April 27, 2023: Sergey Maksimenko "Homotopy types of diffeomorphisms groups of simplest Morse-Bott foliations on lens spaces"

Abstract: Let F be the Morse-Bott foliation on the solid torus T=S¹×D² into 2-tori parallel to the boundary and one singular circle S¹×0. A diffeomorphism h:T→T is called foliated (resp. leaf preserving) if for each leaf ω∈F its image h(ω) is also leaf of F (resp. h(ω)=ω). Gluing two copies of T by some diffeomorphism between their boundaries, one gets a lens space L_p,q with a Morse-Bott foliation F_p,qobtained from F on each copy of T. Denote by D_fol(T,∂T) and D_lp(T,∂T) respectively the groups of foliated and leaf preserving diffeomorphisms of T fixed on ∂T. Similarly, let D_fol(L_p,q and D_lp(L_p,q be respectively the groups of foliated and leaf preserving diffeomorphisms of F_p,q Endow all those groups with the corresponding C^∞ Whitney topologies. The aim of the talk is give a complete description the homotopy types of the above groups D_fol(T,∂T), D_lp(T,∂T), D_fol(L_p,q), D_lp(L_p,q) for all p,q.

April 13, 2023: Liam Kahmeyer " A homotopy invariant of image simple fold maps to oriented surfaces"

Abstract: In 2019, Osamu Saeki showed that for two homotopic generic fold maps $f,g:S^3 \rightarrow S^2$ with respective singular sets $\Sigma(f)$ and $\Sigma(g)$ whose respective images $f(\Sigma)$ and $g(\Sigma)$ are smoothly embedded, the number of components of the singular sets, respectively denoted $\#|\Sigma(f)|$ and $\#|\Sigma(g)|$, need not have the same parity. From Saeki's result, a natural question arises: For generic fold maps $f:M \rightarrow N$ of a smooth manifold $M$ of dimension $m \geq 2$ to an oriented surface $N$ of finite genus with $f(\Sigma)$ smoothly embedded, under what conditions (if any) is $\#|\Sigma(f)|$ a $\Z/2$-homotopy invariant? The goal of this talk is to explore this question. Namely, I will show that for smooth generic fold maps $f:M \rightarrow N$ of a smooth closed oriented manifold $M$ of dimension $m\geq 2$ to an oriented surface $N$ of finite genus with $f(\Sigma)$ smoothly embedded, $\#|\Sigma(f)|$ is a modulo two homotopy invariant provided one of the following conditions is satisfied: (a) $\textrm{dim}(M) = 2q$ for $ q \geq 1$, (b) the singular set of the homotopy is an orientable manifold, or (c) the image of the singular set of the homotopy does not have triple self-intersection points.

March 2, 2023: Miriam Kuzbary "Asymptotic bounds on the d-invariant"

Abstract: As shown by Morita, every integral homology 3-sphere Y has a Heegaard decomposition into two handlebodies where the gluing map along the boundary is an element of the Torelli subgroup of the mapping class group of the boundary composed with the standard gluing map for the 3-sphere. In work in progress with Santana Afton and Tye Lidman, we show that the d-invariant of Y, a homology cobordism invariant of homology spheres defined using Heegaard Floer homology, is bounded above by a linear function of the word length of a corresponding gluing element in the Torelli group for any fixed, finite generating set when the genus is larger than 2. Moreover, we show the d-invariant is bounded for homology spheres corresponding to various large families of mapping classes.

February 23, 2023: Antonio Alfieri "Instanton Floer homology of almost-rational plumbings"

Abstract: Plumbed three-manifolds are those three-manifolds that can be be realized as links of isolated complex surface singularities. Inspired by Heegaard Floer theory Nemethi introduced a combinatorial invariant of complex surface singularities (lattice cohomology) that was recently proved to be is isomorphic to Heegaard Floer homology. I will give an introduction to lattice cohomology, and expose some work in collaboration with John Baldwin, Irving Dai, and Steven Sivek showing that the lattice cohomology of an almost-rational singularity is isomorphic to the framed Instanton Floer homology of its link.

February 16, 2023: Braeden Reinoso "Fixed points, train tracks, and knot Floer detection"

Abstract: A fibered knot K is a knot whose complement admits a fibration over the circle, with fibers given by Seirfert surfaces for K. The return map of this fibration is a mapping class on a Seifert surface for K, and the dynamics of the mapping class are closely related to the geometry of K. I will describe how to leverage this relationship to provide the first knot Floer detection results for knots of genus at least 2, and for knots in manifolds other than S^3. Motivated by knot Floer detection problems, I will also briefly discuss some work in progress regarding a decomposition of mapping class groups into "twist families," with the goal of algorithmically classifying pseudo-Anosov braids with any specified number of fixed points. Many of the techniques in this talk are related to the theory of train track maps for pseudo-Anosov braids.

February 2, 2023: Miguel Paternain "Lie algebras of curves and loop-bundles on surfaces"

Abstract: W. Goldman and V. Turaev defined a Lie bialgebra structure on the Z-module generated by free homotopy classes of loops of an oriented surface. We shall discuss a generalization of this construction replacing homotopies by thin homotopies, based on the combinatorial approach given by M. Chas. We use it to give a geometric proof of a characterization of homotopy classes of simple curves in terms of the Goldman-Turaev bracket, which was conjectured by Chas.

November 10, 2022: Jakob Hedicke "On the existence of integrable Reeb flows"

Abstract: A classical result by Bolsinov and Fomenko implies that 3-dimensional Reeb flows that are Bott-integrable, i.e. that admit a Morse-Bott function invariant under the flow, only exist on graph manifolds. We will discuss different constructions to obtain integrable Reeb flows on these manifolds and examine properties of the underlying contact structures. This talk is based on joint work with Hansjörg Geiges and Murat Sağlam.

September 29, 2022: Colin Adams "Generalizations of Knots to Knotoids and Way Beyond"

Abstract: Knotoids are a generalization of knots by Turaev in 2010 where the circle embedded in space is replaced by an interval embedded in space. They make good models for proteins and have many interesting properties. Many invariants of knots have been extended to knotoids. We extend hyperbolicity and hyperbolic volume to knotoids. We then extend knotoids to generalized knotoids which substantially increase the set of objects under consideration and have many interesting properties and sub-cases.

September 22, 2022: Roman Golovko "On non-geometric augmentations in high dimensions and torsion of Legendrian contact homology"

Abstract: We construct the augmentations of high dimensional Legendrian submanifolds of the contact Euclidean vector space which are not induced by exact Lagrangian fillings. Besides that, for an arbitrary finitely generated abelian group G, we construct the examples of Legendrian submanifolds whose integral linearized Legendrian contact (co)homology realizes G.