Topology seminar

Talks in Fall 2019

• Dec 02 '19

  • Weiyan Chen (University of Minnesota)

  • Topology and Arithmetic Statistics

  Topology studies the shape of spaces. Arithmetic statistics studies the behavior of random algebraic objects such as integers and polynomials. I will talk about a circle of ideas connecting these two seemingly unrelated areas. To illuminate the connection, I will focus on three concrete examples: (1) the Burau representation of the braid groups, (2) analytic number theory for effective 0-cycles on a variety, and (3) cohomology of the space of multivariate irreducible polynomials. These projects are parts of a broader research program, with numerous contributions by topologists, algebraic geometers, and number theorists in the past decade, and lead to many future directions yet to be explored. (PS. This will be a rehearsal of a job talk accessible to the general audience. Any comments or suggestions are appreciated.)

• Nov 11 '19

  • Tyler Lawson (University of Minnesota)

  • Cochain models for the unit group of a differential graded algebra

  Abstract not available

• Oct 28 '19

  • Elden Elmanto (Harvard University)

  • Compactifying the étale topos

  The speaker has long feared the technicalities and intricacies of equivariant stable homotopy theory. Fortunately, beginning with the work of Glasman, major simplification on the foundations of the subject has been made (cf. the work of Ayala-Mazel-Gee-Rozenblyum, Nikolaus-Scholze and the Barwick school). We offer another perspective (that the speaker has a chance of understanding) on equivariant stable homotopy theory, at least for the group C_2, via algebraic geometry. We view it as a way to remedy an infamous annoyance: the 2-étale cohomological dimension of the field of real numbers is infinite. We do this by identifying the genuine C_2-spectra with a category of motives based on Real algebraic geometry ala Scheiderer. This is joint work with Jay Shah.

• Oct 14 '19

  • Craig Westerland (University of Minnesota)

  • Second order terms in arithmetic statistics

  The machinery of the Weil conjectures often allows us to relate the singular cohomology of the complex points of a scheme to the cardinality of its set of points over a finite field. When we apply these methods to a moduli scheme, we obtain an enumeration of the objects the moduli parameterizes. It's rare that we can actually fully compute the cohomology of these moduli spaces, but homological stability results often give a first order approximation to the homology.

  In this talk, we'll explain how to obtain second order homological computations for a class of Hurwitz moduli spaces of branched covers; these yield second order terms in enumerating the moduli over finite fields. We may interpret these as second order terms in a function field analogue of the function which counts number fields, ordered by discriminant. Our second order terms match those of Taniguchi-Thorne/Bhargava-Shankar-Tsimerman in the cubic case, and give new predictions in other Galois settings.

  This is joint (and ongoing) work with Berglund, Michel, and Tran.

• Oct 21 '19

  • Liam Keenan (University of Minnesota)

  • Descent properties of topological Hochschild homology

  Algebraic K-theory is an extremely rich but notoriously difficult invariant to compute. In order to make calculations tractible, topological Hochschild homology and topological cyclic homology were introduced, along with the Dennis and cyclotomic trace maps. A natural question to consider is whether or not these invariants are sheaves for various topologies arising in algebraic geometry. In fact, it turns out that topological Hochschild homology is a sheaf for the fpqc topology on connective commutative ring spectra. In this talk, I plan to introduce the language necessary and sketch the argument of this result.

• Sep 30 '19

  • Sasha Voronov (University of Minnesota)

  • Mysterious Duality

  “Mysterious Duality” was discovered by Iqbal, Neitzke, and Vafa in 2001. They noticed that toroidal compactifications of M-theory lead to the same series of combinatorial objects as del Pezzo surfaces do, along with numerous mysterious coincidences: both toroidal compactifications and del Pezzo surfaces give rise to the exceptional series E_k; the U-duality group corresponds to the Weyl group W(E_k), arising also as a group of automorphisms of the del Pezzo surface; a collection of various M- and D-branes corresponds to a set of divisors; the brane tension is related to the “area” of the corresponding divisor, etc. The mystery is that it is not at all clear where this duality comes from. In the talk, I will present another series of mathematical objects: certain versions of multiple loop spaces of the sphere S^4, which are, on the one hand, directly connected to M-theory and its combinatorics, and, on the hand, possess the same combinatorics as the del Pezzo surfaces. This is a report on an ongoing work with Hisham Sati.

• Sep 23 '19

  • Dongkwan Kim (University of Minnesota)

  • Homology class of Deligne-Lusztig varieties

  Since first defined by Deligne and Lusztig, a Deligne-Lusztig variety has become unavoidable when studying the representation theory of finite groups of Lie type. This is a certain subvariety of the flag variety of the corresponding reductive group, and its cohomology groups are naturally endowed with the action such finite groups, which in turn gives a decomposition of irreducible representations called Lusztig series. In this talk, I will briefly discuss the background of Deligne-Lusztig theory and provide a formula to calculate the homology class of Deligne-Lusztig varieties in the Chow group of the flag variety. If time permits, I will also discuss their analogues.

