Titles and abstracts of plenary talks

Prasit Bhattacharya, New Mexico State University
Equivariant Steenrod Operations
Abstract: Classical Steenrod operations is one of the most fundamental and formidable tools in stable homotopy theory. It led to calculation of homotopy groups of spheres, calculation of cobordism rings, characteristic classes, and many other celebrated applications of homotopy theory to geometry. However, equivariant Steenrod operations are not known beyond the group of order 2. In this talk, I will demonstrate a new geometric construction of the classical Steenrod operations and generalize it to construct G-equivariant Steenrod operations for any finite group G. Time permitting, I will discuss potential applications to equivariant geometry.

Kalina Mincheva, Tulane University
Tropical adic spaces
Abstract: The process of tropicalization associates to an algebraic variety its combinatorial shadow - the tropical variety. Tropical varieties do not come naturally with extra structure such as scheme structure. Moreover, tropicalization as defined in the literature -- in any of its variants (algebraic or analytic) -- is not a morphism in any category. Working towards endowing tropical varieties with extra structure, we study the algebra of convergent tropical power series and the topological spaces (of prime congruences) it corresponds to. The construction so far allows us to see a tropicalization as a natural transformation of functors taking values in the category of topological spaces.

I will not assume any prior knowledge of tropical geometry.

Peter Patzt, University of Oklahoma
Steinberg module in algebraic topology
Abstract: In this talk, I will give an introduction to the applications of the Steinberg module in algebraic topology. The Steinberg module was first introduced in representation theory but later Solomon and Tits discovered a geometric description as the top homology of the Tits building. It has many applications in algebraic topology and I want to highlight a few of them. Borel and Serre found it to be the virtual dualizing module of arithmetic groups. It comes up in spectral sequences Introduced by Quillen and by Rognes that compute algebraic K-theory. This talk includes joint works with Benjamin Brück, Alexander Kupers, Jeremy Miller, Rohit Nagpal, Andrew Putman, Robin Sroka, and Jennifer Wilson.

Maru Sarazola, Johns Hopkins University
Fibrantly-induced model structures
Abstract: Model structures are robust categorical structures that provide an abstract framework to do homotopy theory. Unfortunately, in practice it is often very hard to prove that something satisfies the requirements of a model structure. To this end, there are several results in the literature that explore techniques for constructing model structures on a given category C.

A natural way to do this is to right-induce it through a right adjoint $R: C\to M$ to a known model structure M. This process defines the fibrations and weak equivalences in C as the morphisms whose image under R is one such map in M. Depending on the context, however, this may prove too restrictive. For example, one may be working in a setting where there is a desired class of “fibrant objects” in mind for C, and where the best one can hope for is for these well-behaved classes of fibrations and weak equivalences to hold only between fibrant objects. This leads to what we call a fibrantly-induced model structure.

After a brief review of model structures, this talk will present the idea of fibrantly-induced model structures, and explore some applications. Based on recent work with Leonard Guetta, Lyne Moser and Paula Verdugo.

Shira Tanny, Institute for Advanced Study
Closing lemmas in contact dynamics and holomorphic curves
Abstract: Given a flow on a manifold, how to perturb it in order to create a periodic orbit passing through a given region? While the first results in this direction were obtained in the 1960-ies, various facets of this question remain largely open. I will review recent advances on this problem in the context of contact flows, which are closely related to Hamiltonian flows from classical mechanics. In particular, I'll discuss a proof of a conjecture of Irie stating that rotations of odd-dimensional ellipsoids admit a surprisingly large class of perturbations creating periodic orbits. The proof involves methods of modern symplectic topology including pseudo-holomorphic curves and contact homology. The talk is based on a joint work with Julian Chaidez, Ipsita Datta and Rohil Prasad.

Melissa Zhang, UC Davis
Spectral sequences from Khovanov to Floer
Abstract: Spectral sequences are methods for computing homologies; each successive page is a better approximation of the total homology than the last. When we first learn about spectral sequences, we tend to first learn about examples that begin at a bicomplex, and such that the differentials on each successive page look like increasingly slanted arrows.

In this talk, I want to help familiarize the larger topology community with the types of spectral sequences low-dimensional topologists work with, particularly those sequences relating Khovanov and Floer homologies, which may at first seem more abstract than the spectral sequences we first learned about. We will also discuss applications of such spectral sequences. This talk is based on joint upcoming work with Akram Alishahi and Linh Truong.

Titles and abstracts of rodeo talks

Jonathan Johnson, Oklahoma State University
Non-standard orders on torus bundles with one boundary
Abstract: Consider a torus bundle over the circle with one boundary. Perron-Rolfsen shows that having an Alexander polynomial with real positive roots is a sufficient condition for a surface bundle with one boundary to have bi-orderable fundamental group. This is done by showing the action induced by the monodromy preserves a "standard" bi-ordering of the fundamental group of the surface. In this talk, we discuss if there are other ways to bi-order the fundamental group of a torus bundle with one boundary component. This work is joint with Henry Segerman. This work is partially funded by NSF grant DMS-2213213.

