Student Talks
Talks in A124 (Algebraic Topology)
Noah Wisdom (Northwestern University)
Title: Properties and Examples of A-Landweber Exact Spectra
Abstract: It is classically known that Landweber exact homology theories (complex oriented theories which are completely determined by complex cobordism) admit no nontrivial phantom maps. We propose a definition of $A$-Landweber exact spectra, for $A$ a compact abelian Lie group, and show that an analogous result on phantom maps holds. Also, we show that a conjecture of May on $KU_G$ is false. We do not prove an equivariant Landweber exact functor theorem, and therefore our result on phantom maps only applies to $MU_A$, $KU_A$, their $p$-localizations, and $BP_A$, which are shown to be $A$-Landweber exact by ad-hoc methods.
Chase Vogeli (Cornell University)
Title: Equivariant Homological Stability for Configuration Spaces
Abstract: In an open manifold, an unordered configuration of $n$ points can be extended to a configuration of $n+1$ points by ``adding a point near infinity.'' A classical result of McDuff and Segal shows this induces a phenomenon known as homological stability: after adding sufficiently many points, the low-dimensional homology groups of the unordered configuration space stabilize. In this talk, I’ll discuss an equivariant analogue of this phenomenon for manifolds equipped with an action by a finite group. In an appropriate sense, the Bredon homology of the resulting equivariant configuration spaces exhibits stability with respect to ``adding orbits at infinity.''
Ezekiel Lemann (Binghamton University)
Title: On Scissors Automorphism Groups
Abstract: In this talk we outline how to give a plus construction for scissors congruence K-theory. The arguments rely on K-theory of assemblers and homological stability. This talk is based on joint work with Alexander Kupers, Jeremy Miller, Cary Malkiewich, and Robin Sroka.
Kimball Strong (Cornell University)
Title: Strictification of Infinity Groupoids
Abstract: Grothendieck's "Homotopy Hypothesis" states that the homotopy theory of topological spaces is equivalent to the homotopy theory of weak infinity groupoids. Strict infinity groupoids are a simpler object that captures less information than weak infinity groupoids, but more than chain complexes. We define a functor from simplicial sets to simplicial T-complexes, a simplicial model for strict infinity groupoids, and prove it is left quillen. We further prove that the induced functor on quasicategories is comonadic; that is, induces an equivalence between spaces and coalgebras in strict infinity groupoids. In particular, two simplicial sets are weak homotopy equivalent if and only if the associated coalgebras of strict infinity groupoids are weak homotopy equivalent.
Sofía Martínez (Purdue University)
Title: Using Coalgebras as Models for Equivariant Spaces
Abstract: Given a commutative ring $R$, a $\pi_1$-$R$-equivalence is defined to be a continuous map of spaces inducing an isomorphism on fundamental groups and an R-homology equivalence between universal covers. If $R$ is the ring of integers then this notion coincides with that of a weak homotopy equivalence. When R is an algebraically closed field, Rivera and Raptis described a full and faithful (co)algebraic model for the homotopy theory of spaces up to $\pi_1$-$R$-equivalence by means of simplicial coalgebras considered up to a notion of weak equivalence created by the cobar functor. Their work extends previous algebraic models for spaces considered up to R-homology (Kriz, Goerss, Mandell) by including the information of the fundamental group in complete generality. In this talk, I will describe G-equivariant analogs of this statement obtained through generalizations of a celebrated theorem of Elmendorf.
Lewis Dominguez (University of Kentucky)
Title: Adams Operations on the Burnside Ring from Power Operations
Abstract: Topology furnishes us with many commutative rings associated to finite groups. These include the complex representation ring, the Burnside ring, and the G-equivariant K-theory of a space. Often, these admit additional structure in the form of natural operations on the ring, such as power operations, symmetric powers, and Adams operations. We will discuss two ways of constructing Adams operations. The goal of the talk is to understand these in the case of the Burnside ring.
Valentina Zapata Castro (University of Virginia)
Title: Transfer Systems Can Be Fun!