• Sep 16 '19

  • Peter Webb (University of Minnesota)

  • Transfer in the homology and cohomology of categories

  The cohomology of a category has many properties that extend those that are familiar when the category is a group. Second cohomology classifies equivalence classes of category extensions, first cohomology parametrizes conjugacy classes of splittings, first homology is the abelianization of the fundamental group, and second homology has a theory that extends that of the Schur multiplier. Defining restriction and corestriction maps on the homology of categories is problematic: most attempts to do this require induction and restriction functors to be adjoint on both sides, and this typically does not happen with categories. We describe an approach to defining these maps that includes all the situations where they can be defined in group cohomology, at least when the coefficient ring is a field. The approach uses bisets for categories, the construction by Bouc and Keller of a map on Hochschild homology associated to a bimodule, and the realization by Xu of category cohomology as a summand of Hochschild cohomology.

• Sep 09 '19

  • Weiyan Chen (University of Minnesota)

  • Cohomology of the space of complex irreducible polynomials in several variables

  We will show that the space of complex irreducible polynomials of degree d in n variables satisfies two forms of homological stability: first, its cohomology stabilizes as d increases, and second, its compactly supported cohomology stabilizes as n increases. Our topological results are inspired by counting results over finite fields due to Carlitz and Hyde.

Talks in Spring 2018

• Apr 23 '18

  • Nick Proudfoot (University of Oregon)

  • The Orlik-Terao algebra and the cohomology of configuration spaces

  The Orlik-Terao algebra is the subalgebra of all rational functions in n variables generated by 1/(x_i - x_j). I will explain how to use topological techniques to understand this algebra as a graded representation of the symmetric group. I will also describe two different connections (one proven and one conjectural) between this algebra and the cohomology of configuration spaces.

• May 01 '18

  • Ben Knudsen (Harvard University)

  • Edge stabilization in the homology of graph braid groups

  We discuss a novel type of stabilization map on the configuration spaces of a graph, which increases the number of particles occupying an edge. Through these maps, the homology of the configuration spaces forms a module over the polynomial ring generated by the edges of the graph, and we show that this module is finitely generated, implying eventual polynomial growth of Betti numbers over any field. Moreover, the action lifts to an action at the level of singular chains, which contains strictly more information; indeed, we show that this differential graded module is almost never formal over the ring of edges. These results, along with numerous calculations, arise from consideration of an explicit chain complex, which is a structured enhancement of a cellular model first considered by Swiatkowski. We arrive at this model through a local-to-global approach combining ideas from factorization homology and discrete Morse theory. This is joint work with Byung Hee An and Gabriel Drummond-Cole.

• Apr 03 '18

  • Dylan Wilson (University of Chicago)

  • HF_p is an equivariant Thom spectrum

  Mahowald (and later many others) proved that HF_2 is a Thom spectrum for a bundle over the double loop space on the 3-sphere. This has had lots of applications, including (more recently) some nilpotence results for E_2-algebras and a calculation of THH of F_2. In this talk, we explain how to extend the result to the case of equivariant spectra over a cyclic group of prime power order. The case of HF_2 over C_2 is joint with Mark Behrens, and the general case is joint with Jeremy Hahn. If there's time, we'll discuss how this fits into a program of Hill-Hopkins-Ravenel for solving the 3-primary Kervaire invariant problem.

• Mar 27 '18

  • Jonathan Campbell (Vanderbilt University)

  • Topological Hochschild Homology and Characteristics

  In this talk I'll review the definition of duality in categories and bicategories and how certain functors called "shadows", due to Kate Ponto, can be used to extract Euler characteristic-type invariants from this data. It turns out that topological Hochschild homology (which I'll define) is an example of such a shadow, and this can be used to relate classical invariants from fixed point theory (e.g. the Reidemeister trace) with the image of the cyclotomic trace in K-theory. Time permitting, I'll sketch how anything that one calls a "characteristic" should fit into this story. This is joint work with Kate Ponto.

• Jan 30 '18

  • Weiyan Chen (University of Minnesota, Twin Cities)

  • Topology of enumerative problems: inflection points on cubic curves

  We will focus on one concrete example of a broader program to study certain topological questions arising from classical enumerative problems in algebraic geometry. For example, every smooth cubic plane curve naturally has 9 marked points on it, namely its 9 inflection points. We ask: are there other ways to choose n distinct points on a smooth cubic curve as the curve varies continuously in family? We will show that the 9 inflection points are almost universal among all those choices. The key is to understand the topology of complement of certain discriminant variety.

• Feb 06 '18

  • Kathryn Hess (École polytechnique fédérale de Lausanne)

  • Topological vistas in neuroscience (special colloquium)

  I will describe results obtained in collaboration with the Blue Brain Project on the topological analysis of the structure and function of digitally reconstructed microcircuits of neurons in the rat cortex and outline our on-going work on topology and synaptic plasticity. The talk will include an overview of the Blue Brain Project and a brief introduction to the topological tools that we use. If time allows, I will also briefly sketch other collaborations with neuroscientists in which my group is involved.

