Classifying spaces for commutativity in groups

Omar Antolín Camarena (UNAM) Friday 1:30 Slides

Given a topological group G, we can think of the space of homomorphisms \(\hom(\mathbb{Z}^n,G)\) as the space of n-tuples of elements of G that commute pairwise. These spaces are more subtle than one might think, and even basic invariants such as the number of connected components can lead to surprising results. Fixing G and varying n we get a simplicial space, whose geometric realization is known as the classifying space for commutativity in G. I will survey what is known about these classifying spaces, whose study is still fairly young.

Properties and Examples of A-Landweber Exact Spectra

Noah Wisdom (Northwestern) Friday 2:45

It is classically known that Landweber exact homology theories (complex oriented theories which are completely determined by complex cobordism) admit no nontrivial phantom maps. Herein 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.

Stabilization of 2-Crossed Modules

Milind Gunjal (Florida State) Friday 3:15

For a Waldhausen category, 1-type of the K-theory spectrum is well studied by Muro and Tonks using stable crossed modules. Using the same technique we define the 2-type with the help of a 2-crossed module. Further, we look at its stabilization using symmetric monoidal 2-categories.

Tropical adic spaces

Kalina Mincheva (Tulane) Friday 4:00

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 map as a natural transformation of functors taking values in the category of topological spaces.

Infinity cyclic operads, configuration spaces and Galois symmetries

Marcy Robertson (Melbourne) Saturday 9:00

The idea behind Grothendieck–Teichmüller theory is to study the absolute Galois group via its actions on (the collection of all) moduli spaces of genus g curves. In practice, this is often done by studying an intermediate object: The Grothendieck–Teichmüller group, GT. In this talk, I’ll describe an infinity cyclic operad of configurations on the cylinder which gives a nice homotopical approximation to the genus zero Teichmüller tower. Includes joint work with Luciana Basualdo Bonatto.

Elmendorf's Theorem for Diagrams

Hannah Housden (Vanderbilt) Saturday 10:30

The notion of group action on a topological space can be generalized to the notion of an action by a category. This talk will explore the basic notions and discuss what happens in the case of a category with two objects with one non-identity morphism. As it turns out, there are highly nontrivial homotopical structures here, despite the category being very simple.

From Cubes to Stars: Revisiting the Additivity of the Little Cube Operads

Ben Szczesny (Ohio State) Saturday 11:00

Dunn additivity is a classic statement that tells us that the Boardman–Vogt tensor product of little cube operads is homotopically additive in the dimension of the cubes. This result is a bit of an outlier: because the tensor product isn't homotopical, we aren't guaranteed the same result for other \(\mathbb{E}_k\)-operads, even very similar ones like the little disk operads. In this short contributed talk, we will discuss the main obstacles of extending Dunn's result, some techniques the speaker has come up with to compare embedding operads of different shapes that work in homotopically risky contexts, and extensions of Dunn's result to more general "little star operads". We will also explain equivariant extensions which are related to the \(\mathbb{N}_\infty\)-operads of Blumberg–Hill, and \(\mathbb{E}_\sigma\)-algebras, which (to name one example) have found use in Hahn–Shi's work on Real orientations of Real Morava E-theories.

Contractibility of the orbit space of a saturated fusion system after Steinberg

Jonathon Villareal (UL Lafayette) Saturday 2:00

Recently, Steinberg used discrete Morse theory to give a new proof of a theorem of Symonds that the orbit space of the poset of nontrivial p-subgroups of a finite group is contractible. We extend Steinberg’s argument in two ways, covering more general versions of the theorem that were already known. In particular, following a strategy of Libman, we give a discrete Morse theoretic argument for the contractibility of the orbit space of a saturated fusion system.

Resolutions of spectra in chromatic homotopy theory

Irina Bobkova (Texas A&M) Saturday 2:45

Computation of the stable homotopy groups of spheres is a long-standing open problem in algebraic topology. I will introduce chromatic homotopy theory — a powerful framework for organizing computations and describing large scale patterns in stable homotopy groups. It splits the problem into chromatic levels, and each level can be studied using the theory of formal group laws and their deformations. Levels 0 and 1 are long understood; level 2 is an area of active interest. Here the short resolutions of spectra proved to be a very powerful tool for computations. I will describe some old and new results constructing such short resolutions.

A tropical framework for using Porteous' formula

Andrew Tawfeek (University of Washington) Saturday 4:00

Given a tropical cycle X, one can talk about a notion of tropical vector bundles on X having tropical fibers. By restricting our attention to bounded rational sections of these bundles, one can develop a good notion of characteristic classes that behave as expected classically. We present further results on these characteristic classes and use these properties to prove a Porteous' formula for these bundles, which gives a determinantal expression of the fundamental class of degeneracy loci of a (tropical) bundle morphism in terms of their Chern classes.

The geometric cobordism hypothesis and locality of extended field theories

Dmitri Pavlov (Texas Tech) Saturday 4:30

I will explain my recent work on functorial field theory, including the proof of the Baez–Dolan cobordism hypothesis generalized to nontopological geometric structures, and the proof of a conjecture by Freed and Lawrence on locality of fully extended field theories. The talk will include the necessary background on topology and field theory.

Exodromy in topology and applications

Peter Haine (Berkeley) Sunday 9:00 Slides

Generalizing the monodromy equivalence, the exodromy equivalence says that the ∞-category of constructible sheaves on a nice enough stratified space (X, P) is equivalent to functors out of the exit-path ∞-category of (X, P). Up until recently, the meaning of “nice enough” was quite restrictive; specifically, the exodromy theorem required the stratification of X to be conical. Unfortunately, many stratifications naturally arising in geometry are not conical. In this talk, we’ll discuss joint work with Mauro Porta and Jean-Baptiste Teyssier that allows us to extend the exodromy theorem to a much larger class of stratified spaces. Examples include: stratifications that can be locally refined by conical stratifications, subanalytic stratifications of real analytic spaces, and algebraic stratifications of real varieties. We’ll also explain some applications, such as representability results for moduli of constructible and perverse sheaves.

A Thom Spectrum Model for \(C_2\)-Integral Brown–Gitler Spectra

Sarah Petersen (Colorado) Sunday 10:30

We establish a Thom spectrum model for a \(C_2\)-equivariant analogue of integral Brown–Gitler spectra and show these have a multiplicative property. The \(C_2\)-equivariant spectra we construct enjoy properties analogous to classical nonequivariant integral Brown–Gitler spectra and thus may prove useful for producing splittings of \(BP \langle 1 \rangle \wedge BP \langle 1 \rangle\) and \(bo \wedge bo\) in the \(C_2\)-equivariant setting.

Supersolvable posets and fiber-type arrangements

Christin Bibby (LSU) Sunday 11:00

We present a combinatorial analysis of fiber bundles of generalized configuration spaces on connected abelian Lie groups. These bundles are akin to those of Fadell–Neuwirth for configuration spaces, and their existence is detected by a combinatorial property of an associated finite partially ordered set. We obtain a combinatorially determined class of \(K(\pi,1)\) spaces, and under a stronger combinatorial condition prove a factorization of the Poincaré polynomial when the Lie group is noncompact. In the case of toric arrangements, this provides an analogue of Falk–Randell's formula relating the Poincaré polynomial to the lower central series of the fundamental group. This is joint work with Emanuele Delucchi.