Finite groups of Lie type, such as SL_n(𝔽_q), Sp_n(𝔽_q)..., are ubiquitous in mathematics, and calculating their cohomology has been a central theme over the years. Without any structural reasons as to why, it has calculationally been observed that, when calculable, their mod ℓ cohomology agree with the mod ℓ cohomology of LBG(ℂ), the free loop space on BG(ℂ), the classifying space of the corresponding complex algebraic group G(ℂ), as long as q is congruent to 1 mod ℓ. This is despite that LBG(ℂ) and BG(𝔽_q) are vastly different spaces, also at a prime ℓ, ruling out some space-level equivalence. In recent joint work with Anssi Lahtinen, that combines ℓ-compact groups with string topology à la Chas–Sullivan, we give a general structural relationship between these two cohomologies, which, suitably formulated, even works without any congruence condition on q, as long as it is prime to ℓ. We use this to prove structured versions of previous calculations, and establish isomorphism in new cases. The isomorphism conjecture in general hinges on the fate of a single cohomology class in exceptional Lie groups at small primes. My talk will begin to tell this story, as we know it so far...

Let M be a connected topological manifold of dimension at least 2 with an effective action of a finite group G. Associating with the orbit configuration space F_G(M, n), n ≥ 2 of the G-manifold M, we try to upbuild the theoretical framework of orbit braids in M × I where the action of G on I is trivial, which contains the following contents: we introduce the orbit braid group 𝓑_n^orb(M, G), and show that it is isomorphic to a group with an additional endowed operation (called the extended fundamental group), formed by the homotopy classes of some paths (not necessarily closed paths) in F_G(M, n), which is an essential extension for fundamental groups. The orbit braid group 𝓑_n^orb(M, G) is large enough to contain the fundamental group of F_G(M, n) and other various braid groups as its subgroups. Around the central position of 𝓑_n^orb(M, G), we obtain five short exact sequences weaved in a commutative diagram. We also analyze the essential relations among various braid groups associated to those configuration spaces F_G(M, n), F(M, n), and F(M/G, n). We finally consider how to give the presentations of orbit braid groups in terms of orbit braids as generators. We carry out our work by choosing M = ℂ ≈ ℝ² with typical actions of ℤ_p and (ℤ₂)². We obtain the presentations of the corresponding orbit braid groups, from which we see that the generalized braid group Br(D_n) (introduced by Brieskorn) actually agrees with the orbit braid group 𝓑_n^orb(ℂ\{0}, ℤ₂) and Br(D_n) is a subgroup of the orbit braid group 𝓑_n^orb(ℂ, ℤ₂). This talk is based upon a joint work with Hao Li and Fengling Li.

K-theory is a central object of study that relates algebraic topology, number theory and geometric topology, among other fields. There are many ways of constructing K-theory. Classical results tell us that these are all "the same" in an additive sense. In this talk, I will discuss joint work with Osorno in which we prove a multiplicative comparison of two classic K-theory constructions, those of Segal and Waldhausen. In particular, this produces comparisons of commutative ring spectra and spectrally enriched categories.

We define the double of a simplicial complex and study its connectivity properties. The definition is motivated by homological stability: one can study homological stability for a sequence of groups G₁ → G₂ → G₃ → ... using an associated simplicial complex; the double of this complex is related to stabilizing in steps of two G₂ → G₄ → G₆ → .... Because of this, homological stability suggests that if a simplicial complex is highly connected, so is its double, a result that we prove under appropriate assumptions. (This is joint work with Kathryn Lesh and Bridget Schreiner.)

[Slides] Ningchuan Zhang, Dirichlet character twisted Eisenstein series and J-spectra

[Slides] Foling Zou, Nonabelian Poincaré duality in equivariant factorization homology

Hana Jia Kong, The C₂-equivariant homotopy of ko_C2

[Slides] Yunze Lu, The coefficients of equivariant complex cobordism

Balmer and Sanders show that understanding the problem of determining inclusions among the tensor triangulated thick ideals of finite G-spectra can be reduced to the following problem: given n, and H < G, with G a finite p-group, how big can r be such that there exists a finite G-complex with G-fixed points of chromatic type n, and H-fixed points of chromatic type n+r? Barthel et al. solve this problem when G is abelian by showing that a "blue shift" upper bound that they establish is realized with a family of examples previously analyzed by Arone and Lesh, and basically due to Mitchell.
In my talk I will discuss recent and ongoing joint work with my student Chris Lloyd, on a way to construct examples that is both much more elementary and much more flexible than previous constructions. The idea is to feed lens spaces associated to representations of G into Jeff Smith's "machine" for constructing type n complexes: the examples are thus well chosen stable wedge summands of smash products of lens spaces associated to well chosen G-representations. The method yields alternatives to the Arone–Lesh examples and new examples too, and is well suited for investigation by computer. For example, we are able to resolve the first unknown case, when G is the dihedral group of order 8 and H is a noncentral subgroup of order 2: the maximal shift is 2, not 1, as was previously thought.

