In the last dozen years, topological methods have been shown to produce a new pathway to study arithmetic statistics over function fields, most notably in Ellenberg-Venkatesh-Westerland's work on the Cohen-Lenstra heuristics. More recently, Ellenberg, Tran and Westerland proved the upper bound in Malle's conjecture over function fields by employing a topological observation which identifies the homology of the braid groups with coefficients arising from braided vector spaces with the cohomology of a quantum shuffle algebra, using the Fox-Neuwirth cellular stratification of configuration spaces of the plane. In this talk, we will extend their techniques to study configuration spaces of the punctured plane and prove a similar result for the homology of the Artin groups of type B. As an application, we will discuss computations when the braid representations are one-dimensional over a field, which shed light on a special case of a conjecture about the homology of mixed braid groups due to Ellenberg-Shusterman.
Vincent Hall 570 - Zoom - Meeting ID: 953 2315 3681 - Passcode: uWgLL9
The absolute Galois group of the field of rational numbers and the Grothendieck-Teichmueller (GT) group introduced by V. Drinfeld in 1990 are among the most mysterious objects in mathematics. In my talk, I will introduce (the gentle version) of the Grothendieck-Teichmueller group. I will also introduce the groupoid GTSh of GT-shadows that can be used to study this group. I will explain how the groupoid GTSh acts on child's drawings, describe properties of this action and show some interesting examples. My talk is based on papers in preparation with Jacob Guynee, Jessica Radford and Jingfeng Xia.
Vincent Hall 570 and via Zoom - Meeting ID: 953 2315 3681 - Passcode: uWgLL9
In 2000, Andruskiewitsch and Schneider conjectured that finite dimensional, pointed Hopf algebras over a field of characteristic zero are generated by their grouplike and skew-primitive elements. I’ll explain a new structure theory for braided Hopf algebras which sheds light on this conjecture.
Vincent Hall 570 and via Zoom - Meeting ID: 953 2315 3681 - Passcode: uWgLL9
The plus construction was introduced by Baez-Dolan as a means for defining their notion of an opetope and by another name with another intent by Getzler and Kapranov. The plus construction has since proven to be a key component in different operadic theories such as the Feynman categories of Kaufmann and Ward. In this talk, I will discuss joint work with Ralph Kaufmann where we generalize the plus constructions to an endofunctor of symmetric monoidal categories. A special case is given by unique factorization categories whose plus construction yields Feynman categories. As an upshot, we can use this to connect the plus construction to monoid definitions of operad-like structures.
Vincent Hall 570 and via Zoom - Meeting ID: 953 2315 3681 - Passcode: uWgLL9
In the past several years, there has been a huge amount of progress in our understanding of the Ausoni--Rognes redshift philosophy, which concerns the interaction of algebraic K-theory with the chromatic filtration of the category of spectra. In particular, Land, Mathew, Meier, and Tamme recently established fundamental structural results for chromatically localized algebraic K-theory, which giving a partial solution to the Ausoni--Rognes redshift conjecture. In this talk, I will give an introduction to K-theory and chromatic redshift and explain forthcoming work with Jonas McCandless in which we verify a Land--Mathew--Meier--Tamme style vanishing result for topological restriction homology.
Vincent Hall 570 and via Zoom - Meeting ID: 953 2315 3681 - Passcode: uWgLL9
Andrea Bianchi , University of Copenhagen Recording Slides
A partially multiplicative quandle (PMQ) is a set with two operations, called "conjugation" and "partial multiplication", satisfying some axioms. One can associate with a PMQ Q a Hurwitz space Hur(Q), containing configurations of particles in the plane with Q-valued monodromies around the particles: for specific PMQs one recovers classical Hurwitz spaces Hur^c_{G,n}, as recently studied by Ellenberg-Venkatesh-Westerland, and (up to homotopy equivalences) the moduli spaces M_{g,n} of Riemann surfaces of genus g with n>=1 boundary curves. I will describe in detail the topology of Hur(Q), and give a Fox-Neuwirth-Fuchs-style cell stratification of this space; under mild hypotheses on Q, I will describe the compactly supported cohomology of Hur(Q) as the cohomology of a double bar complex associated with the PMQ-ring Z[Q]: this description invites a comparison with work of Ellenberg-Tran-Westerland and of Hoang. Finally, I will consider Hur(Q) as a topological monoid and state a result about its group completion, under significant additional hypotheses on Q (being finite and "Poincare").
