Titles and abstracts of talks

Anna Marie Bohmann, Vanderbilt University
Free Loop Spaces and Topological coHochschild Homology
Abstract: Free loop spaces arise in many areas of geometry and topology. Simply put, the free loops on a space X is the space of maps from the circle into X. This is a main object of study in string topology and has important connections to geodesics on manifolds. In this talk, we discuss a new approach to computing the homology of free loop spaces via topological coHochschild homology, which is an invariant of coalgebras arising from homotopy theory techniques. This approach produces a spectral sequence for the homology of free loop spaces with algebraic structure that allows us to make new computations. This is joint work with Teena Gerhardt and Brooke Shipley.

Kristen Hendricks, Rutgers University
Homology cobordism and Heegaard Floer homology
Abstract: The homology cobordism group consists of integer homology spheres under connected sum, modulo an equivalence relation called homology cobordism. We review some history of this group and discuss applications of Heegaard Floer homology to its structure. In particular, we show that the homology cobordism group is not generated by Seifert fibered spaces. This is joint work with J. Hom, M. Stoffregen, and I. Zemke.

Robin Koytcheff, University of Louisiana at Lafayette
Integrals, trees, and spaces of pure braids and string links
Abstract: The based loop space of configurations in a Euclidean space R^n can be viewed as the space of pure braids in R^{n+1}. In joint work with Komendarczyk and Volic, we studied its real cohomology using an integration map from a certain graph complex and recovered a result of Cohen and Gitler. Specifically, the map we studied is a composition of Kontsevich’s formality integrals and Chen’s iterated integrals. We showed that it is compatible with Bott-Taubes integrals for spaces of 1-dimensional string links in R^{n+1}. As a corollary, the inclusion of pure braids into string links in R^{n+1} induces a surjection in cohomology for any n>2. More recently, we showed that the dual to the integration map embeds the homotopy groups of the space of pure braids into a space of trivalent trees. We also showed that a certain subspace of these homotopy groups injects into the homotopy groups of spaces of k-dimensional string links in R^{n+k} for many values of n and k.

Jo Nelson, Rice University
Contact Invariants and Reeb Dynamics
Abstract: Contact topology is the study of certain geometric structures on odd dimensional smooth manifolds. A contact structure is a hyperplane field specified by a one form which satisfies a maximum nondegeneracy condition called complete non-integrability. The associated one form is called a contact form and uniquely determines a Hamiltonian-like vector field called the Reeb vector field on the manifold. I will give some background on this subject, including motivation from classical mechanics. I will also explain how to make use of J-holomorphic curves to obtain a Floer theoretic contact invariant, contact homology, whose chain complex is generated by closed Reeb orbits. This talk will feature numerous graphics to acclimate people to the realm of contact geometry.

Paul VanKoughnett, Texas A & M University
Localizations of Morava E-theory
Abstract: The Morava E-theory spectra E_n play essential roles in the study of K(n)-local stable homotopy theory. Many phenomena related to them can be described entirely algebraically in terms of formal groups, and a theorem of Goerss, Hopkins, and Miller uses this relationship to prove that E-theory has an essentially unique commutative ring structure. I will propose a similar formal-group-based picture for the less well-behaved transchromatic localizations L_{K(n-1)}E_n, and discuss the analogue of the Goerss-Hopkins-Miller theorem. Unlike the E-theory spectra themselves, these spectra admit exotic commutative ring structures not equivalent to the localization of the unique commutative ring structure on E-theory.

Abigail Ward, MIT
Symplectic Landau-Ginzburg models from non-Kahler disc counts
Abstract: Homological mirror symmetry is a mathematical phenomenon that exchanges algebro-geometric information on a manifold X with symplectic information on a manifold Y. While originally observed as an involutive exchange of data between pairs of Calabi-Yau manifolds, it is now understood that a large class of spaces enjoy HMS-type correspondences. A frequently occurring tool in the study of these correspondences is the notion of a Landau-Ginzburg model associated to a superpotential. Classically, the superpotential is a holomorphic function on Y which encodes a count of holomorphic discs on X that relies on the Kahler form on X. However, in certain cases when X fails to be Kahler, one can still construct a “symplectic” Landau-Ginzburg model associated to a non-holomorphic superpotential obtained from a "topological" count of holomorphic discs. We will discuss some of these cases, including that of the Hopf surface and some non-Fano toric varieties. The latter is joint work in progress with S. Ganatra, A. Hanlon, and J. Hicks.