Evgeniya Lagoda, FU Berlin
k-regular maps and algebraic topology
Abstract: A map $f: X \rightarrow \mathbb R^n$ is k-regular if whenever $x_1, \ldots, x_k$ are distinct points in $X$, then $f(x_1), \ldots, f(x_k)$ are linearly independent. The study of k-regular maps is thought to be originating in Karol Borsuk’s work in the 50’s and was further fueled by their relationship to approximation theory. In my presentation, I will explain some ideas on how algebraic topology helps to get bounds on (non)existence of k-regular maps.

Peter Marek, Indiana University
Computing with Synthetic Spectra
Abstract: To a sufficiently nice spectrum E, Pstragowski produced a certain stable infinity category Syn_E, called "E-synthetic spectra," whose objects, in a certain sense, remember information about E-Adams spectral sequence calculations. This homotopy theory is bigraded in the sense that there are bigraded spheres and bigraded homotopy groups.

In this talk, we discuss synthetic spectra and some of its basic calculational features in the case of E=HF_2, including how to compute bigraded synthetic homotopy groups and their applications to classical Adams spectral sequence calculations for p=2. In particular, we discuss our computation of the bigraded synthetic homotopy groups of 2-complete tmf, the connective topological modular forms spectrum.

Sarah Petersen, UC Boulder
A Thom Spectrum Model for $C_2$-Integral Brown-Gitler Spectra
Abstract: We make a Thom spectrum model for a $C_2$-equivariant analogue of integral Brown-Gitler spectra precise. Nonequivariantly, integral Brown-Gitler spectra have many computational uses, including splitting of $bo \wedge bo.$ The $C_2$-equivariant spectra we construct share many analogous properties to the nonequivariant integral Brown-Gitler spectra and thus should be useful for producing similar splittings in the $C_2$-equivariant setting. Our main motivation is to use these $C_2$-equivariant Thom Spectra spectra to produce a $C_2$-equivariant spectrum-level splitting of the $BP_\mathbb{R} \langle n \rangle$-cooperations algebra at heights zero and one. This is ongoing joint work with Guchuan Li and Elizabeth Tatum.

Riccardo Pedrotti, UT Austin
Towards a count of holomorphic sections of Lefschetz fibrations over the disc
Abstract: in this brief talk I’d like to report on current work in progress with T. Perutz about a refinement of the Seidel exact triangle to obtain a combinatorial formula for counting holomorphic sections of Lefschetz fibrations over the disk, keeping track of their relative homology classes. Why is this interesting? The reason is that, given a 4-dim Lefschetz fibration over the 2-sphere, one is naturally lead to ask what are its SW invariants. These invariants keep track of Spin^c structures, equivalently - on a closed oriented 4-manifold- second homology classes. A formula for holomorphic sections should lead to a combinatorial formula for the SW-invariants.

Kelly Pohland, Vanderbilt University
The $RO(C_3)$-graded Bredon cohomology of $C_3$-surfaces
Abstract: For spaces with an action of a finite group $G$, there an equivariant cohomology theory called Bredon cohomology, graded on real, finite dimensional $G$-representations. In this talk, we give a brief introduction to this theory and explore a recent family of computations in the case where $G$ is the cyclic group of order 3.

Ignat Soroko, University of North Texas
Property $R_\infty$ for Artin groups
Abstract: A group $G$ has property $R_\infty$ if for every automorphism $\phi$ of $G$ the number of twisted $\phi$-conjugacy classes is infinite. This property is motivated by the topological fixed point theory, and has been a subject of active research. Among the groups which have this property are hyperbolic and relatively hyperbolic groups, mapping class groups, generalized Baumslag-Solitar groups and some others. However, the general picture of groups having this property is quite elusive. In a joint project with Matthieu Calvez, we establish property $R_\infty$ for some spherical and affine Artin groups by utilizing their close relation to certain mapping class groups of punctured surfaces.

Himanshu Yadav, University of Florida
Engineering Innovation: applying topological data analysis to document and topic embeddings to identify persistent gaps in scientific literature
Abstract: Science is growing exponentially, and recognizing substantive areas whose development would yield an important translational impact has become increasingly difficult. We introduce a systematic approach to identifying avenues for innovation inscientific literature by utilizing Topological Data Analysis (TDA), a set of tools used to understand the shape and organization of distributed data. First, we identify holes in embedded scientific literature by applying TDA. We hypothesize that these holes represent untapped or underdeveloped areas of research. Next, persistent homology is used to identify the perimeter of these persistent gaps. We visualize the perimeter of these holes, and their broader context, as networks of substantive relationships. Finally, we present preliminary evaluations of this method, examining the extent that persistent holes which existed from 2001-2010 and filled by new research in 2011-2020. We discuss a range of applications for researchers, administrators, and policymakers, including the guidance of grant solicitations, hiring decisions, etc.