César A. Ipanaque Zapata (IME/USP Brazil and University of Minnesota)
The genus of a fibration was introduced by Schwarz in 1962 [1]. Given a map g : A → B, the usual sectional number sec_u(g) is the least integer m such that B can be covered by m open subsets each of which admits a local section of g [1]. Likewise, the sectional category secat(g) is the least integer m such that B can be covered by m open subsets each of which admits a local homotopy section of g [2].
In the case that g is a fibration, the usual sectional number and the sectional category of g coincide with the Schwarz’s genus of g. In this work we introduce a notion of sectional number of a morphism f in a category with covers, sec(f), which extends the usual sectional number and the sectional category (and, of course, the Schwarz’s notion). We study the invariance of sectional number and the behaviour of sec under weak pullbacks and continuous functors. In addition, we introduce a notion of multiplicative cohomology theory on a category and we use it to present a cohomological lower bound for the sectional number. Our unified concept is relevant in this new setting because it presents a construction of foundations of sectional theory in category theory and build bridges between new areas in mathematics.
Sasha Voronov
G-cyclifications Map(G,X)//G are generalizations of cyclic loop spaces L_c X := Map(S^1, X)//S^1, also known as equivariant or unparameterized free loop spaces. The rational homotopy type of a cyclic loop space has been known since the heyday of rational homotopy theory: a 1985 result of Vigué-Poirrier and Burghelea shows how to construct the Sullivan minimal model of L_c X from that of X. A few years ago, when Berglund gave a talk in this seminar, it was not even clear how to generalize Vigué-Poirrier-Burghelea’s theorem to the toroidification Map(T,X)//T, where T = (S^1)^k is a k-torus. Sati and I needed this generalization, because of a relevance of the toroidifications Map(T,S^4)//T of the 4-sphere to supergravity and a conjectural relevance to del Pezzo surfaces. Now, as we have done this and dramatically simplified the original proof of Vigué-Poirrier and Burghelea, it has opened the way to understanding the rational homotopy theory of a G-cyclification Map(G,X)//G for any compact Lie group G.
VH 570
Craig Westerland
Classically, the Cartier-Milnor-Moore theorem identifies primitively generated Hopf algebras as (restricted) enveloping algebras of their Lie algebra of primitives. The Poincaré-Birkhoff-Witt theorem establishes an isomorphism between an associated graded Hopf algebra of these enveloping algebras with a symmetric algebra. These theorems are fundamental to the study of Hopf algebras and their cohomology.
There is, however, a more general context where one can study Hopf algebra objects in braided monoidal categories. In this setting, these classical results fail to hold. I will speak about recent work that provides analogues of these structure theorems using the notion of a braided operad.
VH 570
Tyler Lawson
Abstract not available
VH 570
Katharine Adamyk (Hamline University)
Transfer systems are combinatorial objects with connections to both equivariant homotopy theory and model categories. (In particular, transfer systems on the subgroup lattice of a group, G, determine both the N-infinity operads associated to G and the possible model structures on the subgroup lattice.) In ongoing work (with Balchin, Barrero, Scheirer, Sulyma, Wisdom, Zapata Castro) we investigate several questions related to the generation of transfer systems. This talk will focus on our results related to the complexity of a finite group, G, which we define to be the maximal size of a minimal generating set for a transfer system on the subgroup lattice of G. In order to determine the complexity of several families of groups, we introduce rainbow diagrams, a new representation of appropriately symmetric generating sets for transfer systems.
VH 570
Tyler Lawson , University of Minnesota
Obstruction theory attempts to classify maps X -> Y inductively, using a "look-ahead" to make sure we don't make immediate bad decisions; similarly, spectral sequences use look-ahead to improve techniques for calculation. In this talk I'll aim to describe a technique for trying to analyze a space from a filtration on it, using a similar look-ahead technique.
Vincent 570 or Zoom Online Meeting Info: Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
Segal's construction of K-theory turns symmetric monoidal categories into connective spectra, and Thomason later showed that every connective spectrum arises this way (up to equivalence). In the setting of equivariant stable homotopy theory, we can ask what the "correct" notion of equivariant symmetric monoidal category models all connective genuine G-spectra via K-theory. This talk will provide an answer to this question, using the K-theory of categorical Mackey functors. Based on joint work with D. Chan and M. Péroux.