Abstract: In this talk, I plan to define what a transfer system is, along with saturation and compatibility concepts. Then I will talk about a result from joint work with Kristen Mazur, Angelica Osorno, Constanze Roitzheim, Rekha Santhanam and Danika Van Niel on compatibility of $C_{p^rq^s}$-transfer systems and how these concepts can be useful in Homotopy Theory.
Guoqi Yan (University of Notre Dame)
Title: The Generalized Tate Diagram of the Equivariant Slice Spectral Sequence
Abstract: The generalized Tate diagram developed by Greenlees and May is a fundamental tool in equivariant homotopy theory. In this talk, I will discuss an integration of the generalized Tate diagram with the equivariant slice filtration of Hill—Hopkins—Ravenel, resulting in a generalized Tate diagram for equivariant spectral sequences. This new diagram provides valuable insights into various equivariant spectral sequences and allows us to extract information about isomorphism regions between these equivariant filtrations.
As an application, we will begin by proving a stratification theorem for the negative cone of the slice spectral sequence. Building upon the work of Meier—Shi—Zeng, we will then utilize this stratification to establish shearing isomorphisms, explore transchromatic phenomena, and analyze vanishing lines within the negative cone of the slice spectral sequences associated with periodic Hill—Hopkins—Ravenel and Lubin—Tate theories. This is joint work with Yutao Liu and XiaoLin Danny Shi.
Lucas Williams (Binghamton University)
Title: Periodic Points and Equivariant Parameterized Cobordism
Abstract: In this talk we investigate invariants that count periodic points of a map. Given a self map f of a compact manifold we could detect n-periodic points of f by computing the Reidemeister trace of f n or by computing the equivariant Fuller trace. In 2020 Malkiewich and Ponto showed that the collection of Reidemeister traces of f^k for varying k|n and the equivariant Fuller trace are equivalent as periodic point invariants, and they conjecture that for families of endomorphisms the Fuller trace will be a strictly richer invariant for n-periodic points.
In this talk we will explain our new result which confirms Malkiewich and Ponto’s conjecture. We do so by proving a new Pontryagin-Thom isomorphism between equivariant parameterized cobordism and the spectrum of sections of a particular parametrized spectrum and using this result to carry out geometric computations.
Talks in A126 (Differential Geometry)
Malik Tuerkoen (University of California, Santa Barbara)
Title: Fundamental Gap Estimates in Various Geometries
Abstract: The fundamental gap is the difference of the first two eigenvalues of the Laplace operator, which is important both in mathematics and physics and has been extensively studied. For the Dirichlet boundary condition the log-concavity of the first eigenfunction plays a crucial role in proving lower bounds, which was established for convex domains in the Euclidean space and the round sphere. In recent works, there has been progress in proving gap estimates on perturbations of the round sphere in dimension two and conformal deformations in higher dimensions. For negatively curved spaces, it turns out that there is no uniform lower bound of the fundamental gap. Hence it is natural to ask whether one can prove a fundamental gap estimate, assuming a stronger notion of convexity. In recent work there has been progress on answering this question. This is based on joint work with G. Khan, H. Nguyen, S. Saha and G. Wei in various subsets.
Joaquin Lema (Boston College)
Title: Surface group representations arising from complex ODEs
Abstract: A complex projective structure on a surface is an atlas of charts to CP^1 such that the coordinate changes are Mobius transformations. This type of geometric structure appears naturally while studying convex cocompact hyperbolic 3-manifolds. In this talk, I'll explain a classical fact that gives a one-to-one correspondence between a particular family of ODEs and this geometric structure and then present another family related to this one that gives rise to representations of surface groups into SL_n (C) for any n. The goal is to state many questions about these representations that I don't know how to answer.
