### Titles and abstracts

Here is a schedule of talks. Return to the conference web-page here.

Here is a schedule of talks. Return to the conference web-page here.

**Title:** Towards a motivic homotopy theory without **A**^{1}-invariance

**Abstract:**
Motivic homotopy theory as conceived by Morel and Voevodsky is based on the crucial observation that the affine line
**A**^{1} plays in algebraic geometry the role of the unit interval in algebraic topology. Inspired by the work of
Kahn-Saito-Yamazaki, we constructed an unstable motivic homotopy category "with modulus", where the affine
line is no longer contractible. In the talk, we will sketch this construction and we will explain why this
category can be seen as a candidate environment for studying representability problems for non **A**^{1}-invariant
generalized cohomology theories.

**Title:** Formality of symmetric monoidal functors and operads via mixed Hodge theory

**Abstract:** It has long been observed that Hodge theory is a powerful tool for proving formality
results, both in the operadic setting or in the setting of rational homotopy theory. In this talk, I will
explain a "purity implies formality" result in the abstract setting of symmetric monoidal functors. I will
then develop some applications to: 1) the formality of certain operads defined in the category of complex
schemes, and 2) the formality of morphisms of schemes in the sense of rational homotopy. This is joint work
with Geoffroy Horel.

**Title:** Moduli Stack of Curves and Motivic Homotopy

**Abstract:**
The stratification of the moduli spaces of curves given by the Deligne-Mumford compactification presents an
important arithmetic combinatoric, which is expressed both in the context of Geometric Etale Galois
representation and of Mixed Tate Motives.
This talk will focus on the stack stratification of the spaces -- such as given by the automorphism groups
of objects. This structure being known to possess similar arithmetic properties, we will present how
the motivic homotopy theory is especially adapted in capturing these properties at the motivic level,
as well as in providing a natural bridge between the motivic and Galois étale theories.

**Title:** The Balmer spectrum of the category of rational equivariant cohomology theories

**Abstract:** We we consider the tensor triangulated category of rational G-equivariant
cohomology theories when G is a torus (or the toral part for a more general compact Lie group).
One may classify thick tensor ideals, and the Balmer spectrum is precisely
the space of subgroups under cotoral inclusion. The first key ingredients are
the Localization Theorem and tom Dieck’s description of the rational Burnside ring.
Since this space of subgroups is the basis for the algebraic model for the category,
it suggests that the strategy that worked in that case might apply elsewhere.

**Title:** Symmetric products and subgroup lattices

**Abstract:**: Let G be a finite group. In this talk I will describe a relationship
between symmetric products of G-representation spheres and the subgroup lattice of G. More concretely,
it turns out that the rational homotopy groups of the symmetric products of the G-sphere spectrum are
naturally isomorphic to the rational homology groups of certain subcomplexes of the subgroup lattice,
which in many cases allows an explicit computation of the former. If time permits, I will further discuss
how this isomorphism can be refined to obtain an algebraic model for the global equivariant homotopy type
of rational symmetric products.

**Title:** Segre classes for algebraic cobordism and their application to
Schubert calculus

**Abstract:** Since the introduction by Levine and Morel of algebraic cobordism, several
attempts have been made to lift to arbitrary oriented cohomology theories the classical results of Schubert
calculus. In this talk I will explain how the use of generalised Segre classes allows one to extend
(among others) the Kempf-Laksov formula, which expresses as a determinant in Chern
classes the fundamental classes of the Schubert varieties of Grassmann bundles. This work is joint with
Tomoo Matsumura.

**Title:** Algebraic K-theory and descent for blow-ups

**Abstract:** We study descent of algebraic K-theory along blow-ups and apply this to Weibel's
conjecture on the vanishing of negative K-groups. This is joint work with F. Strunk and G.
Tamme

**Title:** Homotopy K-theory of E_{∞}-ring spectra

**Abstract:** Weibel’s homotopy K-theory is a variant of algebraic K-theory which satisfies
the property of homotopy invariance with respect to the affine line. One can construct an analogue of
homotopy K-theory for E_{∞}-ring spectra, which satisfies homotopy invariance with respect
to the “brave new affine line”. It turns out that, for connective E_{∞}-ring spectra,
homotopy K-theory is insensitive to derived nilpotent thickenings; that is, the homotopy K-theory of R is
equivalent to that of π_{0}(R).
We will sketch a proof of this fact using the motivic homotopy theory of brave new schemes.