• Jan 23 '18

  • Sasha Voronov (University of Minnesota, Twin Cities)

  • Quantum Deformation Theory

  Classical deformation theory is based on the Classical Master Equation (CME), a.k.a. the Maurer-Cartan Equation: dS + 1/2 [S,S] = 0. Physicists have been using a quantized CME, called the Quantum Master Equation (QME), a.k.a. the Batalin-Vilkovisky (BV) Master Equation: dS + h \Delta S + 1/2 {S,S} = 0. The CME is defined in a differential graded (dg) Lie algebra, whereas the QME is defined in a space V[[h]] of formal power series or V((h)) of formal Laurent series with values in a dg BV algebra V. One can anticipate a generalization of classical deformation theory arising from the QME, quantum deformation theory. Quantum deformation functor and its representability will be discussed in the talk.

Talks in Fall 2017

• Dec 15 '17

  • Kirsten Wickelgren (Georgia Tech)

  • Motivic Euler numbers and an arithmetic count of the lines on a cubic surface

  A celebrated 19th century result of Cayley and Salmon is that a smooth cubic surface over the complex numbers contains exactly 27 lines. Over the real numbers, it is a lovely observation of Benedetti_Silhol, Finashin–Kharlamov and Okonek–Teleman that while the number of real lines depends on the surface, a certain signed count of lines is always 3. We extend this count to an arbitrary field k using an Euler number in A1-homotopy theory. The resulting count is valued in the Grothendieck-Witt group of non-degenerate symmetric bilinear forms. This is joint work with Jesse Kass.

• Dec 08 '17

  • Jesse Wolfson (UC Irvine)

  • The Theory of Resolvent Degree - After Hamilton, Hilbert, Segre and Brauer

  Resolvent degree is an invariant of a branched cover which quantifies how "hard" is it to specify a point in the cover given a point under it in the base. Historically, this was applied to the branched cover P^n/S_{n-1} --> P^n/S_n, from the moduli of degree n polynomials with a specified root to the moduli of degree n polynomials. Classical enumerative problems and congruence subgroups provide two additional sources of branched covers to which this invariant applies. In ongoing joint work with Benson Farb, we develop the theory of resolvent degree as an extension of Buhler and Reichstein's theory of essential dimension. We apply this theory to systematize an array of classical results and to relate the complexity of seemingly different problems such as finding roots of polynomials, lines on cubic surfaces, and level structures on intermediate Jacobians.

• Dec 01 '17

  • Megan Maguire (University of Wisconsin -- Madison)

  • Cohomology algebras of configuration spaces

  For a manifold X with finite-dimensional cohomology, we know that the cohomology algebra of each unordered configuration space of X is finitely generated, but can we say something stronger about its generators? More precisely, does there exist a D (depending only on X) so that the cohomology algebra of each unordered configuration space of X can be generated in degree at most D? We will answer this question in some cases.

• Nov 17 '17

  • Tasos Moulinos (University of Illinois, Chicago)

  • Derived Azumaya algebras and twisted K-theory

  Topological K-theory of dg-categories is a localizing invariant of dg-categories over C taking values in the ∞-category of KU-modules. In this talk I describe a relative version of this construction; namely for X a quasi-compact, quasi-separated C-scheme I construct a functor valued in ShvSp(X(C)), the ∞-category of sheaves of spectra on X(C). For inputs of the form Perf(X,A) where A is an Azumaya algebra over X, I characterize the values of this functor in terms of the twisted topological K-theory of X(C). From this I deduce a certain decomposition, for X a finite CW-complex equipped with a bundle of projective spaces π : P → X, of KU(P) in terms of the twisted topological K-theory of X ; this is a topological analogue of a result of Quillen’s on the algebraic K-theory of Severi-Brauer schemes.

• Nov 10 '17

  • Sasha Voronov (University of Minnesota, Twin Cities)

  • Homotopy Lie algebroids and bialgebroids

  Lie algebroids appear throughout geometry and mathematical physics and realize the idea of a family of Lie algebras parameterized by a smooth manifold. A well-known result of A. Vaintrob characterizes Lie algebroids and their morphisms in terms of homological vector fields on supermanifolds, which might be regarded as a version of derived geometry. This leads naturally to the notion of an L_infty-algebroid, which offers a way to think of a sheaf of L_infty-algebras over a smooth manifold. The situation with Lie bialgebroids and their morphisms is more complicated, as they combine covariant and contravriant features. We approach them in terms of odd symplectic dg-manifolds, building on the work of D. Roytenberg. We extend them to the homotopy Lie case and introduce the notions of an L_infty-bialgebroid and an L_infty-morphism between them. This is a joint work with Denis Bashkirov.

• Nov 03 '17

  • Jennifer Wilson (Stanford University)

  • Stability in the second homology of Torelli groups

  In this talk I will describe stability results for the Torelli subgroups of mapping class groups of surfaces. Specifically, I will explain how to use tools from representation theory to establish patterns their second homology. These "representation stability" results are an application of advances in a general algebraic framework for studying sequences of group representations. This project is joint with Jeremy Miller and Peter Patzt.