Vincent Hall 570 and via Zoom - Meeting ID: 953 2315 3681 - Passcode: uWgLL9
Hana Jia Kong , Institute for Advanced Study Recording Slides
The real motivic stable homotopy category has a close connection to the $C_2$-equivariant stable homotopy category. From a computational perspective, the real motivic computation can be viewed as a simpler version which ``removes the negative cone'' in the $C_2$-equivariant stable homotopy groups. On the other hand, the work of Burklund--Hahn--Senger shows that one can deform the completed $C_2$-equivariant category to get the completed Artin--Tate real motivic category. The $C_2$-effective spectral sequence plays an important role in the deformation point of view; it calculates the Artin--Tate homotopy groups.
In the ongoing project with Gabriel Angelini-Knoll, Mark Behrens, and Eva Belmont, we try to build a $C_p$ analog of this story for an odd prime $p$. We give a new interpretation of the $C_2$-effective spectral sequence, and we show that this interpretation generalizes to the odd prime case. This gives a deformation of the Borel equivariant stable homotopy category for more general groups.Location: Vincent Hall 570 and via Zoom - Meeting ID: 953 2315 3681 - Passcode: uWgLL9
Maximilien Peroux, University of Pennsylvania Recording
Topological Hochschild homology (THH) is an important variant for ring spectra. It is built as a geometric realization of a cyclic bar construction. It is endowed with an action of circle. This is because it is a geometric realization of a cyclic object. The simplex category factors through Connes’ category Λ. Similarly, real topological Hochschild homology (THR) for ring spectra with anti-involution is endowed with a O(2)-action. Here instead of the cyclic category Λ, we use the dihedral category Ξ.
From work in progress with Gabe Angelini-Knoll and Mona Merling, I present a generalization of Λ and Ξ called crossed simplicial groups, introduced by Fiedorwicz and Loday. To each crossed simplical group G, I define THG, an equivariant analogue of THH. Its input is a ring spectrum with a twisted group action. THG is an algebraic invariant endowed with different action and cyclotomic structure, and generalizes THH and THR. Notably, I will introduce a quaternionic refinement that we call hyperreal topological Hochschild homology.
Location: Vincent Hall 570 and via Zoom - Meeting ID: 953 2315 3681 - Passcode: uWgLL9Ningchuan Zhang , University of Pennsylvania Recording Slides
The original version of the Quillen-Lichtenbaum Conjecture, proved by Voevodsky and Rost, connects special values of the Dedekind zeta function of a number field with its algebraic $K$-groups. In this talk, I will discuss a generalization of this conjecture to Dirichlet $L$-functions. The key idea is to twist algebraic $K$-theory spectra of number fields with equivariant Moore spectra associated to Dirichlet characters. Rationally, we obtain a Quillen-Borel type theorem for Artin $L$-functions. This is joint work in progress with Elden Elmanto.
Vincent Hall 570 and via Zoom - Meeting ID: 953 2315 3681 - Passcode: uWgLL9
Phil Tosteson , University of Chicago Recording Slides
Let X be a projective variety, and C be an algebraic curve. The topological problem of computing the homology of the space of algebraic maps from C to X is analogous to the arithmetic problem of counting rational points on X. I will talk about the history of this problem, its connection to the topology of loop spaces, and joint work with Ronno Das considering the case where X is a blowup of projective space at a finite set of points.
Vincent Hall 570 and via Zoom - Meeting ID: 953 2315 3681 - Passcode: uWgLL9
Robin Sroka , McMaster University , Hamilton, Ontario
While the rational cohomology of arithmetic groups such as $\operatorname{SL}_n(\mathbb{Z})$ and $\operatorname{Sp}_{2n}(\mathbb{Z})$ can often be completely computed if the cohomological degree is small compared to $n$, little is known about it in high cohomological degrees. In this talk, I will discuss vanishing results that have recently been obtained for the high-dimensional rational cohomology of $\operatorname{SL}_n(\mathbb{Z})$, $\operatorname{Sp}_{2n}(\mathbb{Z})$ and other arithmetic Chevalley groups. This is related to a conjecture of Church--Farb--Putman and based on joint works with Brück--Miller--Patzt--Wilson, Brück--Patzt and Brück--Santos Rego.