Paul Sweeney (Stony Brook University)
Title: A New Counterexample to Min-Oo's Conjecture via Tunnels
Abstract: Min-Oo's Conjecture is a positive curvature version of the positive mass theorem. Brendle, Marques, and Neves produced a perturbative counterexample to this conjecture. In 2021, Carlotto asked if it is possible to develop a novel gluing method in the setting of Min-Oo's Conjecture and in doing so produce new counterexamples. In this talk, we will build upon the perturbative counterexamples of Brendle-Marques-Neves in order to construct counterexamples that make advances on the theme expressed in Carlotto's question. These new counterexamples are non-perturbative in nature; moreover, we also produce examples with more complicated topology. Our main tool is a quantitative version of Gromov-Lawson Schoen-Yau surgery.
A. Sophie Aiken (University of California, Santa Cruz)
Title: Recent Progress on the Fractional Yamabe Problem
Abstract: Let $(M^n, [\hat{g}])$ be the conformal infinity of an asymptotically hyperbolic Einstein (AHE) manifold $(X^{n+1},g^+).$ We will take the scattering operator associated to the AHE filling in as the fractional conformal Laplacian. Equipped with fractional conformal Laplacians defined via the AHE manifold, we can define a fractional Yamabe problem looking for a conformal metric of $(M^n,[\hat{g}])$ which has constant fractional scalar curvature. We will present some recent developments on the fractional Yamabe problem assuming an AHE filling in.
Yulun Xu (Stony Brook University)
Title: Taking Sections of Plurisubharmonic Functions and Their Application to the Complex Monge-Ampere Equation
Abstract: Let w0 be a bounded, C3, strictly plurisubharmonic function defined on B1⊂ℂn. Then w0 has a neighborhood in L∞(B1). Suppose that we have a function ϕ in this neighborhood with 1−ϵ≤MA(ϕ)≤1+ϵ. Then we can define sections of the ϕ, which have similar properties to the sections of convex functions studied by Caffarelli and Gutierrez. Then we prove the W^{2,p} estimate for ϕ. Suppose that there exists a function u solving the linearized complex Monge-Ampere equation: det(ϕkl¯)ϕIj¯uIj¯=0. Then one has an estimate on |u|C^α(B12) for some α>0.
Arthur Mills (Oregon State University)
Title: A Discrete Curvature Approach to the Drill String Problem
Abstract: In the drilling of a well, the drill string may come into contact with the well bore. The curvature of the well bore determines where these contact points arise. As the drill bit turns, potential contact points are created along the wall of the well. These candidates then become realized as contact points once the bit passes threshold distances related to the surface features of these points. Beyond a certain distance these contact points become permanent in the sense that the drill string will remain in contact with these points for the entirety of the drilling operation. We will use standard techniques from the differential geometry of curves and surfaces to determine these points of contact and compute them in a MATLAB implementation.
Austin Bosgraaf (Oregon State University)
Title: Classification, Symmetry, and Positive Curvature
Abstract: Classification problems are ubiquitous in mathematics; and in the study of positive curvature, classification is a central theme. In 2003, Wilking introduced new tools to the study of positive curvature which lead to new classification results in the presence of torus symmetry. This talk will introduce Wilking's tools and discuss their application in positive curvature. We will see recent results in positive curvature due to Kennard, Khalili Samani, and Searle in the case of discrete actions, and I will present some of my own results.
Alex Xu (Columbia University)
Title: $Pin^-(2)$ Monopoles and Finite Volume Einstein 4-Manifolds
Abstract: We construct infinitely many examples of noncompact 4-manifolds with $T^3$ ends that do not admit any asymptotically hyperbolic Einstein metrics. The techniques for this use estimates coming from Seiberg-Witten theory as well as constructions coming from the $Pin^-(2)$ monopole equations.
Ivo Terek (Ohio State University)
Title: Codazzi Tensors in Homogeneous Spaces
Abstract: Codazzi tensor fields are ubiquitous in geometry, with the most prominent examples being the second fundamental form of a hypersurface of a Riemannian manifold having constant sectional curvature, and the Ricci tensor of a Riemannian manifold with harmonic curvature. We'll discuss the notion of covariant exterior derivative, more examples of Codazzi tensor fields, and then focus on our recent results about Codazzi tensor fields on homogeneous spaces. In particular, we ultimately show that every invariant Codazzi tensor field on a naturally reductive homogeneous space is necessarily parallel. This is joint work with James Marshall Reber.