• Oct 27 '17

  • David Roberts (University of Minnesota, Morris)

  • Monodromy for a large class of Hurwitz-Belyi maps

  Hurwitz varieties covering configuration varieties are important objects in mathematics, with monodromy described via permutation representations of braid groups. In the case of the curves, the maps can be normalized to be Belyi maps, meaning that the base becomes just C-{0,1} and monodromy can be described by giving two permutations. The talk will present a uniform formula explicitly giving the monodromy of a large class of these Hurwitz-Belyi maps simultaneously. Many pictorial examples will be given, illustrating both the monodromy formula itself and related phenomena.

• Oct 06 '17

  • Tyler Lawson (University of Minnesota, Twin Cities)

  • Topology Seminar

  TBA

• Sep 29 '17

  • Craig Westerland (University of Minnesota, Twin Cities)

  • Structure theorems for braided Hopf algebras

  The Milnor-Moore and Poincaré-Birkhoff-Witt theorems are fundamental in studying the structure of Hopf algebras, particularly in characteristic zero. Central to these results is the fact that the space of primitives in a Hopf algebra is a Lie algebra. Part of the classification of pointed Hopf algebras involves a notion of ``braided Hopf algebras;” the primitives in these objects no longer form a Lie algebra.

  I will present work in progress which will establish analogues of the Poincaré-Birkhoff-Witt and Milnor-Moore theorems in this setting. The main new tool is a braided analogue of the notion of a Lie algebra defined in terms of braided operads. This can be used to establish these results, and also presents an unexpected connection to profinite braid groups and related operads.

• Sep 22 '17

  • Saul Glasman (University of Minnesota, Twin Cities)

  • Symmetric power structures on algebraic K-theory

  This is a talk about the lambda-operations and their relatives on algebraic K-theory, which we argue should be thought of as analogous to power operations in a homology theory. Classically constructed as operations on homotopy groups, we extend the lambda-operations to the space level by using the surprising observation that polynomial functors between additive categories give maps of K-theory spaces. Subsequently, we use the language of Lawvere theories to define and study a category of spectral lambda-rings. We end with some speculations concerning how these structures might relate to the trace map. This is joint work in progress with Barwick, Mathew and Nikolaus.

• Sep 15 '17

  • Weiyan Chen (University of Minnesota, Twin Cities)

  • Braids, algebraic curves, and 0-cycles

  We will see how ideas from topology can provide different perspectives on problems in arithmetic via the bridge of étale cohomology. I will present two concrete examples: (1) how braid groups and Burau representations (topology) could help one count points on algebraic curves over finite fields (arithmetic), and (2) how topology would lead to generalizations of the well-known analogy between integers and polynomials, and results about "analytic number theory for effective 0-cycles on varieties over finite fields".

• Dec 11 '17

  • Mike Hill (UCLA)

  • Evenness in equivariant topology

  Motivated by work of Wilson, I will describe a version of ``evenness'' in equivariant homotopy theory. There are dual notions of even cells and even homotopy groups, and one can also ask for things which have both properties. I'll describe some of the ongoing work in this area, all of which is joint with Hopkins.

• Sep 18 '17

  • Sam Nariman (Northwestern University)

  • Friedlander-Milnor's problem for diffeomorphism groups

  Let G be a finite dimensional Lie group and G^delta be the same group with the discrete topology. The natural homomorphism from G^delta to G induces a continuous map from BG^delta to BG. Milnor conjectured that this map induces a p-adic equivalence. In this talk, we discuss the same map for infinite dimensional Lie groups, in particular for diffeomorphism groups and symplectomorphisms. In these cases, we show that the map from BG^delta to BG induces split surjection on cohomology with finite coefficients in "the stable range". If time permits, I will discuss applications of these results in foliation theory, in particular, flat surface bundles.

Talks in Fall 2016

• Dec 05 '16

  • Nitu Kitchloo (Johns Hopkins University)

  • Quantization of the Modular Functor and Equivariant Elliptic Cohomology

  Let M be a compact G-space for a compact Lie group G. I will describe a procedure that can be seen as the categorical quantization of the category of parametrized positive energy representations of the loop group of G. This procedure is described in terms of dominant K-theory of the loop group parametrized over M. More concretely, I will construct a holomorphic sheaf over a universal elliptic curve with values in dominant K-theory of the loop space LM, and show that each stalk of this sheaf is a cohomological functor of M (thereby giving rise to an equivariant cohomology theory). I will also give compelling evidence that this theory is equivalent to equivariant elliptic cohomology of M as constructed by Grojnowski. I will assume very little background, and give a lot of motivation, but some general ideas of what a field theory is may be helpful.

• Nov 14 '16

  • Agnès Beaudry (University of Colorado)

  • The K(2)-local Picard group at p=2

  The K(n)-local categories provide examples of interesting Picard groups. Their importance in chromatic homotopy theory is highlighted by the fact that the dualizing object for Brown-Commenetz duality comes from an invertible element. These groups have been computed at all primes when n=1 and all odd primes when n=2. Mahowald predicted that the torsion in the K(2)-local Picard group at the prime 2 would be very large compared to the torsion in the K(2)-local Picard groups at odd primes. In this talk, I will confirm this and explain our current understanding of the structure of this group.