Vincent 570 or Zoom Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
The homology of various sequences of topological spaces often stabilizes. For instance, classical results of McDuff and Segal imply that the homology of unordered configuration spaces of open manifolds stabilizes as the number of points in the configuration increases. In this talk, I will discuss an equivariant analogue of this phenomenon, Bredon homological stability, where homology is replaced by Bredon homology, and spaces are replaced by G-spaces for some finite group G. This is joint work with Eva Belmont and Chase Vogeli.
VH 570 or Zoom Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
This is joint work with Niels van der Weide, Benedikt Ahrens and Paige Randall North. Category theory can now be found all throughout mathematics. This motivates a proper classification of categories and its various generalizations, such as 2-categories and double categories. However, unlike many algebraic structures, such as groups, categories can be studied both up to isomorphisms and equivalences. Moreover, other categorical structures exhibit an even wider range of equivalences, complicating any classification effort.
In this talk I want to explain that in an alternative mathematical foundation, and concretely a univalent one, we can refine our definition of categories and internalize the desired equivalence type. I will in particular apply this perspective to the case of double categories and discuss some implications thereof. No knowledge beyond the definition of a category is assumed for this talk, and everyone is welcome to attend.
Abstract not available
VH 570 or Zoom link: Meeting ID: 943 1752 6327
Classical enumerative geometry asks geometric questions of the form “how many?” and expects an integral answer. For example, how many circles can we draw tangent to a given three? How many lines lie on a cubic surface? The fact that these answers are well-defined integers, independent upon the initial parameters of the problem, is Schubert’s principle of conservation of number. In this talk we will outline a program of “equivariant enumerative geometry”, which wields equivariant homotopy theory to explore enumerative questions in the presence of symmetry. Our main result is equivariant conservation of number, which states roughly that the orbits of solutions to an equivariant enumerative problem are conserved. We leverage this to compute the S4 orbits of the 27 lines on any smooth symmetric cubic surface.
VH 570Online Meeting Info: Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
Arthur Soulié , CNRS
I will describe a general construction of homological representations for families of groups, including classical braid groups, surface braid groups and mapping class groups. This recovers the well-known previous constructions, in particular those of Lawrence and Bigelow, and in this sense it unifies them. The construction is moreover “global” in the sense that, it defines functors on categories whose automorphism groups are the considered families of groups, and which also carries richer structures such as polynomiality. I will thus discuss polynomiality properties of these homological representation functors, and explain their applications, in particular for twisted homological stability. All this represents a joint work with Martin Palmer.
VH 570 or Zoom Online Meeting Info: Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
Abstract: We introduce a neat way of regarding categories as bimodule monoids. This facilitates the understanding of many constructions in category theory in terms of representation theory. In particular several versions of monoidal categories, such as PROPs, Feynman categories and unique factorization categories can be understood in this way. It also allows one to single out the special properties of operads and algebras over them. From the representation point of view, the notion of algebra or module is very natural. This lends itself to defining bar.and cobar constructions and gives a natural environment for Koszul duality and curvature.
This is joint work with my students Michael Monaco, and Michal Monaco and Yang Mo.
How does the topology of a configuration space of points in a space X behave as the number of points increases? McDuff and Segal proved if X is a manifold of dimension at least 2, then the unordered configuration space is homologically stable, i.e., after enough points are added the homology of the configuration space doesn’t change. While this isn’t the quite the case when X is a graph, An—Drummond-Cole—Knudsen showed that there is a reasonable notion of stability that arises from placing points along the edges of X. Building on their work, we prove that the homology groups of the ordered configuration space of a star graph are representation stable in the sense of Church—Ellenberg—Farb, and we use these results to find generators and relations for homology.
VH 570 - Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
We present a new approach to sheaf theory for data sets by constructing a Grothendieck topology associated to a Cech closure space. A particularly attractive aspect of this theory is that it applies to many of the major classes of interest to applications: directed and undirected graphs, finite simplicial complexes, and metric spaces decorated with a privileged scale, and on topological spaces, the resulting sheaf cohomology is isomorphic to the usual one. In this talk, we will introduce Cech closure spaces and discuss the construction and its basic properties.