Talks in A128 (Geometric Group Theory + TDA)
Ethan Semrad (Florida State University)
Title: Hypernetwork Gromov-Hausdorff Distance: Graphification and Affinity Networks
Abstract: Hypernetworks are a generalization of the standard network that capture multi-way relationships in data. Data with potential hypernetwork structure can show up in many biological systems. Our theoretical foundations for studying the space of hypernetworks will use ingredients from optimal transport. First, we define a hypernetwork distance with a Gromov-Hausdorff structure and examine some of its theoretical properties. Then we define a new type of hypernetwork-to-network transformation in the form of affinity networks and study its Lipschitz bounds. These networks are extensions of the common techniques of clique- and line-graph transformations and may provide systems that capture more of the global relationships between the nodes or hyperedges of the transformed hypernetwork.
Megha Bhat (CUNY Graduate Center)
Title: Orientation-preserving Homeomorphisms of Euclidean Space Are Commutators
Abstract: A uniformly perfect group has commutator width p if every element can be expressed as a product of p commutators. Questions about commutator width have been asked and answered for various groups such as the alternating group and the symmetric group. I will talk about this question for homeomorphism groups of spheres, annuli and Euclidean space, and show that each of these has commutator width one.
Eliot Bongiovanni (Rice University)
Title: Extensions of Finitely Generated Veech Groups
Abstract: Given a closed surface S, a subgroup G of the mapping class group of S has an associated extension group Γ, which is the fundamental group of an S-bundle. A general problem is to infer features of Γ from G: I take G to be a finitely generated Veech group and show that Γ is hierarchically hyperbolic. The focus of this talk is constructing a hyperbolic space Ê on which Γ acts nicely (isometrically and cocompactly). I will briefly survey my subsequent results concerning hierarchical hyperbolicity and quasi-isometric rigidity of Γ, and I will review the broader mathematical discourse of "geometric finiteness" in the context of subgroups of mapping class groups.
The main content of the talk is “proof by picture” and will be accessible to those who are familiar with hyperbolic plane geometry (e.g., the Poincare disk model).
Tomoya Tatsuno (University of Oklahoma)
Title: Abstract Automorphisms of Heisenberg Groups from a Geometric Perspective
Abstract: In 1980, A. Kaplan introduced a class of nilmanifolds attached to Clifford modules, i.e., 2-step nilpotent Lie groups of Heisenberg type. In 1982, Riehm determined their isometry group. In 1996, Saal further determined their Lie group automorphism group, exploiting the Riemannian structure. In this talk, we study their abstract (not necessarily continuous) automorphism group. This is motivated by a classical rigidity result due to Cartan (1930) and van der Waerden (1933): any abstract group automorphism of a compact, simple Lie group must be a Lie group automorphism.
Important examples of 2-step nilpotent Lie groups of Heisenberg type arise as an Iwasawa N-group of symmetric spaces of rank 1. They are precisely the (2n+1)-dimensional Heisenberg groups, (4n+3)-dimensional quaternionic Heisenberg groups, and 15-dimensional octonionic Heisenberg group. For these groups, we show that any group automorphism is a product of a (possibly discontinuous) central automorphism and a Lie group automorphism, using the result of Riehm. Thus, the discontinuity occurs only at the center for these nilpotent Lie groups. The (2n+1)-dimensional Heisenberg group case is already known by Gibbs and Khor, but we give a uniform proof from a geometric perspective.
Mihail Arabadji (University of Massachusetts Amherst)
Title: Nielsen Realization Problem
Abstract: There are many families of non-spin 4–manifolds for which the smooth Nielsen realization problem fails; that is, there are (finite) subgroups of their mapping class groups that cannot be realized by a group of diffeomorphisms. This extends and complements the recent results for spin 4–manifolds. This is joint work with Inanc Baykur.
