Configuration spaces have played an important role in mathematics and its applications. In particular, the question of how their topology changes as the cardinality of the underlying configuration changes has been studied for some fifty years and has attracted renewed attention in the last decade. While classically additional information is associated "locally" to the points of the configuration, there are interesting examples when this additional information is "non- local". With Martin Palmer we have studied homology stability in some of these cases, including Hurwitz space and moduli spaces of asymptotic monopoles.

In this talk, I explain my works and perspectives on the Segal–Stolz–Teichner program, which is one of the most deep and important topics relating homotopy theory and physics. Mathematically, they propose a geometric model of $\rm TMF$, the spectrum of Topological Modular Forms, in terms of supersymmetric quantum field theories. My recent works have been motivated by their proposal. I explain my collaborations with Yuji Tachikawa relating the Anderson self-duality of topological modular forms (in homotopy theory) and absence of anomaly in heterotic string theory (in physics), and my recent collaboration with Theo Johnson-Freyd on giving geometric understanding of the 576-periodicity of $\rm TMF$. I would like to illustrate the exciting interplay between mathematics and physics.

I will review classical methods computing differentials in the Adams spectral sequence and also discuss some recent ones.

In this talk, I will introduce some algorithms which can help us compute the homotopy groups of CW spectra. This involves the computation of the starting pages of spectral sequences, the maps between spectral sequences, differentials and extensions.

I will show how we define the notion of hidden extensions in the Er page of the Adams spectral seqence. This enables us to give a generalized Leibniz rule. I will give some examples on how we use these notions to explore Weinan's data and produce non-trivial differentials in the Adams spectral sequence.

I will discuss the relation between localizations of homotopy theory at v_n-periodic homotopy equivalences and T(n)-homology equivalences. The first can be understood through Lie algebras in spectra, whereas the second turns out to be closely related to Hopf algebras in spectra. I will explain this perspective and state several open questions. Much of this is based on ongoing joint work with Brantner, Hahn, and Yuan.

Equivariant Lazard rings over an abelian compact Lie group were proved by Hausmann to coincide with equivariant stable complex cobordism rings, thus confirming Greenlees's conjecture. Hausmann's proof, however, does not give explicit algebraic information on the rings involved. An explicit presentation of the equivariant Lazard rings for abelian groups was obtained by Kriz and Lu in a form which can be considered a deformation of the non-equivariant Lazard ring by introducing Euler classes. However, one is interested in an even more explicit presentation, where the relations connect more directly with structures familiar in formal group law theory. The payoff of such a presentation is that one can then mimic commutative algebra methods in spectral algebra, thus allowing constructions of equivariant complex-oriented spectra based on algebraic properties of their coefficients. One case when such an algebraic presentation of an equivariant Lazard ring was successful was a classical result by Strickland for the case of Z/2, which was used by Hanke and Wiemeler to prove Greenlees's conjecture for Z/2 before Hausmann. Strickland's description can be thought of as a series of blow-ups, which gives a lot of algebraic control. In this talk, I will discuss how to obtain a "blow-up" description of equivariant Lazard rings for primary cyclic groups, with particular focus on a number of subtleties one encounters in the process.

I will survey some recent advances in equivariant chromatic homotopy theory due to many people in the language of equivariant formal group laws, and discuss computationally how this plays out explicitly, in terms of the equivariant theory of $v_n$-self-maps introduced by Bhattacharya, Guillou, and Li. This is joint work with Jack Carlisle.

The synthetic analogue of the bar comonad controls the universal differentials in the bar spectral sequence of algebras over spectral operads. This can be viewed as a deformation of Koszul duality of such algebras. I will explain work with Burklund and Senger on identifying the universal differentials in the bar spectral sequence for spectral Lie algebras over $\mathbb F_p$. This will also shed light on the mod $p$ homology and Lubin–Tate theory of labeled configuration spaces via a theorem of Knudsen.

I will talk about my joint work with Po Hu, Petr Somberg, and Benjamin Riley on self-conjugate cobordism. Using descent based on complex cobordism in the category of spectral modules over $MSC$, we obtain a new spectral sequence whose initial page is the cohomology of a polynomial algebra acting on the complex cobordism ring. The generators are permanent cycles, corresponding to real projective spaces and Landweber manifolds. The action can be completely described using novel concepts in the theory of formal group laws. This situation is a spectral analogue of the classical context of Gugenheim–May's work on the cohomology of homogeneous spaces. By using motivic loop techniques, we prove that similarly to their situation, our spectral sequence collapses with no integral extensions. This gives a completely algebraic description of self-conjugate cobordism groups.