• Nov 07 '16

  • Tyler Lawson (UMN)

  • Khovanov homotopy

  The Jones polynomial is an invariant of a knot or link, and Khovanov homology is a categorical lift so that the coefficients of the Jones polynomial are ranks of Khovanov homology groups. I'll discuss a refinement of this to a homotopy type due to Lipshitz-Sarkar and Hu-Kriz-Kriz, and discuss ongoing work to refine this to an invariant for tangles.

• Oct 31 '16

  • Nikita Markarian (Dept Math, Higher School of Economics, Moscow)

  • Weyl n-algebras and the Swiss cheese operad

  I will discuss the definition and applications of Weyl n-algebras and connected algebras over the Swiss cheese operad.

• Oct 24 '16

  • Aaron Royer (UCLA)

  • Affinization, Witt vectors and duality

  The Hochschild--Kostant--Rosenberg theorem relates the Hochschild homology and Andre--Quillen homology of smooth commutative algebras. Using ideas from derived algebraic geometry, Ben-Zvi--Nadler proved an HKR theorem for rational E_\infty-rings sans smoothness hypotheses. Their argument ultimately relies on an equivalence between the algebra of rational cochains on the circle and a trivial square-zero extension. We will explain how the situation is more complicated away from characteristic zero, and why this is actually a good thing.

• Oct 17 '16

  • Jack Morava (Johns Hopkins University)

  • Topological Hochschild homology of (some) perfectoid fields

  Beside its familiar topology, the field of complex numbers has complete p-adic topologies for all integral primes. Unlike the Archimedean topology, which has only the real numbers as a complete topological subfield, the p-adic topologies have many interesting (complete, totally ramified) subfields: in particular, the completions of maximal totally ramified extensions of locally compact number fields (such as the p-cyclotomic field obtained by adjoining all p-power roots of unity to the p-adic rationals).

  Recent work of Scholze, Bhatt and others has transformed the study of algebraic geometry over such fields, while similarly recent work of Hesselholt and Madsen has transformed our understanding of their algebraic topology. This talk will focus on connections between Lubin-Tate (ie generalized cyclotomic field) theory and topological Hochschild homology, which seems to be a powerful tool for understanding perfectoid geometry.

• Oct 10 '16

  • Saul Glasman (UMN)

  • Goodwillie calculus and Mackey functors

  This talk concerns a theorem stating that the theory of n-excisive functors from the category of spectra to a target stable oo-category E is equivalent to the theory of Mackey functors on a suitable indexing category - the category of finite sets of cardinality at most n with morphisms surjections - which has many of the salient properties of the orbit category of a finite group. We'll give motivation for this theorem, and give a broad sketch of the proof, which decomposes readily into a sequence of independent assertions. We'll then prove as many of these assertions as time allows.

• Oct 03 '16

  • Saul Glasman (UMN)

  • An overview of equivariant stable homotopy theory and stable functor calculus

  This will be an almost totally expository introduction to certain topics in stable homotopy theory. I'll attempt to motivate and introduce Elmendorf's theorem on G-spaces and the spectral Mackey functor approach to G-spectra. I'll then discuss Goodwillie's functor calculus for functors from spectra to spectra, which answers the question of what it means for such a functor to be "polynomial". If time permits, I'll start exploring the analogies between these two theories, a task which will be taken to its conclusion in next week's talk.

• Sep 26 '16

  • Akhil Mathew (Harvard University)

  • Artin induction for ring spectra and algebraic K-theory

  A theorem of Mitchell states that the chromatic complexity of the algebraic K-theory spectrum of a discrete ring R is bounded by one, i.e., the Morava K-theory vanishes at heights at least two. We give an approach for bounding the chromatic complexity in the algebraic K-theory of ring spectra.

  Let R be a ring spectrum. We say that R-based Artin induction holds for a family of groups if for every finite group G, the rationalized Grothendieck group of the category of perfect R-modules with G-action is induced from the given family. We show that Artin induction theorems can be used to bound chromatic complexity and give several examples in ring spectra, in line with the redshift philosophy of Rognes. This is joint work with Dustin Clausen, Niko Naumann, and Justin Noel.

Talks in Spring 2016

• Jan 25 '16

  • Craig Westerland (University of Minnesota)

  • Iterated algebraic K-theory and T-duality

  We introduce a periodic form of the iterated algebraic K-theory of ku, the (connective) complex K-theory spectrum, as well as a natural twisting of this cohomology theory by higher gerbes. Furthermore, we prove a form of topological T-duality for sphere bundles oriented with respect to this theory. This is joint work with John Lind and Hisham Sati.

• Feb 01 '16

  • Craig Westerland (University of Minnesota)

  • Iterated algebraic K-theory and T-duality, II

  We introduce a periodic form of the iterated algebraic K-theory of ku, the (connective) complex K-theory spectrum, as well as a natural twisting of this cohomology theory by higher gerbes. Furthermore, we prove a form of topological T-duality for sphere bundles oriented with respect to this theory. This is joint work with John Lind and Hisham Sati. I will continue from last week, and discuss T-duality.

