Calendar
Calendar
Posted August 27, 2024
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Nilangshu Bhattacharyya, Louisiana State University
Transverse invariant as Khovanov skein spectrum at its extreme Alexander grading
Olga Plamenevskaya described a transverse link invariant as an element of Khovanov homology. Lawrence Roberts gave a link surgery spectral sequence whose $E^2$ page is the reduced Khovanov skein homology (with $\mathbb{Z}_{2}$ coefficient) of a closed braid $L$ with odd number of strands and $E^{\infty}$ page is the knot Floer homology of the lift of the braid axis in the double branch cover, and the spectral sequence splits with respect to the Alexander grading. The transverse invariant does not vanish in the Khovanov skein homology, and under the above spectral sequence and upon mapping the knot Floer homology to the Heegard Floer homology, the transverse invariant corresponds to the contact invariant. Lipshitz-Sarkar gave a stable homotopy type invariant of links in $S^3$. Subsequently, Lipshitz-Ng-Sarkar found a cohomotopy element in the Khovanov spectrum associated to the Plamenevskaya invariant. We can think of this element as a map from Khovanov spectra at its extreme quantum grading to the sphere spectrum. We gave a stable homotopy type for Khovanov skein homology and showed that we can think of the cohomotopy transverse element as a map from the Khovanov spectra at its extreme quantum grading to the Khovanov skein spectra at its extreme Alexander grading. This is a joint work with Adithyan Pandikkadan, which will be presented in this talk.
Posted August 28, 2024
Last modified September 9, 2024
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Tristan Wells Filbert, Louisiana State University
Whitehead doubles of dual knots are deeply slice
In collaboration with McConkey, St. Clair, and Zhang, we show that the Whitehead double of the dual knot to $1/n$ surgery on the knot $6_1$ in the 3-sphere is deeply slice in a contractible 4-manifold. That is, it bounds a smoothly embedded disc in the manifold, but not in a collar neighborhood of its boundary, the surgered manifold. This is partial progress in answering one of the Kirby questions regarding nullhomotopic deeply slice knots, mentioned in earlier work of Klug and Ruppik. To prove our theorem, we make use of the immersed curves perspective of bordered Floer homology and knot Floer homology.
Posted August 29, 2024
Last modified October 7, 2024
Geometry and Topology Seminar Seminar website
3:30 pm
Bin Sun, Michigan State University
$L^2$-Betti numbers of Dehn fillings
I will talk about a recent joint work with Nansen Petrosyan where we obtain conditions under which $L^2$-Betti numbers are preserved by group-theoretic Dehn fillings. As an application, we verify the Singer Conjecture for certain Einstein manifolds and provide new examples of hyperbolic groups with exotic subgroups. We also establish a virtual fibering criterion and obtain bounds on deficiency of Dehn fillings. A key step in our approach of computations of $L^2$-Betti numbers is the construction of a tailored classifying space, which is of independent interest.
Posted October 7, 2024
Last modified October 28, 2024
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Monika Kudlinska, University of Cambridge
Solving equations in free-by-cyclic groups
A group G is said to be free-by-cyclic if it maps onto the infinite cyclic group with free kernel of finite rank. Free-by-cyclic groups form a large and widely-studied class with close links to 3-manifold topology. A group G is said to be equationally Noetherian if any system of equations over G is equivalent to a finite subsystem. In joint work with Motiejus Valiunas we show that all free-by-cyclic groups are equationally Noetherian. As an application, we deduce that the set of exponential growth rates of a free-by-cyclic group is well ordered.
Posted September 17, 2024
Last modified November 11, 2024
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Arka Banerjee, Auburn University
Urysohn 1-width and covers
A metric space has small Urysohn 1-width if it admits a continuous map to a 1-dimensional complex where the preimage of each point has small diameter. An open problem is, if a space's universal cover has small Urysohn 1-width, must the original space also have small Urysohn 1-width? While one might intuitively expect this to be true, there are strange examples that suggest otherwise. In this talk, I will explore the motivations behind this question and discuss some partial progress we have made in understanding it. This is a joint work with H. Alpert and P. Papasoglu.
Posted December 11, 2024
Last modified January 27, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Akram Alishahi, University of Georgia
Contact invariants in Heegaard Floer homology
Over the past two decades multiple invariants of contact structures have been defined in different variations of Heegaard Floer homology. We will start with an overview of these invariants and their connections. Then, we will discuss one of these invariants that is defined for a contact 3-manifold with a foliated boundary and lives in bordered sutured Floer homology in more details. This is a joint work with Földvári, Hendricks, Licata, Petkova and Vertesi.