Vincent Hall 570 and via Zoom - Meeting ID: 953 2315 3681 - Passcode: uWgLL9
Craig Westerland , University of Minnesota Recording
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
XiaoLin Danny Shi , University of Chicago Recording
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.
Zoom - no in person (Click here to access the talk on Zoom )
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.
Paolo Salvatore , Universita' di Roma "Tor Vergata Recording Slides
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,
Zach Himes, https://zachhimes.github.io Recording Slides
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.
Benjamin Antieau , Northwestern University Recording Slides
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.
Achim Krause , University of Münster Recording Slides
Norihiro Yamada, University of Minnesota Recording Slides
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.
Adrian Diaconu, University of Minnesota - Twin Cities
In this talk, I will discuss recent results, in joint work with Bergström, Petersen, and Westerland, concerning the relationship between the conjectural asymptotic formula for moments of quadratic Dirichlet L-series in the function field setting, and the stable homology of braid groups with coefficients in symplectic representations.
Slide Presentation: https://drive.google.com/file/
Vincent Hall 570 - Zoom
Calista Bernard, University of Minnesota
In the 70s, Fred Cohen and Peter May gave a description of the mod $p$ homology of a free $E_n$-algebra in terms of certain homology operations, known as Dyer--Lashof operations and the Browder bracket. These operations capture the failure of the $E_n$ multiplication to be strictly commutative, and they prove useful for computations. After reviewing the main ideas from May and Cohen's work, I will discuss a framework to generalize these operations to homology with certain twisted coefficient systems and give a complete classification of twisted operations for $E_{\infty}$-algebras. If time permits, I will also explain computational results that show the existence of new operations for $E_2$-algebras.
Vincent Hall 570 - Zoom
Daniel Grady , Texas Tech University
The cobordism hypothesis of Baez--Dolan, whose proof was sketched by Lurie, provides a beautiful classification of topological field theories: for every fully dualizable object in a symmetric monoidal (infinity,d) category, there is a unique (up to a contractible choice) topological field theory whose value at the point coincides with this object. As beautiful as this classification is, it fails to include non-topological field theories. Such theories are important not just in physics, but also in pure mathematics (for example, Yang-Mills). In this talk, I will survey recent work with Dmitri Pavlov, which proves a geometric enhancement of the cobordism hypothesis. In the special case of topological structures, our theorem reduces to the first complete proof of the topological cobordism hypothesis, after the 2009 sketch of Lurie.
Tyler Lawson , University of Minnesota
At the prime 2, the dual Steenrod algebra is a graded ring that is polynomial on infinitely many generators over Z/2, and it appears as the homology of a spectrum H Z/2. In this talk I'll describe a minor mystery about the structure of modules and algebras over it that arose in joint work with Beaudry-Hill-Shi-Zeng, and how this can be resolved by a general result relating pushouts of E_k-algebras with relative tensors over E_{k+1}-algebras. (Joint work with Michael Hill.)
Vincent Hall 570
Nate Bottman , Max Planck Institute for Mathematics, Bonn, Germany
The symplectic (A-infinity,2)-category Symp, which is currently under construction by myself and my collaborators, is a 2-category-like structure whose objects are symplectic manifolds and where hom(M,N) := Fuk(M^- x N). Symp is a coherent algebraic structure which encodes the functoriality properties of the Fukaya category. This talk will begin with the following question: what can we say about the part of Symp that knows only about a single symplectic manifold M, and the diagonal Lagrangian correspondence from M to itself? We expect that the answer to this question should be a chain-level algebraic structure on symplectic cohomology, and in this talk I will present progress toward confirming this. Specifically, I will present a "simplicial version" of the 2-dimensional Fulton-MacPherson operad, which may be of independent topological interest.
