propose a conjecture, possibly upgrading the correspondence in the non-abelian case to a chain isomorphism.