Workshop // 20-22 March 2019 // Berlin
Women in Homotopy Theory and Algebraic Geometry II
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Niny Arcila Maya
Integral cohomology of BPUn and the understanding of topological Azumaya algebras
Abstract: Let X be a CW-complex and let C denote the sheaf of complex-valued functions on X. A topological Azumaya algebra of degree n over X is a sheaf of C-algebras that is locally isomorphic to Matnxn(C). By Skolem-Noether theorem there is a correspondance between isomorphism classes of deg-n Azumaya algebras over X and isomorphism classes of principal PUn-bundles over X, where PUn is the projective unitary group of order n. Therefore, the homotopy theory of the classifying space of PUn is of great interest to study topological Azumaya algebras. Let a be the least prime dividing n. We calculate the integral cohomology Hi (BPUn;Z) in the stable range i<2a+1. We use this computation to classify topological Azumaya algebras of degree pq on low dimensional spaces, where p and q are prime numbers.
Candace Bethea
An example of wild ramification over a non-perfect field
Abstract: In 2017 Marc Levine gave an arithmetic Riemann-Hurwitz formula valued in the Grothendieck-Witt group of non-degenerate symmetric bilinear forms with a technical assumption on the residue fields. In joint work with Jesse Kass and Kirsten Wickelgren, we show by computation that the formula still holds with less restrictive assumptions. In this talk I will explicitly show the result for a specific example suggested by Shuji Saito as a test case for the main theorem.
Natàlia Castellana
Stratification of the module category for homotopical groups
Abstract: Group cohomology is a subject in which both algebra and topology interact in a bidirectional way. One of the most relevant results is that of Quillen stratification theorem on the spectrum of the cohomology ring of a compact Lie group which is described in terms of the information provided by elementary abelian p-subgroups.
Consider the category of modules over kG where G is finite, ch(k)=p with p| |G|, Benson-Iyengar-Krause classified the localizing subcategories by means of subsets of the spectrum of the cohomology of G and where Quillen's stratification result is a key ingredient.
With an appropriate version of the module category, stratification results extend to compact Lie groups (Benson-Grenlees) and classifying spaces of fusion systems (algebraic structures which provide homotopical analogues for p-completions of classifying spaces of finite groups). This is a joint work with Tobias Barthel, Drew Heard and Gabriel Valenzuela.
Eloise Hamilton
Moduli spaces for unstable Higgs bundles of Harder-Narasimhan length 2.
Abstract: The stack of Higgs bundles of a given rank and degree over a curve can be stratified according to Harder-Narasimhan types. In this talk I will explain how, for a Harder-Narasimhan type of length 2, the corresponding stratum can be further stratified using refined invariants, in such a way that each stratum admits a coarse moduli space. After describing these invariants and the construction of the corresponding coarse moduli spaces using non-reductive Geometric Invariant Theory, I will discuss the topology and geometry of these moduli spaces.
Jocelyne Ishak
Rigidity of the K(1)-local stable homotopy category.
Abstract: In some cases, it is sufficient to work in the homotopy category Ho(C) associated to a model category C, but looking at the homotopy level alone does not provide us with higher order structure information. Therefore, we investigate the question of rigidity: If we just had the structure of the homotopy category, how much of the underlying model structure can we recover? For example, the stable homotopy category Ho(Sp) has been proved to be rigid by S. Schwede. Moreover, the E(1)-local stable homotopy category, for p=2, has been shown to be rigid by C. Roitzheim.
In this talk, I will discuss a new case of rigidity, which is the localization of spectra with respect to the Morava K-theory K(1), at p=2. While the K(n)-local spectra can be related to the E(n)-local spectra, there are a lot of main differences to keep in mind while studying the rigidity in the K(1)-local case. Therefore, what might be true and applicable for the E(1)-localization studied by C.Roitzheim might not be true anymore in the K(1)-local world. In this talk, I will emphasis those differences, and sketch the proof of the rigidity of the K(1)-local stable homotopy category at p=2.
Frances Kirwan
Moduli spaces of unstable curves
Abstract: Moduli spaces arise naturally in classification problems in geometry. The study of the moduli spaces of nonsingular complex projective curves (or equivalently of compact Riemann surfaces) goes back to Riemann himself in the nineteenth century.
The construction of the moduli spaces of stable curves of fixed genus is one of the classical applications of Mumford's geometric invariant theory (GIT), developed in the 1960s. Here a projective curve is stable if it has only nodes as singularities and its automorphism group is finite. The aim of this talk is to describe these moduli spaces and outline their GIT construction, and then explain how recent methods from non-reductive GIT can help us to classify the singularities of unstable curves in such a way that we can construct moduli spaces of unstable curves (of fixed singularity type).
