09:00 to 10:00 Marc Levine (Universität Duisburg-Essen)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 Grothendieck-Witt ring over the base-field. Here we define an invariant in the Grothendieck-Witt 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 anti-canonical class $-K_S$), we define a class in the Grothendiek-Witt 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 $-D\cdot K_S-1$. 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 $Sym^{3d-1}(S)^0(k)$, where $Sym^{3d-1}(S)^0$ is the open subscheme of $Sym^{3d-1}(S)$ parametrizing geometrically reduced 0-cycles. 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 G-equivariant 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 Co-authors: Elden Elmanto (Northwestern University), Adeel A. Khan (Universität Regensburg), Vladimir Sosnilo (St. Petersburg State University), Maria Yakerson (Universität Duisburg-Essen)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 E-infinity-spaces. 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 Urbana-Champaign)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 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 K-theory of a field F to the Milnor K-theory 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 four-slices 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 invariant-one 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 Holler-Kriz 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 A1-enumerative geometry We will discuss several applications of A1-homotopy 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, p-typical) Cartier module and its higher generalization to spectra (topological Cartier modules). We present examples such as Witt vectors and K-theory 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