Titles & Abstracts

Neda Bagherifard (U Oregon)

An Excision Theorem in Heegaard Floer Theory

In my talk, I will discuss the excision construction for 3-manifolds, which relates two closed, oriented 3-manifolds, Y and Y′, through a process of cutting and regluing along surfaces. I will explain how this construction leads to the isomorphism of twisted Heegaard Floer homology groups of Y and Y'. Additionally, I will present applications of our excision formula, including examples that demonstrate certain manifolds cannot be related by this construction. Furthermore, I will show how twisted Heegaard Floer homology groups for specific families of 3-manifolds can be computed using our excision formula.

——————————————— 

Ipsa Bezbarua (CUNY)

Contractibility of the knot complex of incompressible spanning surfaces

One of the fundamental structures studied in knot theory is a compact surface whose boundary is the link under consideration, called a spanning surface. Osamu Kakimizu constructed two closely related simplicial complexes using the spanning surfaces of a given link - the incompressible complex and the Kakimizu complex - to study the properties of the link. In 2012, Piotr Przytycki and Jennifer Schultens showed that the Kakimizu complex is contractible for any link. In this talk, we will see that their arguments can be modified to show contractibility of the incompressible complex as well.

——————————————— 

Holt Bodish (UIUC)

Bordered bimodule for the 2 component earring link

We discuss an interpretation of the action of the bordered bimodule for the 2-component link L, formed by adding a meridian to a given knot K, on immersed curves in the punctured torus. This bimodule recovers both the connected sum of two knots as well as information about splicing knot complements. This is joint work in progress with J. Rasmussen. 

——————————————— 

Michele Capovilla-Searle (UIowa)

Quasipositive fiber surfaces and well-quasi-orderedness 

Finite graphs are well-quasi-ordered under the graph minor relation by a famous result of Robertson-Seymour. In joint work with J. Breen, we investigate the open question of whether quasipositive fiber surfaces (i.e. pages of open books supporting tight S^3) are well-quasi-ordered under the surface minor relation. 

——————————————— 

Nicholas Cecil (UIowa)

Duality for Categorical Wreath Products

In this talk, I'll describe a categorical wreath product introduced by Clemens Berger to describe an object of higher categorical interest dual to an object first studied by Joyal.  I'll discuss how this duality is a special case of a more formal duality connected to Max Kelly's notion of a club.

——————————————— 

Jonathan Delgado (UCI)

The Mapping Cone Yang-Mills Functional: A Form-Sensitive Gauge Theory

We introduce a mapping cone Yang-Mills functional: a functional on the space of mapping cone connection forms which couple the usual connection forms from gauge theory with any distinguished closed 2-form on a closed manifold (e.g. a symplectic or Kahler form). We study properties of these cone connection forms and cone curvature. We find natural (anti)-self-duality equations of the cone curvature in the case of 3-manifolds which generalize the Bogomolny monopole equations. We also explore the regularity of the cone Yang-Mills functional including a monotonicity formula and ε-regularity.

——————————————— 

Mingyuan Hu (Northwestern)

Skeins, Clusters and Wavefunctions

 For a Lagrangian submanifold lives in a 3 Calabi-Yau manifold, Ekholm and Shende defined a wavefunctions living in the HOMFLY-PT skein module of the Lagrangian, which encode open Gromov-Witten invariants in all genus and arbitrarily many boundary components. We consider a class of Lagrangians living in $\bC^3$, generalizing the Aganagic-Vafa branes. We develop a skein valued cluster theory to compute these wavefunctions. In some simple cases, our computation matches up with the topological vertex. Along the way we define a skein dilogarithm and prove a pentagon relation, which will imply the 5-term relation of Garsia and Mellit, originally formulated in terms of Macdonald polynomials. This talk is based on arxiv:2312.10186 (joint with Gus Schrader and Eric Zaslow) and arXiv:2401.10817.

——————————————— 

Roman Krutowski (UCLA)

Morse theory of the loop space of S^2 and derived Hecke algebras

In the work of Honda, Tian, and Yuan it was shown that higher-dimensional Heegaard Floer homology of k cotangent fibers in T^*\Sigma for an oriented surface \Sigma of positive genus is quasi-equivalent to the braid HOMFLY-PT skein algebra of \Sigma. In this talk, I will present the computation of the HDHF homology for T^*S^2, which is based on a Morse-theoretic based multiloop A_\infty-algebra that one can assign to any oriented manifold. Not only is it quasi-equivalent to the HDHF A_\infty-algebra, but it also presents a new string topological invariant of manifolds. This talk is based on a joint work with Ko Honda, Yin Tian, and Tianyu Yuan.

——————————————— 

Guanyu Li (Cornell)

Lie (co)homology and representation homology

Introduced by Berest, Ramadoss, and Yeung, the representation homology of a (pointed, connected) topological space detects representation-theoretical features of the space. In this talk, we will construct a map relating the (co)homology of Lie algebras and the representation homology of a space. In the torus case, this shows a strong connection to the classical Bott-Kostant theorem of cohomology of nilpotent Lie algebras.

——————————————— 

Yun Liu (IU Bloomington)

Relative Join Construction and Towers of Borel Fibrations

In a recent work, Yu. Berest and A. C. Ramadoss formulated and studied the realization problem for rings of quasi-invariants of finite reflection groups in terms of classifying spaces of compact Lie groups. The main tool used in their work is the fiber-cofiber construction introduced in topology by T. Ganea.

——————————————— 

Shashini Marasinghe (Michigan State)

Seifert fibered 3-manifolds and Turaev-Viro invariants volume conjecture

We study the large r asymptotic behaviour of the Turaev-Viro invariants of oriented Seifert fibered 3-manifolds at the root q=e^\frac{2\pi i}{r}. As an application, we prove the volume conjecture for large families of oriented Seifert fibered 3-manifolds with empty and non-empty boundary.

——————————————— 

Ivan So (Michigan State)

Filtered Instantons and Concordance of Satellite Knots 

Using filtered instanton Floer homology, Nozaki-Sato-Taniguchi defined a homology cobordism invariant r_s for invariant for integer homology spheres. In this talk, I will talk about some new concordance results derived from this invariant. 

——————————————— 

Yi Wang (U Illinois)

Detected Seifert surfaces and intervals of orderable Dehn fillings

Motivated by the L-space conjecture, Culler-Dunfield and Gao utilized arcs of PSL(2, R) representations in knot groups to establish left-orderability for intervals of Dehn fillings on one-cusped hyperbolic 3-manifolds. We extend the techniques of Gao by establishing such arcs originating at ideal points of SL(2, C) character varieties, finding new intervals of orderable Dehn fillings. These arcs come from ideal points detecting Seifert surfaces in the knot complement via Culler-Shalen theory.

——————————————— 

Randy Van Why (Georgia Tech)

Symplectic geometry, surface configurations, and non-algebraic manifolds 

We will discuss the local geometry of symplectic normal crossing divisors and how they can be used to produce interesting examples of symplectic 4-manifolds. We will discuss their relation to the question of which symplectic manifolds can be algebraic or Stein. This talk will be motivated by classical algebraic geometry and assume no familiarity with symplectic manifolds.