VH 570 Online Meeting Info: Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
While the stable homotopy groups of spheres are quite complicated, they are also quite structured. In particular, they contain certain periodic families of elements which can be thought of as breaking up the stable homotopy groups into smaller pieces. Chromatic homotopy theory concerns a categorification of this decomposition, which can be implemented through Bousfield localizations of the category of spectra. There are two such families of localization functors, one with deep connections to geometry (localization at Morava E-theory and K-theory), and the other with excellent categorical properties (localization at a telescope of a type n complex). Ravenel's telescope conjecture asks whether these two families of localization functors are equivalent. In this talk, I will attempt to introduce this circle of ideas and hopefully get around to saying something about how algebraic K-theory allows you to produce specific counterexamples to the telescope conjecture.
VH 570 - Online Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
André Weil proposed a beautiful connection between algebraic topology and the number of solutions to equations over finite fields in a celebrated paper from 1948: the zeta function of a variety over a finite field is simultaneously a generating function for the number of solutions to its defining equations and a product of characteristic polynomials of endomorphisms of cohomology groups. The ranks of these cohomology groups are the Betti numbers of the associated complex manifold. We enrich the logarithmic derivative of the zeta function to a power series with coefficients in the Grothendieck--Witt group of stable isomorphism classes of unimodular bilinear forms, using traces of powers of Frobenius in A1-homotopy theory. Building off of work of Morel--Sawant and Bondarko, we construct a symmetric monoidal chain functor from smooth schemes to bounded complexes of homotopy modules. We show the quadratically enriched logarithmic zeta function to be connected to the Betti numbers of the associated real manifold under various restrictions. This is joint work in progress with Tom Bachmann and joint work with Margaret Bilu, Wei Ho, Padma Srinivasan and Isabel Vogt.Location: VH 570 - Online Meeting Info: Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
Allen Yuan
The Bhatt—Scholze theory of prismatic cohomology is a fundamental breakthrough in the study of p-adic cohomology theories in arithmetic geometry. Recently, Drinfeld and Bhatt—Lurie have introduced stacks which give natural categories of coefficients for prismatic cohomology. The goal of this talk will be to discuss these stacks from a homotopy theoretic perspective and explain the central role played by the equivariant complex cobordism spectrum. This is joint work with Devalapurkar, Hahn and Raksit.
VH 570 Online Meeting Info: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
The theory of resolvent degree draws from from Klein's "hypergalois" program and broader industries of reducing numbers of coefficients, having developed into an invariant measuring the complexity of algebraic functions, field extensions, groups, and moduli problems. We offer a concrete introduction to RD through focusing on finite groups, emphasizing the notion of versality paradigmatic of Klein's program. We also discuss the general framework implemented in joint work with Alexander Sutherland and Jesse Wolfson, building on those of Farb–Kisin–Wolfson and Duncan–Reichstein, that permits us to address resolvent questions via classical invariant theory in new ways. We will conclude by reflecting on the rich history of solving polynomials, along with what we do and don't know about resolvent degree.
VH 570 - Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
In this talk we leverage Henry’s recent connection between abstract homotopy theory and formal languages to show a result on the equivalence invariance of formal category theory. To build the bridge, we focus on equipments: a special kind of double categories that have shown be a powerful environment to express formal category theory. We build a model structure on the category of double categories and double functors whose fibrant objects are the equipments and combine this together with Makkai’s early approach to equivalence invariant statements in higher category theory via FOLDS (First Order Logic with Dependent Sorts).
VH 570 - Online Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
Anh Trong Nam Hoang , University of Minnesota Recording
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 conjecture. More recently, Ellenberg, Tran and Westerland proved the upper bound in Malle’s conjecture for function fields by studying stability of the homology of braid groups with certain exponential coefficients. In this talk, we will give an overview of their framework and extend their techniques to study other questions in arithmetic statistics. As an example, we will demonstrate how this extension can be used to study character sums of the resultant of monic square-free polynomials over finite fields, answering and generalizing a question of Ellenberg and Shusterman.
