Mondays at 2:30 in Vincent Hall 570
The Milnor–Moore theorem identifies a large class of Hopf algebras as enveloping algebras of the Lie algebras of their primitives. If we broaden our definition of a Hopf algebra to that of a braided Hopf algebra, much of this structure theory falls apart. The most obvious reason is that the primitives in a braided Hopf algebra no longer form a Lie algebra. In this talk, we will discuss recent work to understand what precisely is the algebraic structure of the primitives in a braided Hopf algebra in order to “repair” the Milnor–Moore theorem in this setting. It turns out that this structure is closely related to the dualizing module for the braid groups, which implements dualities in the (co)homology of the braid groups.Location: Zoom
In this talk, we will present Real-oriented models of Lubin-Tate theories at p=2 and arbitrary heights. For these models, we give explicit formulas for the action of certain finite subgroups of the Morava stabilizer groups on the coefficient rings. This is an input necessary for future computations. The construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory. As a consequence, we will describe how we can use these models to prove periodicity theorems for Lubin-Tate theories and set up an inductive approach to prove differentials in their slice spectral sequences. This talk is based on several joint projects with Agnès Beaudry, Jeremy Hahn, Mike Hill, Guchuan Li, Lennart Meier, Guozhen Wang, Zhouli Xu, and Mingcong Zeng.
In this talk, I will describe an explicit computation, in terms of generators and relations, of the coefficient ring of the equivariant stable complex cobordism spectrum MU_G in the case where G is a primary cyclic group. I will also discuss some applications of these calculations, including construction of equivariant complex-oriented spectra via their equivariant formal group laws, and calculations of the coefficient ring of MU_G for some non-abelian groups G, such as the symmetric group on three elements.
Certain p-adic Lie groups have the property that their cohomology admits a finite-length resolutions in terms of the cohomology of their finite subgroups. This phenomenon was first observed in stable homotopy theory by Goerss-Henn-Mahowald-Rezk, who used such a resolution of the height 2 Morava stabilizer group at the prime 3 to construct a topological resolution for the K(2)-local sphere. I'll describe a new resolution for the analogous case of the groups SL_2(Z_3) and GL_2(Z_3), as well as some attempts to construct such resolutions for general p-adic Lie groups. This is joint work with Eva Belmont.
I will discuss various forms of the evenness conjecture for equivariant complex cobordism and some of their broader context. Then I will describe my recent counterexample to the homotopical version of the conjecture, which complements a recent theorem by Samperton and Uribe disproving the geometric version. Proving my example hinges on a certain generalization of orientation in equivariant homology, which leads to a new completion theorem for Morava K(n)-theory, whose statement does not involve higher derived functors.
For a commutative ring spectrum R, there are two natural candidates for the "multiplicative group" of R. One is the spectrum of units in R, denoted gl_1(R), and the other is the spectrum of "strict units" in R, denoted G_m(R). The latter is obtained from the former by taking the mapping spectrum out of the Eilenberg-McLane spectrum Z. The spectrum gl_1(R) is closely related to R itself. For example, the homotopy groups of R and gl_1(R) agree in all degrees above 0. On the other hand, the spectrum G_m(R) is a more subtle object and the subject of active research. The initial example of a commutative ring spectrum is the sphere spectrum S. In my talk, I will describe a computation of G_m(S). I will also explain how to compute the connective spectrum of maps from Z to the Picard spectrum of S, which gives a (non-trivial) delooping of G_m(S), and discuss extensions of the computation to other commutative ring spectra, such as the algebras of spherical Witt vectors associated with perfect F_p-algebras.
For a simply connected finite CW-complex X, we construct an algebraic model for the rational homotopy type of Baut(X), the classifying space of fibrations with fiber X. This space is in general far from nilpotent, so its rational homotopy type cannot be modeled by a dg Lie algebra over Q as in Quillen's theory. Instead, we work with dg Lie algebras in the category of algebraic representations of a certain reductive algebraic group associated to X. A consequence of our results is that the rational cohomology ring of Baut(X) can be computed in terms of cohomology of arithmetic groups and Lie algebra cohomology. In special cases the computation reduces to invariant theory and calculations with modular forms. We moreover show that the representations of the homotopy mapping class group of X in the higher rational homotopy groups of Baut(X) are algebraic in a suitable sense. This extends a classical result of Sullivan and Wilkerson to higher homotopy groups. Our results also improve and generalize certain earlier results due to Ib Madsen and myself on Baut(M) for highly connected manifolds M. This is joint work with Tomas Zeman. arXiv:2203.02462,
Bestvina--Feighn proved that Aut(F_n) is a rational duality group, i.e. there is a Q[Aut(F_n)]-module, called the rational dualizing module, and a form of Poincare duality relating the rational cohomology of Aut(F_n) to its homology with coefficients in this module. Bestvina--Feighn's proof does not give an explicit combinatorial description of the rational dualizing module of Aut(F_n). But, inspired by Borel--Serre's description of the rational dualizing module of arithmetic groups, Hatcher--Vogtmann constructed an analogous module for Aut(F_n) and asked if it is the rational dualizing module. In work with Miller, Nariman, and Putman, we show that Hatcher--Vogtmann's module is not the rational dualizing module.
Joint work with Achim Krause and Thomas Nikolaus computes the algebraic K-groups of rings such as Z/p^n using algorithms to compute syntomic cohomology. This talk will give background on the problem as well as an overview of the techniques connecting prismatic cohomology, cyclotomic spectra, TC, and K-theory.
Topology is arguably the study of "computational nearness." For instance, a limit point of a subset S of a space A can be seen as a point p in A such that any finite approximation of p cannot be separated from S. Instead of directly working on these computations that gradually approach to points, however, general topology axiomatically defines spaces in terms of the algebra of open sets. This axiomatic approach turns out to be too general, e.g., one needs the Hausdorff axiom for expected properties of spaces. The mathematical structure of open sets does not behave well either, e.g., Hausdorff spaces are not closed under quotient, and the category of topological spaces is not closed. Besides, the excessive use of power sets in general topology is undesirable for constructive mathematics. Motivated by these problems, I propose a new combinatorial foundation of general topology based on game semantics, in which spaces are given by finite trees or games, and points in spaces by algorithms or strategies about how to walk on games. This combinatorics reformulates general topology in a quite intuitive fashion by capturing computational nearness in terms of strategies, and its mathematical structure behaves well, overcoming the above problems. Moreover, the finiteness of games and the computability of strategies are preferred for the constructive standpoint. In this talk, I present the overview and main ideas, not technical details, of this research paradigm.
Marshall Smith, University of Minnesota
Central to the chromatic approach to homotopy theory is the category of K(n)-local spectra. Recently (in 2021), Heard proved the existence of a descent spectral sequence computing the Picard group of this category, a spectral sequence previously belonging to folklore. Using a modification of Morava E-theory due to Davis, we construct a similar, perhaps more computable spectral sequence, and believe we can use this to show that the rank of the K(n)-local Picard group is equal to that of the algebraic Picard group of invertible Morava modules. This is work in progress.