• Feb 08 '16

  • Tyler Lawson (University of Minnesota)

  • Derived Morita theory

  In this talk I'll discuss joint work with David Gepner on the study of Morita theory and Azumaya algebras in the derived context.

• Feb 15 '16

  • John Lind (University of Regensburg)

  • Duality in bicategories and the THH transfer

  Associated to a fibration E --> B with homotopy finite fiber is a stable wrong way map LB --> LE of free loop spaces coming from the transfer map in THH. This transfer is defined under the same hypotheses as the Becker-Gottlieb transfer, but on different objects. I will use duality in bicategories to explain why the THH transfer contains the Becker-Gottieb transfer as a direct summand. The corresponding result for the A-theory transfer may then be deduced as a corollary. When the fibration is a smooth fiber bundle, the same methods give a three step description of the THH transfer in terms of explicit geometry over the free loop space. (Joint work with C. Malkiewich)

• Mar 21 '16

  • Shigeo Koshitani (Chiba University, Japan)

  • On endotrivial modules for dihedral defect groups

  We will be discussing endotrivial modules for the group algebra kG of a finite group G over a field k of characteristic p > 0 (cf. [Koshitani-Lassueur, Manuscripta Math. 148 (2015), 265--282] and [Koshitani-Lassueur, J.Group Theory, 27 pages, in press]). Endotrivial modules go back to E.C. Dade [1978, Ann. of Math] and also J.L.Alperin. Endotrivial modules show up often and play a very important role in the representation theory of finite groups; for example an endotrivial module for kG induces a self stable equivalence of the module category. Our result is a kind of complement to [Carlson-Mazza-Thevanaz, European J. Math. 15 (2013), 157--177].

• Apr 11 '16

  • TriThang Tran (University of Oregon)

  • Topology Seminar - TBA

  TBA

• Apr 25 '16

  • Søren Galatius (Stanford University and University of Copenhagen)

  • Topology Seminar - TBA

  TBA

Talks in Fall 2015

• Dec 14 '15

  • Martin Markl (Ordway visitor) (Institute of Mathematics of the Czech Academy)

  • Deformation theory IV

  A mini-course on deformation theory based on L_\infty-algebras and the Maurer-Cartan equation, culminating with an outline of Kontsevich's deformation quantization proof.

• Nov 23 '15

  • Martin Markl (Ordway visitor) (Institute of Mathematics of the Czech Academy)

  • Deformation theory I

  A mini-course on deformation theory based on L_\infty-algebras and the Maurer-Cartan equation, culminating with an outline of Kontsevich's deformation quantization proof.

• Dec 07 '15

  • Martin Markl (Ordway visitor) (Institute of Mathematics of the Czech Academy)

  • Deformation theory III

  A mini-course on deformation theory based on L_\infty-algebras and the Maurer-Cartan equation, culminating with an outline of Kontsevich's deformation quantization proof.

• Nov 30 '15

  • Martin Markl (Ordway visitor) (Institute of Mathematics of the Czech Academy)

  • Deformation theory II

  A mini-course on deformation theory based on L_\infty-algebras and the Maurer-Cartan equation, culminating with an outline of Kontsevich's deformation quantization proof.

• Nov 23 '15

  • Martin Markl (Ordway visitor) (Institute of Mathematics of the Czech Academy)

  • Deformation theory I

  A mini-course on deformation theory based on L_\infty-algebras and the Maurer-Cartan equation, culminating with an outline of Kontsevich's deformation quantization proof.

• Nov 09 '15

  • Pedram Hekmati (IMPA, Brasil)

  • Topological T-duality and Hodgkin’s theorem

  A famous problem in topology, first solved by Hodgkin in 1967, is to determine the K-theory of compact simply-connected Lie groups. Hodgkin's original proof was extremely technical, motivating the discovery of a number of simpler proofs. In this talk I will present a new, surprisingly simple proof of Hodgkin's theorem using topological T-duality, an idea that originated in physics. This is based on joint work with David Baraglia.

• Oct 26 '15

  • Jesse Wolfson (University of Chicago)

  • Counting Problems and Homological Stability

  In 1970, Arnold showed that the i'th homology of the space of un-ordered configurations of n points in the plane becomes independent of n for large n. A decade later, Segal extended Arnold's method to show that the i'th homology of the space of degree n holomorphic maps from P^1 to itself also becomes independent of n for large n, and, moreover, that both sequences of spaces have the same limiting homology. We explain how, using Weil's number field/function field dictionary, one might have predicted this topological coincidence from easily verifiable statements about specific counting problems. We then discuss ongoing joint work with Benson Farb and Melanie Wood in which we use other counting problems to predict and discover new instances of homological stability in the topology of complex manifolds.