VH 570 Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
Søren Galatius ,University of Copenhagen
It is well known that the homeomorphism group of a disk relative to its boundary is contractible. This is known as the Alexander trick, and was published 100 years ago. I will discuss joint work with Randal-Williams on the homeomorphism group of a compact contractible manifold relative to its (not necessarily simply connected) boundary, which we prove to be contractible if the dimension is at least 6.
VH 570 Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
The quest we have started with Hisham Sati for physically motivated structures on certain loop spaces continues. In this talk I will describe our latest finds: the identification of the rational homotopy type (represented by a minimal model) of the k-fold free loop space divided by the k-torus T^k action, Map(T^k, S^4)//T^k and finding the E_k-symmetry of that minimal model, where E_k stands for the exceptional series of simple Lie algebras.Location: VH 570 Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
"Classically", Mackey and Tambara functors are equivariant generalizations of abelian groups and commutative rings, respectively. What this means is that, in equivariant homotopy theory, Mackey functors appear wherever one would expect to find abelian groups, and Tambara functors appear wherever one would expect to find commutative rings. More recently, work by Bachmann has garnered interest in related structures which appear in motivic homotopy theory -- these Motivic Mackey Functors and Motivic Tambara Functors do not have anything to do with group-equivariance, but have the same axiomatic. In this talk, I'll introduce a general context for interpreting the notions of Mackey and Tambara Functors, which subsumes both the equivariant and motivic notions. The aim of this approach is to translate theorems between contexts, enriching the theory and providing cleaner proofs of essential facts. To this end, I'll discuss recent progress in boosting a foundational result about norms from equivariant algebra to this more general context.
VH 570 Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
Classical resolvent problems (essential dimension, essential p-dimension, resolvent degree, . . .) ask some form of "How complex is . . . a polynomial, an enumerative problem, a branched cover, a variation of Hodge structure, . . .?" An idea going back to Arnold is that characteristic classes should be able to detect this intrinsic complexity. However, to make this work one must show that the relevant characteristic class remains nonzero under restriction to arbitrary Zariski open subvarieties. In this talk, we describe a new method for solving this restriction problem in many cases using prismatic cohomology. As an application, we prove a conjecture of Brosnan that for a complex abelian variety A, the essential p-dimension of the p-isogeny cover A\to A equals dim A for all but finitely many p. This is joint work with Benson Farb and Mark Kisin.
VH 570 Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
I will describe a categorical framework for homological algebra which simultaneously generalizes categories of R-modules (moreover exact categories) and the category of finite sets (moreover extensive categories). The homological theory of chain complexes of finite sets provides a very simple model of homological concepts equipped with visualization tools that may provide helpful insights for more general homological algebra. With a few minor adjustments, this framework also serves as a setting for generalized algebraic K-theory, extending several classical K-theory theorems to more combinatorial objects such as finite sets and algebraic varieties, resulting in a chain complex model for the K-theory of finite sets (and moreover any extensive category). Based on joint work with Maru Sarazola.Location: VH 570 Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
Andres Mejia , University of Pennsylvania
The Algebraic K-theory of a smooth manifold is a receptacle for many sensitive invariants. The driving example is the classical fact that the H-cobordism type of a manifold is completely controlled by only the fundamental group of its Algebraic K-Theory space. In fact, there is a reduction to a related invariant that only depends on the fundamental group of the manifold M. Turning to higher invariants, we are not so lucky, but there is still a comparison map called the linearization map that lets us compute parts of the Algebraic K-theory space in good situations. This talk will discuss a new construction of the linearization map when we are presented with a manifold together with the action of a finite group. If time permits, we will also discuss future directions with a view towards equivariant stable cobordism. These results are joint with D. Chan and M. Calle.
VH 570 Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
Viktor Burghardt , University of Michigan
Let C be a stable $\infty$-category. A Poincaré structure on C is a contravariant quadratic functor out of C into spectra, which satisfies a non-degeneracy condition and provides a duality $C^{op}\xrightarrow{\simeq} C$. When applied to module categories of rings, among other things, this encodes involutions of the base ring. In this talk we want to venture into the world of Poincaré $\infty$-categories, which Is due to Lurie, and explore what it can say about Brauer groups. This is joint work with Ben Antieau, Noah Riggenbach and Lucy Yang.