Titles and abstracts of rodeo talks

Leo Digiosia, Rice University
Cylindrical contact homology of links of simple singularities
Abstract: We compute the cylindrical contact homology of the links of the simple singularities. These manifolds are contactomorphic to $S^3/G$ for finite subgroups $G\subset\text{SU}(2)$. The contact homology realizes information of the links as Seifert fiber spaces. The ranks of these groups are given in terms of $|\text{Conj}(G)|$, demonstrating a form of the McKay correspondence.

Daniel Grady, Texas Tech
The geometric cobordism hypothesis
Abstract: We rope and hog-tie a geometric enhancement of the cobordism hypothesis.

Rok Gregoric, UT Austin
The moduli stack of oriented formal groups
Abstract: The moduli stack of oriented formal groups is a spectral-algebro-geometric avatar of chromatic homotopy theory. We will sketch how it provides a bridge between classical formal groups and the stable category of spectra, and how certain central aspects of chromatic homotopy theory manifest in terms of its geometry.

Ang Li, University of Kentucky
A comparison between C_2-equivariant and classical squaring operations
Abstract: For any C_2-equivariant spectrum, we can functorially assign two non-equivariant spectra - the underlying spectrum and the geometric fixed point spectrum. They both induce maps from the RO(C_2)-graded cohomology to the classical cohomology. In this talk, I will compare the RO(C_2)-graded squaring operations with the classical squaring operations along the induced maps. This is joint work with Prasit Bhattacharya and Bertrand Guillou.

Elizabeth Tatum, University of Illinois at Urbana-Champaign
Towards Splitting $BP \langle 2 \rangle \wedge BP\langle 2 \rangle$ at Odd Primes
Abstract: In the 1980s, Mahowald and Kane used Brown-Gitler spectra to construct splittings of $bo \wedge bo$ and $l \wedge l$. These splittings were important input for the $bo$- and $l$-based Adams spectral sequences. I will discuss progress towards an analogous splitting for $BP\langle 2 \rangle \wedge BP \langle 2 \rangle$ at odd primes.

Misha Tyomkin, Dartmouth College
Numbers on barcode of a strong Morse function
Abstract: Morse function f on a manifold M is called strong if all its critical values are pairwise distinct. For a given field F, Barannikov decomposition (a.k.a. barcode) is a canonical pairing of some critical points of such f with neighboring indices. We present a construction which naturally associates a number (i.e. an element of F) to each Barannikov pair (a.k.a. bar in the barcode), defined up to a sign. It turns out that if homology of M over F is that of a sphere, then the product of all the numbers is independent of f (up to sign). We then proceed to consider homology with twisted coefficients (still in some field F). It is in this setting where the Reidemeister torsion is defined. We construct the twisted barcode and prove that the mentioned product equals to the Reidemeister torsion, whenever the latter is defined; in particular, it's again independent of f. Based on a joint work with Petya Pushkar, https://arxiv.org/abs/2012.05307.

Shawn Williams, Rice University
A Fox-Milnor Condition for 1-Solvable Boundary Links
Abstract: A well known result of Fox and Milnor states that the Alexander polynomial of slice knots factors as $f(t)f(t^{-1})$, providing us with a useful obstruction to a knot being slice. In 1978 Kawauchi extended this result to the case of the multivariable Alexander polynomial of slice links. In this talk, we will present an extension of this result for the multivariable Alexander polynomial of 1-solvable boundary links. This extends a previous result for certain localizions of the Alexander module.

Yilong Zhang, Ohio State University
Monodromy of vanishing cycles
Abstract: A vanishing cycle on a compact complex surface is a topological 2-sphere that contracts to a point as the surface degenerates and acquires an ordinary singularity. As a complex surface varies in the universal family of smooth hyperplane sections of a 3-fold, the monodromy of vanishing cycles carry topological information of the 3-fold. We investigate this problem for smooth hypersurfaces in CP^4.