Ang Li (University of Kentucky)
For any C_2-equivariant spectrum, we can functorially assign two non-equivariant spectra - the underlying spectrum and the geometric fixed point spectrum. They both induce maps from the RO(C_2)-graded cohomology to the classical cohomology. In this talk, I will compare the RO(C_2)-graded squaring operations with the classical squaring operations along the induced maps. This is joint work with Prasit Bhattacharya and Bertrand Guillou.
Akhil Mathew, University of Chicago
I will explain some descent and vanishing results in the algebraic K-theory of ring spectra, motivated by the redshift philosophy of Ausoni-Rognes. These results are all proved by considering group actions on stable ∞-categories and their K-theory, as well as some tools coming from chromatic homotopy theory. Joint work with Dustin Clausen, Niko Naumann, and Justin Noel.
Jenny Wilson , University of Michigan, Ann Arbor
In this talk I will describe some current efforts to understand the high-degree rational cohomology of SL_n(Z), or more generally the cohomology of SL_n(R) when R is a number ring. Although the groups SL_n(R) do not satisfy Poincare duality, they do satisfy a twisted form of duality, called Bieri--Eckmann duality. Consequently, their high-degree rational cohomology groups are governed by an SL_n(R)-representation called the Steinberg module. The key to understanding these representations is through studying the topology of certain associated simplicial complexes. I will survey some results, conjectures, and ongoing work on the Steinberg modules, and the implications for the cohomology of the special linear groups. This talk includes work joint with Brück, Kupers, Miller, Patzt, Sroka, and Yasaki. The talk is geared for topologists and will not assume prior expertise on the cohomology of arithmetic groups!
Vincent Hall 570 - Zoom
Pedro Tamaroff (Max Planck Institute for Mathematics in the Sciences in Leipzig (MPIMiS))
In the classical Batalin–Vilkovisky formalism, the BV operator is a differential operator of order two with respect to a commutative product; in the differential graded setting, it is known that if the BV operator is homotopically trivial, then there is a genus zero level cohomological field theory induced on homology; a non-commutative analogue of this was developed in arXiv:1510.03261. In this talk, we will explore generalisations of non-commutative Batalin–Vilkovisky algebras for differential operators of arbitrary order, showing that homotopically trivial operators of higher order also lead to interesting algebraic structures on the homology. There will in fact be a nice geometrical story lurking behind. This is joint work with Vladimir Dotsenko and Sergey Shadrin.
Amit Sharma , Kent State University, Ohio
An algebraic version of the cobordism hypothesis states that the symmetric monoidal quasi-category of 1-cobordisms Bord_1 is a model for the free compact closed quasi-category on one generator. Our goal is to construct a combinatorial model for the latter. We propose a symmetric monoidal bicategory and conjecture that it is the desired model. Recall that the space of endomorphisms of the unit object of Bord_1 is the free E_∞-space generated by the classifying space of the topological group Diff^+(S^1). In support of our conjecture, we show that the category of endomorphisms of the unit object of our symmetric monoidal bicategory is a free symmetric monoidal category generated by the cyclic category Λ. Recall that both BDiff^+(S^1) and BΛ are K(Z, 2) spaces. This is a joint work with André Joyal.Location: Vincent Hall 570 Click here to access the talk on Zoom
Peter Haine , University of California, Berkeley
Given a (nice) connected topological space T, local systems on T can be understood as representations of the fundamental group of T. Similarly, given a connected scheme X, étale local systems on X can be understood as representations of the étale fundamental group of X. This suggests a general phenomenon: in many situations, a category of sheaves should be expressible in terms of representations of a more simple/combinatorial object (an “exit-path category"). In this talk, we’ll give a survey of where exit-path categories arise in topology and geometry. In particular, we’ll discuss joint work with Barwick and Glasman on understanding exit-path categories in algebraic geometry and how to use them to classify schemes.
Mike Hill , UCLA
In equivariant homotopy, additive transfers and multiplicative norms are parameterize by the same combinatorial data: a transfer/indexing system. The transfer systems form a poset, and this reflects properties in the homotopy. We can ask also about compatibility between the additive and multiplicative structure. In this talk, I’ll describe this structure and compatibility, and I’ll sketch some recent work with undergraduates counting the number of compatible pairs for cyclic p-groups.
Vincent Hall 570 - Mode: In person