Conference: Higher Invariants in Equivariant and Geometric Topology

WELCOME

Enrique Becerra, University of Miami: The stringy spectrum of orbifolds 

In this talk I will introduce the stringy spectrum as an additive invariant of orbifolds and explain its basic properties. Moreover, I will also show its relation with the classical spectrum of isolated hypersurface singularities and give some examples.  Video

Carmen Rovi, Loyola University: Invariants of manifolds and relations to cobordism

The classical problem of scissor's congruence asks whether two polytopes can be obtained from one another through a process of cutting and pasting. In the 1970s this question was posed instead for smooth manifolds: which manifolds M and N can be related to one another by cutting M into pieces and gluing them back together to get N? In recent work with Renee Hoekzema, Mona Merling, Laura Murray,  and Julia Semikina, we upgraded the group of cut-and-paste invariants of manifolds with boundary to an algebraic K-theory spectrum and lifted the Euler characteristic to a map of spectra. I will discuss how cut-and-paste invariants relate to cobordism of manifolds and how the novel construction categorifies these invariants. I will also discuss new results on the categorification of cobordism cut-and-paste invariants: the group of invariants preserved by both cobordism and cut-and-paste equivalence.  Video

Mona Merling, University of Pennsylvania: Equivariant A-theory and spaces of equivariant h-cobordisms

Waldhausen's algebraic K-theory of manifolds satisfies a homotopical lift of the classical h-cobordism theorem and provides a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. I will give an overview of joint work with Goodwillie, Igusa and Malkiewich about the equivariant homotopical lift of the h-cobordism theorem.  Video

Manuel Rivera, Purdue University: Homotopy theories of coalgebras

I will discuss the construction and meaning of different homotopy theories on the category of simplicial coalgebras over a field of arbitrary characteristic. One of these provides a full and faithful model for the homotopy theory of spaces considered under a notion of weak equivalence generated by continuous maps inducing an isomorphism on fundamental groups and, at the level of universal covers, an isomorphism on homology with coefficients on a fixed algebraically closed field. This extends known algebraic models for the p-adic homotopy theory of spaces (Kriz, Goerss, Mandell) by including the fundamental group in complete generality. They key idea is to combine Koszul duality between (dg) coalgebras and algebras to obtain a homological formulation of the fundamental group in terms of chain level non-linear structure. Time permitting, I will discuss G-equivariant analogs of these models.  Video

Igor Kriz, University of Michigan, The Z/p-equivariant Steenrod algebra and its applications

I will discuss recent joint work with Hu, Somberg, and Zou calculating the dual of the Z/p-equivariant Steenrod algebra with constant coefficients, using the previously known partial results of Sankar and Wilson. I will describe the menagerie of Mackey functors one encounters, and also talk about a recent joint application with Hu, Li, Somberg, and Zou to constructing a version of a Z/p-equivariant spectrum BPR at an odd prime p, the existence of which was conjectured by Hill, Hopkins, and Ravenel.  Video

Yuri Tschinkel, Courant Institute of Mathematical Sciences: New Invariants in equivariant birational geometry

I will discuss new obstructions to equivariant birationality of higher-dimensional algebraic varieties, introduced in joint work with A.  Kresch.

Bernardo Villarreal, CIMAT Merida: Topological components of commuting elements of nilpotent Lie groups

Pettet and Souto showed that the space of commuting elements in a reductive connected Lie group G, Hom(Z^k,G), admits a deformation retract into the space of commuting elements of its maximal compact subgroup. As was previously noted by Goldman, the reductive assumption can not be weakened, as he showed that such a deformation retraction does not exist for H = H(R)/Z, the reduced Heisenberg group. In a similar fashion, one can show that the deformation retraction does not exist for any n-by-n reduced upper unitriangular matrix group with real entries, UT_n/Z, which are nilpotent Lie groups of class n-1. In this talk I will argue that instead of taking commuting elements of a maximal compact subgroup, one should consider commuting elements of a maximal nilpotent Lie group of class 2. In the UT_n/Z case, these are the reduced generalized Heisenberg groups G_n. The components of Hom(Z^k,G_n) have a very nice description (e.g. some are diffeomorphic to real symplectic Stiefel manifolds), and are parametrized by the cohomology classes H^2(Z^k,Z) for sufficiently large n. This is joint work with O. Antolín-Camarena.

Kyoung Seog Lee, University of Miami: Logarithmic transformations and change of geometric invariants via o-minimal geometry

Logarithmic transformation is an important analytic operation introduced by Kodaira in the 1960s. One can obtain a new elliptic fibration by performing logarithmic transformations on a given elliptic fibration and usually, the transforms change certain invariants of the underlying manifolds. In this talk, I will discuss how certain geometric invariants of elliptic fibrations are changed under logarithmic transformations from the perspective of o-minimal geometry.

Bernhard Hanke, Augsburg University: Large and small group homology

Group homology is defined in either algebraic or topological terms, depending on the application and taste. It is therefore somewhat surprising that it can also encode metric properties that one usually studies on smooth manifolds, such as enlargeability or having positive scalar curvature. The key role is played by the image of the homological fundamental class of the given manifold under the classifying map. We will explain this line of thought and present some applications.