Quillen proves that the universal ring for formal group laws is isomorphic to the homotopy of complex cobordism MU. I will discuss equivariant complex oriented cohomology theories and discuss an equivariant version of Quillen's theorem for abelian compact Lie groups.

Classifying the tensor-invertible objects in categories with G-actions is an old question in mathematics. In my talk I’ll survey some of the things we know, and wish to know, about such objects, e.g., when the category in question is spaces, spectra, or various algebraic avatars.

I will talk about the subject of pre-Tannakian categories, which means locally finite symmetric tensor categories with strong duality linear over a field. Interest in this subject came from mixed motives. Deligne proved that in characteristic 0, pre-Tannakian categories are categories of representations of super-group schemes if and only if they satisfy a moderate growth condition. He also developed a method called interpolation for constructing families of new examples. Especially semisimple examples are highly striking and interesting. I will talk about many examples, including those coming from Harman and Snowden’s recent theory of measures on oligomorphic groups. I will then discuss my own universal approach using a structure called T-algebra, which, under certain conditions, is equivalent to commutative algebras in a certain fixed symmetric tensor category. I will talk about new examples I constructed using these methods, including examples involving the oscillator representation and Howe’s theta-correspondence. I will also discuss a variant of Quillen cohomology which can be used to study deformations of T-algebras.

For any prime p and integer n > 0 the K(n)-local determinant sphere S(det) is an explicit K(n)-local spectrum whose Morava E-theory satisfies E∗(S(det)) = E∗ ⊗_ℤp ℤ_p(det). Here ℤ_p(det) denotes the Morava module ℤ_p on which the Morava stabilizer group acts via the determinant homomorphism. The spectrum S(det) plays a crucial role in understanding the K(n)-local Brown Comenetz dual I_n of the sphere. In fact, by work of Hopkins and Gross the Morava module of the spectrum I_n for a prime p is given by the Morava module of S(det) up to a shift. This implies that there exists a unique K(n)-local spectrum P_n which lies in the exotic Picard K(n)- local group κ_n,p such that the homotopy type of I_n agrees with S(det) ∧ P_n up to a shift. In the case of chromatic level 2 the prime 2 is the only one for which P₂ remained mysterious. This talk is a report on joint work in progress with Paul Goerss. We focus on the case n = p = 2. We first show that the homotopy type of the K(1)-localization of S(det) agrees with that of the K(1)-localization of I₂ up to a shift. This is then used to give a complete characterization of P₂ within κ_2,2, which is an explicitly known abelian group of order 2⁹.

In this talk, we will construct the resolution of the motivic fundamental group of a punctured elliptic curve in the DG category of elliptic motives. As a byproduct, we find some elliptic analogue of Bloch-Totaro cycles. This is joint work with Tomohide Terasoma.

Rational $G$-equivariant cohomology theories are represented by rational $G$-spectra. The talk will report progress on the conjecture that there is a Quillen equivalence $$ G\text{-spectra}/{\mathbb Q} \cong \text{DG-}{\mathcal A}(G) $$ for an abelian category ${\mathcal A}(G)$. I will explain a little about the general nature of categories concerned (joint work with Balchin and Barthel) but focus on the proof of the conjecture for many new groups, but especially for toral subgroups of $SU(3)$. This involves classifying subgroups of $G$ up to conjugacy, construction of ${\mathcal A}(G)$ and the proof of the Quillen equivalence.

In the nineties, Michael Wiess introduced a Taylor tower corresponding to any functor $E$ from $\mathcal J$, the category of finite dimensional inner product spaces, to the category of pointed topological spaces. The Weiss tower restricted to $V$ in $\mathcal J$, approximates $E(V)$ as an inverse limit. A fundamental feature of Weiss calculus is that the homogeneous layers are determined using cohomology theories. Thus, using this theory one can study ‘unstable problems’ using techniques from stable homotopy theory.  In a joint work with Yang Hu, we introduce an equivariant analog of Weiss calculus (for a finite group $G$) in which $\mathcal J$ is the functor from the orbit category that sends $G/H$ to the category of finite $H$-representations, for any subgroup $H$ of $G$. Consequently, we get a Taylor tower indexed by orthogonal $G$-representations in which the homogeneous layers are understood using genuine equivariant cohomology theories. Time permitting, we will discuss potential applications.