Speaker: Morgan Opie - UCLA
Abstract: In this talk, I will talk about the subtleties of studying unstable complex vector bundles on complex projective spaces using stable homotopy theory. I’ll start with a classical example of techniques used by Atiyah-Rees to classify all complex rank $2$ topological vector bundles on $\mathbb{C}P^5$ using real K theory and discuss my work on complex rank $3$ topological vector bundles on $\mathbb{C}P^5$ using topological modular forms....
Speaker: Sil Linskens - University of Bonn
Abstract: Global equivariant stable homotopy theory is a formalism which admits applications to orbifold cohomology, elliptic cohomology and importantly equivariant stable homotopy theory. Many important equivariant spectra admit global refinements, and these refinements have led to theoretical advances and have aided in calculations. However not only does global homotopy theory admit applications to equivariant homotopy theory, but it seems reasonable to expect that global homotopy theory is in some sense determined by equivariant homotopy theory....
Speaker: Shaul Barkan - Hebrew University of Jerusalem
Abstract: Stable homotopy theory is intimately related to the geometry of formal groups through the Adams Novkiov spectral sequence. Franke took a step towards making this analogy precise by introducing a derived category of certain sheaves on the moduli stack of formal groups as an analog of the $\infty$-category of spectra of chromatic height $\leq h$. He conjectured that at primes sufficiently larger than the height the homotopy truncation of the two categories coincide....
Speaker: Alicia Lima – University of Chicago
Abstract: Brauer groups and Azumaya algebras play an important role in many areas of mathematics. For instance, Antieau and Gepner showed that the Brauer group of the sphere spectrum $\mathbb{S}$ is zero, giving us the rigidity result for $\mathbb{S}$-modules. In this talk, I will walk you through Hopkins and Lurie’s computation (up to a filtration) of the Brauer groups of Lubin-Tate spectra, which are important building blocks in Chromatic Homotopy Theory....
Speaker: Andrea Lachmann - University of Wuppertal
Abstract: The Localisation Theorem is a classical result by Quillen, which for any localisation of rings yields a long exact sequence of K-groups. In this talk we will define the algebraic K-theory functor from stable $\infty$-categories to spaces or spectra, and produce an analogue of the localisation theorem for this setting.
I have already given a short talk on this topic at the YTM, but this time I will go a lot more into detail, so you can still learn something new even if you attended my previous talk....
Speaker: Maxime Ramzi – Unversity of Copenhagen
Abstract: We use the Hochschild-homological interpretation of dimensions and more generally traces in symmetric monoidal $\infty$-categories to produce formulas for dimensions and traces of colimits over sufficiently nice spaces.
These formulas specialize on the one hand to the very classical character-theoretic formulas for dimensions of coinvariants in the case of vector spaces in characteristic $0$ and when the space is $BG$ for some finite group $G$; and on the other hand to the Blumberg-Cohen-Schlichtkrull formula for topological Hochschild homology of Thom spectra....
Speaker: Jonas McCandless - University of Münster
Abstract: I will introduce the notion of a polygonic spectrum which is a multi-object version of the notion of a cyclotomic spectrum indexed by polygons rather than the circle. This is designed to capture the structure on topological Hochschild homology of a ring with coefficients in a bimodule. Using this, I will explain the construction of TR with coefficients and explain how this is equipped with compatible Frobenius and Verschiebung maps....