Zihao Liu (Rice University)
Title: Scaled Homology and Topological Entropy
Abstract: In this talk, I will introduce a scaled homology theory, lc-homology, for metric spaces such that every metric space can be visually regarded as “locally contractible” with this newly-built homology as well as its connection to classic singular homology theory. In addition, after briefly introducing topological entropy, I will discuss how to generalize one of the existing results of entropy conjecture, relaxing the smooth manifold restrictions on the compact metric spaces, by using lc-homology groups. This is joint work with Bingzhe Hou and Kiyoshi Igusa.
Dalton Sconce (Indiana University, Bloomington)
Title: Why do we Care about Geodesic Currents
Abstract: Using tools from geometric measure theory might seem strange if we are concerned with the geometric and topological structure of a surface; however it turns out that by various means we can turn things like curves and metrics into the same type of object — a geodesic current. This provides us the ability to actually compare these disparate elements in a meaningful way. This expository talk will be primarily explaining what sort of creatures geodesic currents are and why we might care about them. If time permits, we’ll discuss in particular how the geometric intersection number that is defined for curves on a surface can be extended to the space of currents to recover other interesting geometric data.
Manisha Garg (University of Illinois, Urbana-Champaign)
Title: Asymptotic Dimension and Assouad-Nagata Dimension
Abstract: The concept of dimension has been studied for centuries. Topological dimension (or Lebesgue covering dimension) of a metric space is at most n if it is at most n-dimensional at microscopic scales, and asymptotic dimension of a metric space is at most n if it appears n-dimensional upon zooming out as far as needed. The notion of Assouad-Nagata dimension captures both small-scale and large-scale behaviors of metric spaces. Specifically, the Assouad-Nagata dimension of a metric space is at most n if it is n-dimensional at every scale (whether zooming in or out), maintaining its n-dimensional appearance.
In this talk, we will introduce asymptotic dimension and Assouad-Nagata dimension along with their basic properties. We will explore the relationship between these dimensions and a few others, discussing what information they provide about the space itself.
Talks in A106 (Geometric Topology)
Haochen Qiu (Brandeis University)
Title: A Surgery Formula for Families Seiberg-Witten Invariants
Abstract: We prove a surgery formula for the ordinary Seiberg-Witten invariants, and a surgery formula for the families Seiberg-Witten invariants of families of $4$-manifolds obtained through fibrewise surgery. Our formula expresses the original Seiberg-Witten moduli space cut down by a cohomology class in the configuration space in terms of the Seiberg-Witten invariants of the surgery manifold, under certain assumptions on the families. We use these surgery fomula to investigate the homotopy type of the space of positive scalar curvature metrics of $4$-manifolds that are not simply connected.
Michele Capovilla-Searle (University of Iowa)
Title: Birman-Ko-Lee Left Canonical form and it’s Applications
Abstract: The dual Garside left-canonical form and the fractional dehn twist coefficient is used to solve the conjugate problem in braid theory; introduced by Xu, Kang-Ko-Lee, Birman-Ko-Lee. The fractional Dehn twist coefficient (FDTC) is a measurement of boundary twist of a surface homeomorphism, introduced by Honda-Kazez-Matic. In contact geometry it is a powerful tool to detect tightness/overtwistedness of a given contact structure. In this talk I will give applications of the left-canonical form to the FDTC of braids and Bennequin inequality of transverse links. This is joint work with Keiko Kawamuro and Rebecca Sorsen.
Ian Sullivan (University of California, Davis)
Title: Kirby Belts and the Skein Lasagna Module of S2xS2
Abstract: The lasagna module of a chosen link homology theory is a new kind of smooth $4$-manifold invariant. In this talk, we define and investigate a certain homotopy colimit of a directed system of tangle complexes whose homology is the skein lasagna module for $\mathfrak{gl}_{2}$ Khovanov-Rozansky homology of $S^2\times B^2$ with geometrically essential links in the boundary. We then use this new computational technique to compute the skein lasagna module of $S^2\times S^2$, and to show that the skein lasagna module of $S^2\times B^2$ recovers Khovanov homology for links in $S^1\times S^2$.