• Oct 19 '15

  • Inna Zakharevich (University of Chicago)

  • The annihilator of the affine line

  The Grothendieck ring of varieties is defined to be the free abelian group generated by k-varieties, modulo the relation that for any closed subvariety Y of a variety X, we impose the relation that [X] = [Y] + [X \ Y]; the ring structure is defined by [X][Y] = [X x Y]. Last December two longstanding questions about the Grothendieck ring of varieties were answered: 1. If two varieties X and Y are piecewise isomorphic then they are equal in the Grothendieck ring; does the converse hold? 2. Is the class of the affine line a zero divisor? Both questions were answered by Borisov, who constructed an element in the kernel of multiplication by the affine line; coincidentally, the proof also constructed two varieties whose classes in the Grothendieck ring are the same but which are not piecewise isomorphic. In this talk we will investigate these questions further by constructing a topological analog of the Grothendieck ring and analyzing its higher homotopy groups. Using this extra structure we will sketch a proof that Borisov's coincidence is not a coincidence at all: that any element in the annihilator of the Lefschetz motive can be represented by a difference of varieties which are equal in the Grothendieck ring but not piecewise isomorphic.

• Oct 12 '15

  • Andy Putman (Rice University)

  • Stability in the homology of congruence subgroups

  I will explain how to use categorical tools arising from the Lannes-Schwartz Artinian conjecture (recently proved by myself and Sam, and independently by Sam-Snowden) to understand patterns in the cohomology of congruence subgroups of SL(n,Z). This is joint work with Steven Sam.

• Oct 05 '15

  • Alexander A. Voronov (University of Minnesota)

  • Categorification of Dijkgraaf-Witten theory

  This is a new perspective on an old topic: Dijkgraaf-Witten theory, i.e., gauge theory with a finite gauge group, producing a TQFT, i.e., a functor from the category of cobordisms to that of vector spaces. I will recall earlier approaches to Dijkgraaf-Witten theory and then describe a categorification of it. This is based on a cap product between homology and cohomology with coefficients in Picard groupoids. This is a joint work with Amit Sharma.

• Sep 28 '15

  • Tyler Lawson (University of Minnesota)

  • Homotopy limits and colimits

  Ordinary limits and colimits behave poorly from the point of view of homotopy equivalences, but they have generalizations which do. This will be an informal talk discussing these limits and colimits in algebraic topology, and in particular the impact of some of the "new" category theory from Lurie's "Higher Topos Theory".

• Sep 21 '15

  • Craig Westerland (University of Minnesota)

  • Fox-Neuwirth cells, quantum shuffle algebras, and the homology of braid groups

  The notion of a braided vector space V comes from the Hopf algebra community, and examples abound, from conjugacy classes in groups to braidings coming from Cartan matrices. From this definition, the tensor powers of V form a family of braid group representations. They also may be assembled into a non-commutative, non-cocommutative braided Hopf algebra called a quantum shuffle algebra. Our main result identifies the homology of the braid groups with these coefficients as the cohomology of this algebra. Using change of rings spectral sequences, we begin to get a handle on these homology groups. This is joint work with TriThang Tran.

Talks in Spring 2015

• Jan 26

  • Christian Haesemeyer (UCLA)

  • Algebraic invariants of singularities

  I will discuss my work with Cortinas, Walker, Weibel on the K-theory and cyclic homology of singular algebraic varieties from the point of view of a concrete application to questions about the existence of vector bundles with certain properties.

• Feb 02

  • Tyler Lawson (University of Minnesota)

  • The Blakers-Massey excision theorem

  The Blakers-Massey excision theorem describes the homotopy groups of a union of two spaces through a range. I'll discuss a recent new proof of this result, with the unusual property that it was reverse-engineered by Rezk from a proof by Favonia in a computer proof assistant.

• Feb 09

  • Tyler Lawson (University of Minnesota)

  • The Blakers-Massey excision theorem II

  The Blakers-Massey excision theorem describes the homotopy groups of a union of two spaces through a range. I'll discuss a recent new proof of this result, with the unusual property that it was reverse-engineered by Rezk from a proof by Favonia in a computer proof assistant.

• Feb 23

  • Nathan Perlmutter (U. Oregon)

  • Homological stability for the diffeomorphism groups of odd-dimensional manifolds

  I will present a new homological stability result for the diffeomorphism groups of manifolds of dimension 2n+1 ≥ 9, with respect to forming the connected sum with copies of an arbitrary (n − 1)-connected, (2n + 1)-dimensional manifold that is stably parallelizable. This result is an odd dimensional analogue of a recent theorem of Galatius and Randal-Williams regarding the homological stability of the diffeomorphism groups of manifolds of dimension 2n ≥ 6, with respect to forming connected sums with S^n × S^n .

• Feb 25

  • Martin Markl (Mathematical Institute of the Czech Academy)

  • Structures of string field theory via modular envelopes

  Abstract not available

• Mar 23

  • Fabian Hebestreit (University of Notre Dame)

  • Twisted Spin cobordism and positive scalar curvature

  After a brief exposition of twisted (co)homology and more generally parametrised homotopy theory, I want to explain the relevance of twisted Spin cobordism to the search for Riemannian metrics of positive scalar curvature and in particular the Gromov-Lawson-Rosenberg conjecture. I will then present a homotopical analysis of the underlying parametrised spectrum and, should time permit, report on ongoing joint work with Michael Joachim and Stephan Stolz aimed at proving an existence theorem for such metrics on manifolds whose universal cover admit a spin structure.