**Title:** Real spectra and their duals

**Abstract:**
Real bordism is a C2-equivariant spectrum constructed by
Landweber and Araki and is in some sense analogous to algebraic
cobordism. There are many C2-spectra built from it, like the BPR

**Title:** Proper equivariant stable homotopy theory

**Abstract:**In this talk we will present a setup for doing equivariant stable homotopy
theory with infinite discrete groups and finite isotropy. We develop a
theory of proper G-spectra for a discrete group G and show that various
equivariant cohomology theories on proper G-spaces are represented in this
category of G-spectra. Our main examples include equivariant K-theory and
equivariant stable cohomotopy. This is a joint work with Degrijse,
Hausmann, Lück and Schwede. Next, we will also mention connections to
geometric group theory and relate proper G-spectra to classical finiteness
questions for groups. The latter is a joint work with Barcenas and
Degrijse. At the end we will also indicate how proper
**Q**_{p}/**Z**_{p}-spectra are
related to cyclotomic spectra and topological cyclic homology.

**Title:** Constructible 1-motives

**Abstract:** Thanks to the work of Voevodsky, Morel, Ayoub, Cisinski and Déglise,
we have at our disposal a mature theory of triangulated categories of
mixed motivic sheaves with rational coefficients over general base
schemes, with a "six operations" formalism and the expected
relationship with algebraic cycles and algebraic K-theory. A parallel
development has taken place in the study of Voevodsky's category of
mixed motives over a perfect field, where the subcategory of 1-motives
(i.e., generated by motives of curves) has been completely described
by work of Orgogozo, Barbieri-Viale, Kahn and Ayoub. We explain how to
combine these two sets of ideas to study the triangulated category of
1-motivic sheaves over a base. Our main results are the definition of
the motivic t-structure for 1-motivic sheaves, a precise relation with
Deligne 1-motives, and the extraction of the "1-motivic part" of a
general motivic sheaves via a "motivic Picard functor".

**Title:** Logarithmic topological Hochschild homology

**Abstract:** Logarithmic ring spectra form a common generalization of the concept
of a structured ring spectrum appearing in homotopy theory and the
concept of an affine log scheme appearing in algebraic geometry. In
this talk I will explain the definition of logarithmic ring spectra
and the construction of their logarithmic topological Hochschild
homology. Topological K-theory spectra give rise to logarithmic ring
spectra that sit between the corresponding connective and periodic
K-theory spectra, and I will present both structural and computational
results about the logarithmic topological Hochschild homology of these
logarithmic ring spectra. (This is report on joint work with John
Rognes and Christian Schlichtkrull.)

**Title:** Global homotopy theory

**Abstract:** This is an overview lecture on aspects of global homotopy theory.

**Title:** Cohomological invariants of algebraic groups and the Morava K-theory

**Abstract:** In the talk I will discuss an approach to cohomological
invariants of algebraic groups over a field based on the Morava
K-theories which are universal generalized oriented cohomology theories
in the sense of Levine-Morel with respect to the Lubin-Tate formal group
law. I will explain that the second Morava K-theory detects the
triviality of the Rost invariant and I will describe a method to compute
the Morava K-theory of some affine varieties.

**Title:** The Gross-Hopkins duals of higher real K-theory spectra

**Abstract:**
The Hopkins-Mahowald higher real K-theory spectra are generalizations of real K-theory; they are ring spectra
which give some insight into higher chromatic levels while also being computable. This will be a talk based on
joint work with Barthel and Beaudry, in which we compute that higher real K-theory spectra at prime `p` and
height `p-1` are Gross-Hopkins self-dual with shift `(p-1) ^{2}`. We expect this will allow us to detect exotic
invertible

**Title:** The symmetric complex over projective space

**Abstract:** Projective spaces were the first truly geometric objects over which symmetric
vector bundles were studied. In 1980, Arason showed that essentially all symmetry is inherited from the
ground field: up to the usual K-theoretic stabilization process and up to hyperbolic factors, any symmetric
bundle over a projective space can be pulled back from a point. For complexes, Gille, Walter and later
Nenashev showed by a variety of methods that the situation is markedly different: on each projective
space, there is essentially one additional symmetric complex that is not visible over the point.
Unfortunately, the elegant simplicity of the result is in stark contrast to the intricacies of the
existing calculations, and, in particular, to the sharp distinctions between several cases each relies on.
We will present a short historic survey and offer a similarly brief remedy for this conceptual deficiency.