VH 570 Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
The free loop space of a topological space has a canonical circle action given by rotating loops, making it an S^1-space. The work of Jones, Goodwillie, and others, relates the equivariant homology of the free loop space to the cyclic homology of the algebra of singular chains on the topological monoid of based loops. One can model the free loop space in terms of the chains on the underlying space considered as a categorical coalgebra, a notion Koszul dual to a non-negatively graded dg category. This construction is "as small as possible", has no hypotheses on the underlying space and is suitable for computations in (non-simply connected) string topology. I will present a cyclic theory for categorical coalgebras and dg-categories extending the theory of cyclic homology for (dg) algebras and coalgebras. In particular, the cyclic chains of the categorical coalgebra of chains on a simplicial set provides a model for the S^1-equivariant chains on the free loop space that is suitable for computations. The proofs of these results can be understood in terms of a combinatorial model for the unit of the Bar-Cobar adjunction.
VH 570 Online Meeting Info: Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
Rune Haugseng, Norwegian University of Science and Technology
Strict n-categories can be defined inductively as categories enriched in strict (n-1)-categories, starting with sets when n=0. Similarly, $(\infty,n)$-categories can be described as $\infty$-categories enriched in $(\infty,n-1)$-categories, starting with spaces when n=0. I will describe an extension of this to more general enrichments: if V is an $E_n$-monoidal $\infty$-category, then V-enriched $(\infty,n)$-categories can be defined both by iterated enrichment starting with V and as a V-enriched version of Barwick’s n-fold Segal spaces. Time permitting, I will attempt to explain how this follows quite formally from $(infty,2)$-categorical manipulations of fibrations, lax transformations and Gray tensor products.
VH 570Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
I will discuss recent joint work with Tom Bachmann and Mike Hopkins wherein we establish a version of the Freudenthal suspension theorem in motivic homotopy theory. Along the way, I will try to give a gentle introduction to motivic homotopy theory and explain why one might be interested in establishing such a result.Location: VH 570 Online Meeting Info: Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
In recent years, the connection between topology, homological algebra, and function field arithmetic has become increasingly apparent. Works by Ellenberg-Tran-Westerland and Kapranov-Schechtman have established a connection between arithmetic statistics of covers of the affine line and the cohomology of certain Hopf algebras called Nichols algebras. In this talk, I will discuss recent work that makes this connection very precise in the case of covers of the affine line of degrees 3, 4, and 5 and suggests what this connection might look like for other kinds of covers. The key ingredient is the construction of certain smooth Hurwitz space compactifications inspired by the work of Anand Deopurkar. Using the geometry of these compactifications, I will explain how the study of arithmetic statistics over local fields, such as Igusa's computations of his local zeta functions, can surprisingly be used to obtain global arithmetic statistics over the affine line and new computations of the cohomology of special Nichols algebras.
VH 570 Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
We describe a new homological stability result for a generalized version of Hurwitz spaces. This builds on previous work of Ellenberg-Venkatesh-
VH 570 Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327
I will talk about a series X_k, k ≥ 0, of topological spaces whose rational homotopy type admits an action of the Lie group of exceptional type E_k. These rational homotopy types govern supergravity in (11-k)-dimensional spacetime. This extends Mysterious Duality of Iqbal, Neitzke, and Vafa (2001) as a mysterious connection between M-theories in various dimensions and del Pezzo surfaces to a triality which adds a connection to rational homotopy theory. The connection between physics and topology in our work is not mysterious but rather explicit, and if a conjectural connection between topology and algebraic geometry is clarified, it will unveil the whole mystery of the triality at once. This is based on published [arXiv:2111.14810, arXiv: 2212.13968] and ongoing work with Craig’s and mine friend Hisham Sati.