Posted January 23, 2025
Last modified January 27, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Matthew Stoffregen, Michigan State University
Pin(2) Floer homology and the Rokhlin invariant
In this talk, we describe a family of homology cobordism invariants that can be extracted from Pin(2)-equivariant monopole Floer homology (using either Manolescu or Lin's definitions), that have some properties in common with both the epsilon and upsilon invariants in knot Floer homology. We'll show a relationship of this family to questions about torsion in the homology cobordism group, and to triangulation of higher-dimensional manifolds. This is joint work in progress with Irving Dai, Jen Hom, and Linh Truong.
Posted November 12, 2024
Last modified February 10, 2025
Geometry and Topology Seminar Seminar website
2:30 pm
Porter Morgan, University of Massachusetts Amherst
Irreducible 4-manifolds with order two fundamental group
Let R be a closed, smooth, oriented 4–manifold with order two fundamental group. The works of Freedman and Hambleton-Kreck show that R is determined up to homeomorphism by just a few basic properties. That said, there are often many different manifolds that are homeomorphic to R, but not diffeomorphic to it or each other. In this talk, we’ll describe how to construct irreducible copies of R; roughly speaking, these are smooth manifolds that are homeomorphic to R, and don’t decompose into non-trivial connected sums. We’ll show that if R has odd intersection form and non-negative first Chern number, then in all but seven cases, it has an irreducible copy. We’ll describe some of the techniques used to realize these irreducible smooth structures, including torus surgeries, symplectic fiber sums, and a novel approach to constructing Lefschetz fibrations equipped with free involutions. This is joint work with Mihail Arabadji.
Posted February 6, 2025
Last modified February 12, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Neal Stoltzfus, Mathematics Department, LSU
Discrete Laplacians, Ribbon Graphs, and Link Polynomials
The Whitney homology of the independence lattices of the state space of a ribbon graphs supports three independent anti-commuting discrete Laplacians. They relate to the three fundamental combinatorial invariants of independent subsets: rank, nullify and genus. We explore the combinations that give link invariants.
Posted January 14, 2025
Last modified February 18, 2025
Geometry and Topology Seminar Seminar website
3:30 pm
Dave Auckly, Kansas State University
Restrictions on the genus of trivial families of surfaces in twisted families of 4-manifolds
Several notions of equivalence in topology may be expressed via the existence of families. Thus, asking when an untwisted family of surfaces can be placed in a twisted family of manifolds in a natural question. This talk will describe a generalized adjunction inequality for families.
Posted March 4, 2025
Last modified March 10, 2025
Geometry and Topology Seminar Seminar website
2:30 pm Lockett 233
Maarten Mol, University of Toronto
Constructibility of momentum maps and variation of singular symplectic reduced spaces (Joint with Mathematical Physics and Representation Theory Seminar)
Proper maps in various categories studied in singularity theory (for example, the real analytic category) are known to be constructible, in the sense that the image of the map can be stratified in such a way that the map is a topological fiber bundle over each stratum. Such stratifications provide insight into how the fibers of the map vary. In this talk we will discuss the existence of such a stratification for momentum maps of Hamiltonian Lie group actions (a natural class of maps studied in symplectic/Poisson geometry), which provides insight into how the so-called symplectic reduced spaces of the Hamiltonian action vary. Along the way we will also try to give an overview of some more classical results on the geometry of such maps.
Posted March 10, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Scott Baldridge, Louisiana State University
A new way to prove the four color theorem using gauge theory
In this talk, I show how ideas coming out of gauge theory can be used to prove that certain configurations in the list of "633 unavoidable's" are reducible. In particular, I show how to prove the most important initial example, the Birkhoff diamond (four “adjacent" pentagons), is reducible using our filtered $3$- and $4$-color homology. In this context reducible means that the Birkhoff diamond cannot show up as a “tangle" in a minimal counterexample to the 4CT. This is a new proof of a 111-year-old result that is a direct consequence of a special (2+1)-dimensional TQFT. I will then indicate how the ideas used in the proof might be used to reduce the unavoidable set of 633 configurations to a much smaller set. This is joint work with Ben McCarty.
Posted March 26, 2025
Geometry and Topology Seminar Seminar website
2:30 pm – 3:30 pm Lockett 233
Scott Baldridge, Louisiana State University
A new way to prove the four color theorem using gauge theory, Part 2
This is a continuation of last week’s talk in which we explain the definition of the homology theory used to prove that Birkhoff’s diamond is reducible. I will quickly summarize last week's discussion before heading into new material, so people can attend this week even if they couldn’t attend last week. This is joint work with Ben McCarty at University of Memphis.
Posted January 23, 2025
Last modified April 29, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Annette Karrer, The Ohio State University
Connected Components in Morse boundaries of right-angled Coxeter groups
Every finitely generated group G has an associated topological space, called a Morse boundary, that captures the hyperbolic-like behavior of G at infinity. It was introduced by Cordes generalizing the contracting boundary invented by Charney--Sultan. In this talk, we study subgroups arising from connected components in Morse boundaries of right-angled Coxeter groups and of such that are quasi-isom