Ollie Thakar (Harvard University)
Title: L-Space Conjecture and Left-Orderable Knot Surgeries
Abstract: A group is left-orderable if it admits a total order which is invariant under left-multiplication. Hence, the left-orderability of the fundamental group of a 3-manifold is a topological invariant. I will discuss the L-Space Conjecture, a recent conjecture of Boyer-Gordon-Watson which (amazingly!) hopes to demonstrate this invariant is interesting through a unification of the algebraic, geometric, and analytical aspects of 3-manifolds: it predicts that for a rational homology sphere, the notions of a Heegaard Floer L-space, taut foliations, and left-orderability coincide. I will then review an original strategy to prove a knot surgery has left-orderable fundamental group, featuring some surprising techniques including group cohomology.
Yikai Teng (Rutgers University, Newark)
Title: Infinite Order Knot Traces
Abstract: In 2017, Robert Gompf proved the existence of infinite order corks~\cite{Gom17}. Namely, there exists a cork $(C,f)$ such that $f^k$ does not extend to a self-diffeomorphism of the cork for any $k$. In fact, if we do a cork twist via such maps $f^k$, it is possible to result in pairwise non-diffeomorphic 4-manifolds. In this talk, we will adopt this idea to show that knot traces can actually behave as infinite order plugs.
Shuo Zhang (University of Minnesota, Twin Cities)
Title: Floer Homology of Iterated Dehn Twist
Abstract: Seidel conjectured that the fix point Floer homology of the global monodromy of a Lefschetz fibration is the Hochschild homology of the Fukaya category generated by the vanishing cycles. We prove a slightly more general result on twisted Hochschild homology by using the Lagrangian cobordism result of Mak-Wu.
Roman Krutowski (University of California, Los Angeles)
Title: Higher-dimensional Heegaard Floer Fukaya Category of Cotangent Bundles
Abstract: Higher-dimensional Heegaard Floer homology introduced by Collin, Honda and Tian is an invariant of Liouville domains, which is defined in the spirit of Lipshitz's cylindrical reformulation. Further, one may associate an HDHF Fukaya category to any Liouville domain. The first computations of this A_\infty-category was obtained in the work of Honda, Tian and Yuan in the case of cotangent bundles of orientable surfaces of genus at least 1. In this talk I will tell about Morse-theoretic model which allows one to compute HDHF A_\infty algebra naturally associated with any cotangent bundle and I will illustrate it in case of the cotangent bundle of the two-dimensional sphere. This talk is based on a joint work with Honda, Tian and Yuan.
Melody Molander (University of California, Santa Barbara)
Title: Skein Theory of Subfactor Planar Algebras
Abstract: Subfactor planar algebras first were constructed by Vaughan Jones as a diagrammatic axiomatization of the standard invariant of a subfactor. Planar algebras can be conveniently encoded by diagrams in the plane. These diagrams satisfy some skein relations and have an invariant called an index. The Kuperberg Program asks to find all diagrammatic presentations of subfactor planar algebras. This program has been completed for index less than 4. In this talk, I will introduce subfactor planar algebras and find presentations for subfactor planar algebras of index 4 associated with the affine A Dynkin diagram.
Randy Van Why (Northwestern University)
Title: Symplectic and Contact Geometries in Low Dimensions
Abstract: Using the complex projective plane CP2 (a 4-manifold), I will explain, by example, a number of notions from contact and symplectic geometry such as symplectic normal crossing divisors, Boothby-Wang Bundles, symplectic concavity/convexity, and Weinstein manifolds. At times, I will touch on peripheral concepts such as symplectic Lefschetz fibrations, open book decompositions, and allusions to Floer theories. No experience in symplectic or contact geometry is required.
Talks in A108 (Knot Theory + TDA)
Ipsa Bezbarua (CUNY Graduate Center)
Title: Contractibility of the Knot Complex of Incompressible Spanning Surfaces
Abstract: 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 very 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.