Suppose G is a finite group, and f is a map from a CW complex F to the fixed point of a G-CW complex Y. Is it possible to extend F to a finite G-CW complex X satisfying X^G = F, and extend f to a G-map g: X → Y, such that g is a homotopy equivalence after forgetting the G-action?
In case Y is a single point, the problem becomes whether a given finite CW complex F is the fixed point of a G-action on a finite contractible CW complex. In 1942, P.A. Smith showed that the fixed point of a p-group action on a finite ℤ_p-acyclic complex is still ℤ_p-acyclic. In 1971, Lowell Jones proved a converse for semi-free cyclic group action on finite contractible X. In 1975, Robert Oliver proved that, for general action on finite contractible X, if the order of G is not prime power, then the only obstruction is the Euler characteristic of F.
We extend the classical results of Lowell Jones and Robert Oliver to the general setting. For semi-free action, we encounter a finiteness type obstruction. For general action by group of not prime power order, the obstruction is the Euler characteristics over components of Y^G. We calculate such obstructions for various examples.
This is a joint work with Sylvain Cappell of New York University, and Shmuel Weinberger of University of Chicago.

Representation spheres and universes are prominent in classical equivariant stable homotopy theory, but more recent approaches to the subject emphasize algebraic structure over the representation theory. In this talk, I will discuss how these two perspectives are reflected on the level of N_∞ operads.
Roughly speaking, a N_∞ operad is a structure that parametrizes homotopy commutative monoids equipped with additional equivariant transfer maps. Since Blumberg and Hill's groundbreaking work, it has been known that the homotopy theory of N_∞ operads is essentially algebraic. However, this viewpoint ignores the peculiarities of the natural geometric examples, namely the Steiner and linear isometries operads over incomplete universes. I will describe a few cases where one can characterize such operads in purely algebraic terms, and I will explain why for most groups, there are N_∞ operads that are not equivalent to any Steiner or linear isometries operad.

A fundamental problem in 4-dimensional topology is the following geography question: which simply connected topological 4-manifolds admit a smooth structure? After the celebrated work of Kirby–Siebenmann, Freedman, and Donaldson, the last uncharted territory of this geography question is the "11/8-Conjecture." This conjecture, proposed by Matsumoto, states that for any smooth spin 4-manifold, the ratio of its second Betti number and signature is least 11/8.
Furuta proved the "10/8 + 2"-Theorem by studying the existence of certain Pin(2)-equivariant stable maps between representation spheres. In this talk, we will present a complete solution to Furuta's problem by analyzing the Pin(2)-equivariant Mahowald invariants. In particular, we improve Furuta's result into a "10/8 + 4"-Theorem. Furthermore, we show that within the current existing framework, this is the limit. This is a joint work with Mike Hopkins, Jianfeng Lin and XiaoLin Danny Shi.

The homotopy theory of orbispaces is a generalization of the homotopy theory of G-spaces in which the group G is allowed to vary. More precisely, orbispaces model the homotopy theory of topological Artin stacks (a.k.a. unstable global homotopy theory) in much the same way as Kan complexes model the homotopy theory of topological spaces. We will show how certain geometrically defined cohomology theories, such as topological K-theory and elliptic cohomology, extend to orbispaces, and how this extension is useful for both theoretical and computational purposes. Finally, we will discuss some calculations in equivariant elliptic cohomology.

Waldhausen's algebraic K-theory of spaces provides a critical link in the classification of manifolds and their diffeomorphisms. We will explain this connection and discuss work in progress with Cary Malkiewich on the analogous story for manifolds with group action.

The symplectic K-theory groups KSp_∗(ℤ) may be defined similarly to usual algebraic K-theory K_∗(ℤ), replacing the group GL_n(ℤ) by the symplectic group Sp_2g(ℤ). Much is known about the relationship between KSp_∗(ℤ) and K_∗(ℤ), due to work of Karoubi and others. I will explain that large cyclic subgroups of Sp_2g(ℤ) may be used to detect an important part of KSp_∗(ℤ), and furthermore to understand an action of field automorphisms of the complex numbers on the p-adic completion. This is joint work with T. Feng and A. Venkatesh.

Following the rational homotopy theory of Quillen and Sullivan, one can compute the rational homotopy groups of a space from a commutative algebra model of its cochains by taking (derived) indecomposables. More abstractly, this procedure implements a form of Koszul duality between commutative algebras and Lie algebras. The Lie algebra model of rational homotopy theory generalizes to v_n-periodic homotopy theory; however, the above procedure generalizes only partially. Work of Behrens–Rezk shows that the v_n-periodic homotopy groups of spheres and of certain compact Lie groups can be computed from their cochains valued in Morava E-theory (or some variant thereof), but in general it is unclear for which spaces this works. I will report on ongoing joint work with Brantner, Hahn, and Yuan, which provides a large class of spaces for which "Koszul duality works" also in v_n-periodic homotopy.

This is joint work with Vitaly Lorman and James D. Quigley. The ℤ/p-Tate cohomology spectrum of the n'th Johnson–Wilson theory splits as a wedge of (n-1)'th Johnson–Wilson theories (after completion). We construct a C₂-equivariant lifting of this splitting for Real Johnson–Wilson theories. The C₂-fixed points of this splitting is a higher height analogue to Davis and Mahowald's splitting of the Tate cohomology spectrum of ko as a wedge of Hℤ.

Red shift is the philosophy that the chromatic height n part of a ring spectrum controls the chromatic height (n+1) part of its algebraic K-theory. A recent result of Bhatt, Clausen and Mathew shows that this philosophy is precisely true if the ring spectrum is a discrete ring. I will report on joint work with Markus Land and Georg Tamme, where we find a new proof of this result and provide generalizations to ko- and tmf-algebras.

J.D. Quigley, The motivic kq-resolution

Hood Chatham, An orienation map for height p-1 real E-theory

Weinan Lin, On the E₂-page of the May spectral sequence