Posted November 29, 2023
Last modified March 20, 2024
Geometry and Topology Seminar Seminar website
3:30 pm – 4:30 pm Lockett 233
Katherine Raoux, University of Arkansas
A 4-dimensional rational genus bound
The minimal genus question asks: “What is the minimum genus of a surface representing a particular 2-dimensional homology class?” Historically, minimal genus questions have focused on 2-dimensional homology with integer coefficients. In this talk, we consider a minimal genus question for homology classes with Q mod Z coefficients. We define the rational 4-genus of knots and present a lower bound in terms of Heegaard Floer tau invariants. Our bound also leads to PL slice genus bounds. This is joint work with Matthew Hedden.
Posted December 1, 2023
Last modified March 22, 2024
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Katherine Goldman, Ohio State University
CAT(0) and cubulated Shephard groups
Shephard groups are common generalizations of Coxeter groups, Artin groups, graph products of cyclic groups, and (certain) complex reflection groups. We extend a well-known result that Coxeter groups are CAT(0) to a class of Shephard groups that have “enough” finite parabolic subgroups. We also show that in this setting, if the underlying Coxeter diagram is type FC, then the Shephard group is cubulated (i.e., acts properly and cocompactly on a CAT(0) cube complex). Our method of proof combines the works of Charney-Davis on the Deligne complex for an Artin group and of Coxeter on the classification and properties of the regular complex polytopes. Along the way we introduce a new criteria (based on work of Charney) for a simplicial complex made of simplices of shape A_3 to be CAT(1).
Posted December 1, 2023
Last modified April 1, 2024
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Neal Stoltzfus, Mathematics Department, LSU
The Heart of the Braid Group
The ubiquitous braid group can be approached from many perspectives (algebraic geometrically, combinatorially, geometric group theoretically). This talk will concentrate on developing a description of the image of the known injective (finite dimensional faithful) representation of Lawrence/Krammer/Bigelow. Recalling Artin's faithful infinite dimensional representation and his "combing of pure braids", we first develop an analog for the (unfaithful) Burau representation case using covering spaces, local coefficients and the Reidemeister homotopical intersection theory for the braid action on one-point configurations. Next we introduce the braid action on the (unordered) two-point configuration space utilized by Krammer and Bigelow. For an easier description and computation, we will utilize the two-fold covering space of ordered pair configurations. The complements of these (discriminantal) arrangements are fibrations whose fundamental groups are semi-direct products from pure braid combing. Computing Blanchfield duality of the complements of open tubular neighborhood of the hyperplane arrangements we determine the first restriction on the image of the LKB representation: Hermitian form invariance under the intersection form discovered by Budney and Song. Additional conditions are determined by arithmetic monoidal conditions arising from matrix invariants over polynomial monoids, N[q, 1-q, t] related to (Dehornoy-Paris-Garside) group structures within the braid group. We conclude with a discussion of potential applications.
Posted December 6, 2023
Last modified April 1, 2024
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Joseph Breen, University of Iowa
The Giroux correspondence in arbitrary dimensions
The Giroux correspondence between contact structures and open book decompositions is a cornerstone of 3-dimensional contact topology. While a partial correspondence was previously known in higher dimensions, the underlying technology available at the time was completely different from that of the 3-dimensional theory. In this talk, I will discuss recent joint work with Ko Honda and Yang Huang on extending the statement and technology of the 3-dimensional correspondence to all dimensions.
Posted April 1, 2024
Last modified April 15, 2024
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Kevin Schreve, Louisiana State University
Homology growth and aspherical manifolds
Suppose we have a space X and a tower of finite covers that are increasingly better approximations to the universal cover. In this talk, we will be interested in how classical homological invariants grow as we go up the tower. In particular, I will survey various conjectures about the rational/F_p-homology growth and integral torsion growth in these towers. We'll discuss constructions of closed aspherical manifolds that have F_p-homology growth outside of the middle dimension, and give some applications to (non)-fibering of high-dimensional manifolds. This is joint work with Grigori Avramidi and Boris Okun.
Posted February 1, 2024
Last modified April 30, 2024
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Jean-François Lafont, The Ohio State University
Strict hyperbolizations produce linear groups
Strict hyperbolization is a process developed by Charney--Davis, which inputs a simplicial complex, and outputs a negatively curved piecewise hyperbolic space. By applying this process to interesting triangulations of manifolds, one can create negatively curved manifolds with various types of pathological large scale behavior. I will give a gentle introduction to strict hyperbolization, and will explain why the fundamental groups of the resulting spaces are always linear over Z. This is joint work with Lorenzo Ruffoni (Tufts University).
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 February 4, 2025
Last modified February 10, 2025
Geometry and Topology Seminar Seminar website
Lockett 233
Matthew Haulmark, Cornell University
TBA
Posted January 27, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Galen Dorpalen-Barry, Texas A&M
TBA
Posted January 23, 2025
Last modified January 27, 2025
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Annette Karrer, The Ohio State University
TBA