Abigail Linton
Massey products and moment-angle complexes
Abstract: Massey products are higher cohomology operations that are often important to the study of formality of spaces, among other applications. Moment-angle complexes, one of the main objects in toric topology, have a natural underlying combinatorial structure that allows us to study combinatorial obstructions to Massey products in the cohomology of moment-angle complexes. I present two frameworks of combinatorial operations on simplicial complexes that create non-trivial $n$-Massey products on classes of any given degree.
Ingrid Membrillo Solis
Homotopy types of gauge groups over high dimensional manifolds
Abstract: Let G be a topological group, and let X be a path-connected pointed topological space. Given a principal G-bundle P over X, the gauge group of P is the group of its bundle automorphisms covering the identity on X. The topology of gauge groups have been extensively studied due to their connections to differential geometry and mathematical physics. In this talk I will give an introduction to gauge groups from a homotopy theoretical point of view. I will present some results on the homotopy classification of gauge groups of principal G-bundles over connected sums of high dimensional manifolds, for G a simply connected simple compact Lie group.
Morgan Opie
Localization in homotopy type theory
Abstract: We study localization at a prime in homotopy type theory, using self maps of the circle. We seek to show that analogues for classical results about the effect of localization of spaces on algebraic invariants hold for localization of types. Our main result is that for a pointed, simply connected type X, the natural map X→X(p) induces algebraic localizations on all homotopy groups. This is joint work with J. D. Christensen, E. Rijke, and L. Scoccola.
Viktoriya Ozornova
Tmf-algebras and associated stacks
Abstract: In a joint project with Lennart Meier, we look at the cohomology theory of topological modular form and its relatives. Its representing spectrum is the global sections of a certain sheaf (in a sense of derived algebraic geometry) on the moduli stack of elliptic curves, and this gives a possibility to attack problems about topological modular form using the geometry of this stack. Variants of this are known, related e.g. to the moduli stack of cubical curves and to the moduli stack of elliptic curves with level structure. Our work in progress seeks to combine these generalizations, yielding potentially a better understanding of tmf-modules. In my talk, my main goal would be to explain the vocabulary used in this abstract and to give evidence for our conjecture.
Sabrina Pauli
A^1-contractible varieties
Abstract: The affine line A^1 is the only smooth, complex, A^1 -contractible curve and it is conjectured that A^2 is the only smooth, complex, A^1 -contractible surface. However, there exist nontrivial examples of smooth, complex, A^1 -contractible varieties in dimension three and higher, one being the Koras-Russell cubic {x^2y = z^2 + t^3 + x} ⊂ A^4 . In my talk I will give a short survey on A^1 -contractible varieties and explain how one can construct some of them using affine modifications.
Charanya Ravi
Rigidity for equivariant pseudo pretheories
Abstract: This talk is a report on joint work with Jeremiah Heller and Paul Arne Østvær. We establish versions of the Suslin and Gabber rigidity theorems in the setting of equivariant pseudo pretheories of smooth schemes over a field with an action of a finite group. Examples of equivariant pseudo pretheories include equivariant algebraic K- theory and presheaves with equivariant transfers.
Lena Vos
The motivic Igusa zeta function of a space monomial curve with a plane semigroup
Abstract: The motivic Igusa zeta function of a polynomial f was introduced by Denef and Loeser as a generalization of Igusa's p-adic zeta function, using motivic integration instead of p-adic integration. It is a rational function and has an equivalent definition in terms of the jet schemes of the hypersurface defined by f, which can be generalized to an ideal generated by some polynomials.
In this talk, we are interested in the motivic zeta function of a space curve Y appearing as the special fibre of an equisingular family whose generic fibres are complex plane branches. The curve Y is called `monomial' because it has a monomial parametrization, which involves the generators of the semigroup associated to the family. We will show the main results for such space monomial curves: the structure of the jet schemes and the resulting motivic zeta function, both in terms of the generators of the semigroup. We will also discuss the poles of the motivic zeta function; this interest is motivated by the monodromy conjecture for ideals, which relates the poles of the motivic zeta function to the eigenvalues of monodromy associated to an ideal. This is joint work with H. Mourtada and W. Veys.
Sarah Whitehouse
Geometric and combinatorial models for generalised homotopy associativity
Abstract: Consideration of associativity up to homotopy leads to the idea of an A-infinity structure, important in many areas of mathematics, including topology, representation theory and mathematical physics. There are various well-known geometric models for these structures, most famously the Stasheff polytopes or "associahedra". The relevant combinatorics can be encoded using trees and involves Catalan numbers. I will review these ideas and talk about a more recent generalisation, including work of my former PhD student, Gemma Halliwell.
Susanna Zimmermann
Birational symmetries of projective spaces
Abstract: The projective space is the simplest projective variety and its group of birational symmetries, called Cremona group, has been studied since the 19th century. The group of birational plane symmetries is well studied while less is known about the Cremona groups of higher dimensional projective spaces since we do not know any reasonable generating set. In this talk we will look at how to construct group homomorphisms of the Cremona group to finite groups