Mario Velasquez, Universidad Nacional de Colombia :On the group cohomology and topological K-theory of groups $\Gamma=\mathbb{Z}^n\rtimes \Z/m$ with $m$ free of squares

Let $\rho:\mathbb{Z}/m\to GL(n,\Mathbb{Z})$ be a homomorphism, it defines a semidirect product $\Gamma=\mathbb{Z}^n\rtimes\mathbb{Z}/m$, when $m$ is free of squares we provide explicit computations of the group cohomology of $\Gamma$, we also provide computations for the associated toroidal orbifold. For this, we proof that every $\mathbb{Z}/m$-lattice can be decomposed in a unique way (up to isomorphism) as direct sum of $\mathbb{Z}/m$-lattices with constant isotropy and then use the Lyndon-Hoschild-Serre spectral sequence associated to a certain extension. On the other hand, using the Baum-Connes conjecture we prove that topological K-theory of the reduced group C*-algebra of $\Gamma$ is torsion free. This generalizes results of Adem-Gomez-Pan-Petrosyan and Davis-Luck-Langer. This is joint work with Luis Jorge Sánchez.

Craig Westerland, University of Michigan: Cohomology of Hurwitz spaces

Hurwitz moduli spaces parameterize branched covers of algebraic curves.  In this talk, I’ll focus on branched covers of the affine line and explain several approaches to understanding the cohomology of these moduli spaces using connections to braided Hopf algebras and a new family of braided operads.

Sophie Kriz, University of Michigan: Some computations on FI-modules. 

FI-modules were introduced by Church, Ellenberg, and Farb to address  numerous questions of representation stability, such as the cohomology of moduli spaces of pointed curves, configuration spaces, and Torelli groups. While FI-modules arewell-understood over fields of characteristic 0, I will talk about some of my recent results on FI-modules in positive and mixed characteristic, including computations of local cohomology. I will also discuss examples of simple generic FImodules in positive characteristic which only contain reducible representations in high degrees, thus solving a problem of R. Nagpal.

Eugenia Ellis, Universidad de la República: Algebraic kk-theory and the KH-isomorphism conjecture

We relate the Davis-Lück homology with coefficients in Weibel´s homotopy K-theory to the equivariant algebraic kk-theory using homotopy theory and adjointness theorems. We express the left hand side of the assembly map for the KH-isomorphism conjecture introduced by Bartels-Lück in terms of equivariant algebraic kk-groups.

Ivan Cheltsov, University of Edinburgh: Equivariant birational geometry of the three-dimensional projective space

In this talk, we will discuss G-equivariant birational geometry of the three-dimensional projective space for a finite group G acting biregularly on the projective space. In particular, we will describe all possibilities for the group G such that the projective space is not G-equivariantly birational to fibrations into rational curves or surfaces.

Kaiqi Yang, Courant Institute of Mathematical Sciences: Equivariant birational geometry of linear actions

Based on the Burnside group formalism developed by Kontsevich, Kresch, and Tschinkel, we study linear actions on projective spaces. Our main results are as follows:
1. In dimension 2, the Burnside formalism produces new non-birational linear and projectively linear actions.
2. In dimension 3, we exhibit new types of non-birational linear actions on $P^3$ as well as non-linearizable actions on smooth quadrics.

Bernardo Uribe (on behalf of Dev Sinha), Universidad del Norte: Computation in unitary equivariant bordism (on behalf of Dev Sinha)

Dev Sinha could not make it to the workshop but he wanted to share some results he had obtained on explicit computations of the unitary equivariant bordism ring. I will present these results on his behalf.

José Manuel Gómez, Universidad Nacional de Colombia: On the second homotopy group of spaces of commuting elements in Lie groups

In this talk we will explore the problem of computing the second homotopy group of the spaces of the form Hom(Z^n, G), where G is a compact Lie group. We will show in particular, that if G is compact, simply connected and simple, then \pi_{2}( Hom(Z^2, G))=Z and furthermore one can obtain geometric generators. This talk is based in a joint work with Alejandro Adem and Simon Gritschacher

Gisela Tartaglia, Universidad de la Plata: Coinduction for comodule algebras

Given $G$ a discrete group and $H$ a subgroup, it is known how to induce $G$-algebras from $H$-algebras. We will present a construction of $C_c(G)$-comodule algebras starting from $C(H)$-comodule algebras, for finite $H$.  This construction gives a Morita equivalence between the crossed products of the corresponding dual actions, and also recovers the group case. For this purpose we will consider $C_c(G)$ as an algebraic (discrete) quantum group. Joint work with Eugenia Ellis and Ana Karina González de los Santos.

Eric Samperton, Purdue University: Examples and counterexamples in 2-d equivariant bordism

Fix a finite group G and an oriented surface S.  Given an action of G on S, when does this action extend to an action on a 3-manifold M with  boundary S?  What can we say about M or the qualities of the action of G on M?  I’ll quickly review the concrete answer that Angel, Segovia, Uribe and I gave to the first question.  I’ll then report on various examples and counterexamples regarding the second question; this part of  the talk is based on joint work with Marco Boggi and Carlos Segovia.

Omar Antolin, UNAM: Classifying crystalline interacting topological phases through equivariant cohomology

I'll describe joint work with Daniel Sheinbaum in which we explain how to compute how many distinct topological phases of matter there are for crystalline interacting gapped systems given their symmetry group. Our answer is given in terms of equivariant homotopy classes of maps, and in some cases can be simplified to Borel equivariant cohomology. I'll explain in broad strokes how this classification arises from natural assumptions and will compare both the classification and those assumptions with those from other proposals you can find in the literature.