Vincent Hall 570 or Zoom: https://umn.zoom.us/j/95323153681?pwd=SVFzbVVwTnJnN0l0VVJJdzJ5dmVFUT09 - Meeting ID: 953 2315 3681Passcode: uWgLL9
In this talk I will discuss what is and isn't known about the homology of 3-manifolds and their finite covers. In particular, we will consider how homology can grow in towers of finite covers.
VinH 570 or via Zoom - Meeting ID: 953 2315 3681 - Passcode: uWgLL9
A quandle is an algebraic structure satisfying laws like the conjugation operation in a group. In this talk I'll discuss how a knot naturally has an associated quandle, and a homotopy-theoretic interpretation of where this structure comes from. I'll also discuss an analogue of a theorem of Milnor, identifying the homotopy type of the free quandle on a topological space, and how this impacts the study of "cohomology for quandles". This is joint work with Markus Szymik.Location: In person in VinH 570, broadcast on Zoom: https://umn.zoom.us/j/95323153681?pwd=SVFzbVVwTnJnN0l0VVJJdzJ5dmVFUT09
The moduli spaces of smooth projective curves with marked points have cohomology that attaches characteristic classes to surface bundles with disjoint sections. As such, this cohomology is of fundamental importance in algebraic geometry and topology. However, only a tiny fraction of the cohomology is understood. I will present joint works with Bibby, Chan and Yun, and with Hainaut, in which we gain access to the least algebraic part of the cohomology for curves of genus 2, using tropical geometry and configuration spaces on graphs. In particular we find the first examples of families of cohomology classes in the top cohomological dimension, which seem to tell a geometric story that is yet to be understood.
VinH 570, broadcast on Zoom
After introducing several ways to think about the cohomology of the moduli space of curves, I will discuss a recent theorem of mine saying that in a stable range, the rational cohomology of the moduli space of curves with level structures is the same as that of the ordinary moduli space of curves: a polynomial algebra in the Miller-Morita-Mumford classes.
VinH 570 or via Zoom : Meeting ID: 953 2315 3681 - Passcode: uWgLL9
In this talk I will discuss the topological structure of the space Hom(Z^n,G) of commuting n-tuples in G, where G is a unitary group or general linear group over C. I will explain how the equivariant homology of Hom(Z^n,G) is related to Hochschild homology, and how this can be used in homology calculations. In particular, I will explain how to obtain the rational homology, homology stability, and some results on p-torsion.
VinH 570, broadcast on Zoom: https://umn.zoom.us/j/95323153681?pwd=SVFzbVVwTnJnN0l0VVJJdzJ5dmVFUT09
The homotopy groups of topological modular forms are very interesting and most computations need a lot of nontrivial topology information. In this talk, we present two new approaches of the 2-primary computation based on new techniques from equivariant homotopy theory and motivic homotopy theory respectively. The new approaches use more algebraic input and provide new information. In particular, the equivariant approach avoids the use of Toda brackets. The motivic approach settles a sign in the multiplicative structure, which is the last unresolved detail about the multiplicative structure in Bruner and Rognes' book. This talk is based on joint projects with Zhipeng Duan, Dan Isaksen, Hana Jia Kong, Yunze Lu, Yangyang Ruan, Guozhen Wang, and Heyi Zhu.
VinH 570, broadcast on Zoom: https://umn.zoom.us/j/95323153681?pwd=SVFzbVVwTnJnN0l0VVJJdzJ5dmVFUT09
Hochschild homology of a ring has a topological analogue for ring spectra, topological Hochschild homology (THH), which plays an essential role in the trace method approach to algebraic K-theory. For ring spectra with an anti-involution there is a theory of Real topological Hochschild homology (THR), which is an O(2)-equivariant spectrum that receives a trace map from Real algebraic K-theory. In this talk, I will introduce Real topological Hochschild homology and discuss how it can be characterized as an equivariant norm. This informs a new definition of Real Hochschild homology of rings with anti-involution, which is the algebraic analogue of Real topological Hochschild homology. This is joint work with Gabriel Angelini-Knoll and Mike Hill.
