Equivariant and motivic homotopy theory
Monday 13th August 2018 to Friday 17th August 2018
09:00 to 09:50  Registration  
09:50 to 10:00  Welcome from David Abrahams (INI Director)  
10:00 to 11:00 
Daniel Dugger (University of Oregon) Surfaces with involutions
I will talk about the classification of surfaces with involution, together with remarks on the RO(Z/2)graded Bredon cohomology of these objects.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Jeremiah Heller (University of Illinois) A motivic Segal conjecture 
INI 1  
12:30 to 13:30  Lunch at Churchill College  
13:30 to 14:30  Free Time  
14:30 to 15:30 
Agnes Beaudry (University of Colorado) Pic(EZ/4) and Tools to Compute It
In this talk, we describe tools to compute the Picard group of the category of EGmodule spectra where E is Morava Etheory at n=p=2 and G is a finite cyclic subgroup of the Morava stabilizer of order 4. We discuss two tools: 1) A group homomorphism from RO(G) to Pic(EG) and methods for computing it. 2) The Picard Spectral sequence and some of its equivariant properties.
This work is joint with Irina Bobkova, Mike Hill and Vesna Stojanoska

INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00 
Clark Barwick (University of Edinburgh) Exodromy and endodromy
On the étale homotopy type of a scheme, there is a natural stratification. We describe both it and its dual, and we discuss some of the ramification information it captures. Joint work with Saul Glasman, Peter Haine, Tomer Schlank.

INI 1  
17:00 to 18:00  Welcome Wine Reception at INI 
09:00 to 10:00 
Marc Levine (Universität DuisburgEssen) Quadratic Welschinger invariants
This is report on part of a program to give refinements of numerical invariants arising in enumerative geometry to invariants living in the GrothendieckWitt ring over the basefield. Here we define an invariant in the GrothendieckWitt ring for ``counting'' rational curves. More precisely, for a del Pezzo surface S over a field k and a positive degree curve class $D$ (with respect to the anticanonical class $K_S$), we define a class in the GrothendiekWitt ring of k, whose rank gives the number of rational curves in the class D containing a given collection of distinct closed points $\mathfrak{p}=\sum_ip_i$ of total degree $D\cdot K_S1$. This recovers Welschinger's invariants in case $k=\mathbb{R}$ by applying the signature map. The main result is that this quadratic invariant depends only on the $\mathbb{A}^1$connected component containing $\mathfrak{p}$ in $Sym^{3d1}(S)^0(k)$, where $Sym^{3d1}(S)^0$ is the open subscheme of $Sym^{3d1}(S)$ parametrizing geometrically reduced 0cycles.

INI 1  
10:00 to 11:00  Poster Session in Discussion Room  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Magdalena Kedziorek (Universiteit Utrecht) Algebraic models for rational equivariant commutative ring spectra
I will recall different levels of commutativity in Gequivariant stable homotopy theory and show how to understand them rationally when G is finite or SO(2) using algebraic models. This is joint work with D. Barnes and J.P.C. Greenlees.

INI 1  
12:30 to 13:30  Lunch at Churchill College  
13:30 to 14:30  Free Time  
14:30 to 15:30 
Marc Hoyois (Massachusetts Institute of Technology) Motivic infinite loop spaces and Hilbert schemes
Coauthors: Elden Elmanto (Northwestern University), Adeel A. Khan (Universität Regensburg), Vladimir Sosnilo (St. Petersburg State University), Maria Yakerson (Universität DuisburgEssen)We prove a recognition principle for motivic infinite loop spaces over a perfect field of characteristic not 2. This is achieved by developing a theory of framed motivic spaces, which is a motivic analogue of the theory of Einfinityspaces. Our main result is that grouplike framed motivic spaces are equivalent to the full subcategory of motivic spectra generated under colimits by suspension spectra. As a consequence, we deduce some representability results for suspension spectra of smooth varieties and various motivic Thom spectra in terms of Hilbert schemes of points in affine spaces.Related Linkshttps://arxiv.org/abs/1711.05248  preprint

INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00 
Vesna Stojanoska (University of Illinois at UrbanaChampaign) Galois extensions in motivic homotopy theory
There are two notions of homotopical Galois extensions in the motivic setting; I will discuss what we know about each, along with illustrative examples. This is joint work in progress with Beaudry, Heller, Hess, Kedziorek, and Merling.

INI 1 
09:00 to 10:00 
Doug Ravenel (University of Rochester) Model category structures for equivariant spectra
We
will discuss methods for constructing a model structure the category of
orthogonal equivariant spectra for a finite group G. The most convenient
one is two large steps away from the most obvious one.

INI 1  
10:00 to 11:00 
Oliver Roendigs (Universität Osnabrück); (Universitetet i Oslo) On very effective hermitian Ktheory
We argue that the very effective cover of hermitian Ktheory in the sense of motivic homotopy theory is a convenient algebrogeometric generalization of the connective real topological Ktheory spectrum. This means the very effective cover acquires the correct Betti realization, its motivic cohomology has the desired structure as a module over the motivic Steenrod algebra, and that its motivic Adams and slice spectral sequences are amenable to calculations. The latter applies to provide the expected connectivity for its unit map from the motivic sphere spectrum. This is joint work with Alexey Ananyevskiy and Paul Arne Ostvaer.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Teena Gerhardt (Michigan State University) Hochschild homology for Green functors
Hochschild homology of a ring has a topological analogue for ring spectra, topological Hochschild homology (THH), which plays an essential role in the trace method approach to algebraic Ktheory. For a C_nequivariant ring spectrum, one can define C_nrelative THH. This leads to the question: What is the algebraic analogue of C_nrelative THH? In this talk, I will define twisted Hochschild homology for Green functors, which allows us to describe this algebraic analogue. This also leads to a theory of Witt vectors for Green functors, as well as an algebraic analogue of TRtheory. This is joint work with Andrew Blumberg, Mike Hill, and Tyler Lawson.