Chung-Ping Lai (Oregon State University)
Title: Group Ring and the Homology of a Simplicial G-Complex
Abstract: Simplicial complexes offer a way to combinatorically study many topological spaces of interest and can be used to compute the homology with any coefficients of a topological space. A simplicial complex becomes a simplicial G-complex when there is a group acting on it. In this talk, we will first give some background of simplicial G-complexes. Then we will introduce a new approach to computing the homology of a simplicial G-complex that is inspired by representation theory, which studies groups by representing their elements as linear transformations. In particular, we utilize the construct of a group ring. In the case when G is a cyclic group of prime order and we are computing homologies with field coefficients, an explicit algorithm to compute the homology using this approach will be provided.
John Carney (Virginia Commonwealth University)
Title: 2-adjacent Knots
Abstract: An $n$-adjacent knot is a knot that contains some $n$ crossings such that changing any non-empty subset of those crossings results in the unknot. This will introduce the concept of $n$-adjacency, discuss previous results where $n$ is greater than 2 and show some of our results on the case of 2-adjacency, including the Conway polynomial and 4-genus of 2-adjacent knots. This talk will explore how the $n=2$ case is more complicated than when $n>2$ and ways that we can approach this case.
Peyton Wood (University of California, Davis)
Title: Using Quandle Invariants to Distinguish Classical and Legendrian Knots
Abstract: We will review the quandle State-Sum Invariant and discuss emerging strategies for using this invariant to distinguish knots. We will share progress made towards using the Quandle State-Sum Invariant to classify prime knots up to reversal and mirror imaging. Then, we will briefly overview Legendrian knots, define Legendrian racks, and show how one can use them to distinguish Legendrian knot isotopy classes using a counting invariant defined by the number of Legendrian rack colorings of the front projection. Lastly, we will discuss how Legendrian racks give rise to a similar Sum-State Invariant and demonstrate its potential to be used in distinguishing Legendrian knot isotopy classes.
Susan Rutter (CUNY Graduate Center)
Title: Links in Thickened Surfaces
Abstract: In 1978, Gordon and Litherland introduced a pairing on a spanning surface of a knot, generalising the Goeritz matrix for unorientable spanning surfaces. In 2022, Boden, Chrisman, and Karimi introduced a generalisation of the GL pairing for links in thickened surfaces. Moreover, the relation of the GL form to the intersection form of the double branched cover extends to a similar theorem for the case of thickened surfaces. I will present the classical and recent work.
Yan Tao (University of California, Los Angeles)
Title: Stable Homotopy Invariants of Link Floer Homology
Abstract: I will give an overview of combinatorial Link Floer homology and how it is used to construct a Link Floer stable homotopy type. I will then present some of my own work in working towards computing stable homotopy invariants thereof.
Aakash Parikh (Rutgers University, New Brunswick)
Title: Localization Spectral Sequences for Strongly Invertible Knots
Abstract: A knot K is said to be "strongly invertible" if there is a Z/2 symmetry of S^3 that fixes K setwise and reverses the orientation of K. In this talk we will discuss two new spectral sequences in knot Floer homology associated to a strongly invertible knot. A novel feature of one of the spectral sequences is that its E^1 page is given by the singular knot Floer homology of the singularized butterfly link of K. These constructions were inspired by work of Hendricks on the link Floer homology of doubly periodic knots, and work of Lipshitz and Sarkar on the Khovanov homology of strongly invertible knots.
Miguel Lopez (University of Pennsylvania)
Title: Cellular Sheaves of Lattices
Abstract: Cellular sheaves have seen a wide variety of uses from the modeling of opinion dynamics, delay tolerant networks, graphic statics and even in machine learning. In many such instances one considers sheaves valued in vector spaces which are both computable and amenable to traditional homological algebra. In this talk, we introduce sheaves of lattices and Galois connections, discuss what they can be good for, and how to make sense of sheaf cohomology and a sheaf Laplacian in this exotic setting. We conclude with applications to formal concept analysis for data mining.