VH 570
David Favero (University of Minnesota) Recording
Given a topological space stratified by a poset, we can study "entrance paths'' which flow in the direction of the poset. These paths form a type of directed fundamental groupoid (which is no longer a groupoid). Choosing base points, we get a non-commutative algebra which is an analog of the fundamental group. I will make some basic observations on the global dimension of this algebra. Surprisingly, through a form of homological mirror symmetry, these observations imply a conjecture of Orlov about the homological complexity of the derived category of coherent sheaves for all toric varieties. This is based on joint work with Jesse Huang ( https://arxiv.org/abs/2302.09158 ).
In person - VinH 570, broadcast on Zoom: https://umn.zoom.us/j/95323153681?pwd=SVFzbVVwTnJnN0l0VVJJdzJ5dmVFUT09
The classical Thomason’s theorem says that a homotopy colimit of a diagram of categories (considered as spaces by their nerves) is captured by the (classifying space of) the Grothendieck construction of the diagram. We extend this classical result to higher categories namely we show that the homotopy colimit of a diagram of (marked) quasi-categories is captured by the (marked) Lurie's Grothendieck construction of the diagram. Moreover our result implies a stronger version of the classical Thomason’s theorem: A homotopy colimit of a diagram of categories (considered as marked quasi-categories) is captured by its Grothendieck construction (also considered as a marked quasi-category). In order to prove our result we revisit Lurie’s theory of straightening and unstraightening which relates coCartesian fibrations and diagrams of higher categories. Our approach not only simplifies the aforementioned theory but constructs a new (Quillen) pair of straightening and unstraightening functors which go in directions opposite to that of Lurie’s functors. Our unstraightening functor is a (marked version of) a classical homotopy colimit functor for diagram of spaces and therefore, unlike Lurie’s unstraightening functor, is a left adjoint. Finally, our result implies that Lurie’s Grothendieck construction of a diagram of quasi-categories is it’s opLax colimit. This result was first proved by Gepner, Haugseng, and Nikolaus.
VinH 570 or via Zoom : Meeting ID: 953 2315 3681 - Passcode: uWgLL9
Non-equivariantly, the dual Steenrod algebra spectrum is a wedge of suspensions of HZ/p. I will talk about the computation of the equivariant dual Steenrod algebra for G = C_p, the cyclic group of order p. It turns out that when p is odd, the dual Steenrod algebra spectrum is a wedge of suspensions of HZ/p and another spectrum, which we call HT. I will take about how to obtain the generators of these summands. This is joint work with Po Hu, Igor Kriz, and Petr Somberg.
VinH 570, broadcast on Zoom: https://umn.zoom.us/j/95323153681?pwd=SVFzbVVwTnJnN0l0VVJJdzJ5dmVFUT09
Joint work with Madeline Brandt and Siddarth Kannan. We use moduli spaces of G-admissible covers and tropical geometry to give a sum-over-graphs formula for the weight-0 compactly supported Euler characteristic of the moduli spaces H_{g,n} of n-marked hyperelliptic curves of genus g, as a virtual representation of S_n. Computer calculations then enable fully explicit formulas for the above in small genus. My aim is to make this talk accessible to anyone with passing familiarity with M_g and its Deligne-Mumford compactification, which I will also review.
Vincent Hall 570 - Zoom Meeting
John Milnor's attempt at the higher algebraic K-groups of a field uses the field units as generators and the Steinberg relation as the only relation. The resulting Milnor K-groups map to the higher algebraic K-groups Daniel Quillen defined slightly later as homotopy groups of a certain topological space, and the map is an isomorphism in degrees 0, 1, and 2. A decade later Andrei Suslin constructed a Hurewicz-type homomorphism from Quillen's algebraic K-groups to the Milnor K-groups of a field and proved that the resulting endomorphism on the n-th Milnor K-group is multiplication by (n-1)! if n>0. He conjectured that the image of the Hurewicz-type homomorphism is the same as the image of this endomorphism (hence as small as possible) and proved the degree 3 case. Recently Aravind Asok, Jean Fasel, and Ben Williams ingeniously used A1-homotopy groups of algebraic spheres to settle the degree 5 case. Also using A1-homotopy groups, but now for the projective plane, which almost coincides with the third special linear group, I will explain how to treat the fourth degree.
VH 570 - Zoom
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