INI 1  
12:30 to 13:30  Lunch at Churchill College  
13:30 to 17:00  Free Afternoon  INI 1  
19:30 to 22:00  Formal Dinner at Gonville & Caius College 
09:00 to 10:00 
Mark Behrens (University of Chicago) C_2 equivariant homotopy groups from real motivic homotopy groups
The Betti realization of a real motivic spectrum is a genuine C_2 spectrum. It is well known (c.f. the work of DuggerIsaksen) that the homotopy groups of the Betti realization of a complex motivic spectrum can be computed by "inverting tau". I will describe a similar theorem which describes the C_2equivariant RO(G) graded homotopy groups of the Betti realization of a cellular real motivic spectrum in terms of its bigraded real motivic homotopy groups. This is joint work with Jay Shah.

INI 1  
10:00 to 11:00 
Tom Bachmann (Universität DuisburgEssen) Motivic normed spectra and Tambara functors
Motivic normed spectra are a motivic analogue of the Gcommutative ring spectra from genuine equivariant stable homotopy theory. The category of Gcommutative ring spectra such that the underlying genuine equivariant spectrum is concentrated in degree zero (i.e. is a Mackey functor) is in fact equivalent to the category of GTambara functors. I will explain an analogous motivic result: the category of normed motivic spectra (over a field), such that the underlying motivic spectrum is an effective homotopy module, is equivalent to a category of homotopy invariant presheaves with generalized transfers and étale norms.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Anna Marie Bohmann (Vanderbilt University) Graded Tambara functors
Let E be a Gspectrum for a finite group G. It's long been known that homotopy groups of E have the structure of "Mackey functors." If E is G commutative ring spectrum, then work of Strickland and of Brun shows that the zeroth homotopy groups of E form a "Tambara functor." This is more structure than just a Mackey functor with commutative multiplication and there is much recent work investigating nuances of this structure.
I will discuss work with Vigleik Angeltveit that extends this result to include the higher homotopy groups of E. Specifically, if E has a commutative multiplication that enjoys lots of structure with respect to the G action, the homotopy groups of E form a graded Tambara functor. In particular, genuine commutative G ring spectra enjoy this property.

INI 1  
12:30 to 13:30  Lunch at Churchill College  
13:30 to 14:30  Free Time  
14:30 to 15:30 
Mona Merling (Johns Hopkins University) Toward the equivariant stable parametrized hcobordism theorem
Waldhausen's introduction of Atheory of spaces revolutionized the early study of pseudoisotopy theory. Waldhausen proved that the Atheory of a manifold splits as its suspension spectrum and a factor Wh(M) whose first delooping is the space of stable hcobordisms, and its second delooping is the space of stable pseudoisotopies. I will describe a joint project with C. Malkiewich aimed at telling the equivariant story if one starts with a manifold M with group action by a finite group G.

INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00 
Fabien Morel (LudwigMaximiliansUniversität München) Etale and motivic variations on Smith's theory on action of cyclic groups
In this talk, after recalling some facts on Smith's theory in classical topology, I will present some analogues in algebraic geometry using \'etale cohomology first, and I will also give analogues in \A^1homotopy theory and motives of Smith's theory concerning the multiplicative group \G_m, which behaves like a "cyclic of order 2".

INI 1 
09:00 to 10:00 
Aravind Asok (University of Southern California) On Suslin's Hurewicz homomorphism
I will discuss some recent progress on an old conjecture of Suslin about the image of a certain ``Hurewicz" map from Quillen's algebraic Ktheory of a field F to the Milnor Ktheory of F. This is based on joint work with J. Fasel and T.B. Williams.

INI 1  
10:00 to 11:00 
Carolyn Yarnall (University of Kentucky) Klein fourslices of HF_2
The slice filtration is a filtration of equivariant spectra analogous to the Postnikov tower that was developed by Hill, Hopkins, and Ravenel in their solution to the Kervaire invariantone problem. Since that time there have been several new developments, many dealing with cyclic groups. In this talk, we will focus our attention on a noncyclic group! After recalling a few essential definitions and previous results, we will investigate some computational tools that allow us to leverage homotopy results of HollerKriz to determine the slices of integer suspensions of HF_2 when our group is the Klein four group. We will end with a few slice tower and spectral sequence examples demonstrating the patterns that arise in the filtration.
This is joint work with Bert Guillou.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Kirsten Wickelgren (Georgia Institute of Technology) Some results in A1enumerative geometry
We
will discuss several applications of A1homotopy theory to enumerative
geometry. This talk includes joint work with Jesse Kass and Padmavathi
Srinivasan.

INI 1  
12:30 to 13:30  Lunch at Churchill College  
13:30 to 14:30  Free Time  
14:30 to 15:30 
Thomas Nikolaus (Universität Münster) Cyclotomic spectra and Cartier modules
We start by reviewing the notion of cyclotomic spectra and basic examples such as THH. Then we introduce the classical algebraic notion of an (integral, ptypical) Cartier module and its higher generalization to spectra (topological Cartier modules). We present examples such as Witt vectors and Ktheory of endomorphisms. The main result, which is joint work with Ben Antieau, is that there is a close connection between the two notions.

INI 1 