• Apr 13

  • Dylan Wilson (Northwestern University)

  • Genera, Orientations, and tmf with level structure

  We will review the work of Ando-Hopkins-Rezk on the Witten genus and show how their construction carries over to the case of topological modular forms with level structure. We also indicate how to improve genera on String manifolds to genera on Spin manifolds and construct the Ochanine genus.

• Apr 20

  • Omar Ortiz (University of Western Ontario)

  • Moment graphs and the wonderful compactification

  The wonderful compactification X of a semisimple group of adjoint type was first constructed by De Concini and Procesi in 1983 and has been extensively studied in algebraic geometry. The equivariant cohomology of X resembles in many ways that of the flag variety, allowing some Schubert calculus to be performed. This talk will be a review of these ideas and an indication of current work on a moment graph theory for X.

• Apr 27

  • Kirsten Wickelgren (Georgia Tech)

  • Motivic desuspension

  Certain problems such as classifying manifolds up to cobordism are stable in the sense that they are solved in categories where it is possible to desuspend. Other problems, such as classifying algebraic vector bundles on schemes, require analogous unstable information. The EHP sequence in algebraic topology is a tool for turning stable information into unstable information. We will discuss the situation from algebraic topology, and present an EHP sequence in A^1 homotopy theory of schemes. This is joint work with Ben Williams.

• May 04

  • John Terilla (Queens College, CUNY)

  • Homotopy Probability Theory with an application to Fluids

  Homotopy probability theory is a theory in which the vector space of random variables is replaced with a chain complex. I'll discuss how to use homotopy algebra (rather than analysis) to extract meaningful expectations and correlations among random variables. Some natural examples will be introduced, including an application to fluid flow.

Talks in Fall 2014

• Oct 06

  • Alexander A. Voronov (University of Minnesota and IPMU)

  • Ambidexterity I: Dijkgraaf-Witten theory

  I will continue the series of talks, started by talks on the Morava K-theory of Eilenberg-Mac Lane spaces by Tyler Lawson and Craig Westerland in the Spring 2014, aimed at ambidexterity after Jacob Lurie and Mike Hopkins, and go over the Dijkgraaf-Witten Topological Quantum Field Theory as a motivational example.

• Oct 13

  • Alexander A. Voronov (University of Minnesota and IPMU)

  • Ambidexterity II: DW theory and ambidextrous spaces

  This is the second talk in a series, started by talks on the Morava K-theory of Eilenberg-Mac Lane spaces by Tyler Lawson and Craig Westerland in the Spring 2014, aimed at ambidexterity after Jacob Lurie and Mike Hopkins. I will show how Dijkgraaf-Witten Topological Quantum Field Theory leads to a phenomenon called ambidexterity and talk about ambidextrous maps and spaces.

• Oct 20

  • Benjamin Ward (Simons Center for Geometry and Physics, Stony Brook University)

  • Maurer-Cartan elements and natural operations

  The process of twisting by a Maurer-Cartan element in an operadic context produces cochain complexes computing, for example, the Chevalley-Eilenberg cohomology of a Lie algebra, the Hochschild cohomology of an A-infinity algebra, the singular cohomology of a topological space, the cyclic cohomology of a Frobenius algebra or the equivariant cohomology of an S^1 space. A byproduct of this perspective is the existence of certain families of cohomology and cochain operations, as I will discuss.

• Oct 22

  • TriThang Tran (University of Oregon)

  • Configurations spaces and symmetric complements

  In this talk, we discuss homological stability for spaces that are defined as the complements of the closures of certain strata in the n-fold symmetric product — named "symmetric complements". Symmetric complements are closely related to configuration spaces where homological stability was previously known to hold. Homological stability for symmetric complements, in particular, answers a conjecture by Vakil and Wood, which goes part way to understanding the relationship between homological stability, and the existence of certain limits in the Grothendieck ring of varieties.

  The talk will proceed by providing a brief introduction to homological stability using configuration spaces as an example. We will then discuss how to leverage stability for configuration spaces to obtain stability for symmetric complements. This is joint work with A. Kupers and J. Miller.

• Oct 27

  • Rune Haugseng (Max-Planck Institute for Mathematics)

  • The higher Morita category of E_n-algebras

  I will discuss a construction of a higher category of E_n-algebras and iterated bimodules, generalizing the classical bicategory of algebras and bimodules. This leads to generalizations of the Picard and Brauer groups, which have been studied in stable homotopy theory as interesting invariants of ring spectra, and should also lead to an "algebraic" construction of factorization homology as an extended topological quantum field theory.

• Nov 03

  • Craig Westerland (University of Minnesota)

  • Torsion in the homology of random chain complexes and the Cohen-Lenstra distribution

  The expected Betti numbers of "random" simplicial complexes have been studied for some time, beginning with remarkable properties of the Erdos-Renyi random graph. Very little is known about the expected torsion in the homology of a random simplicial complex X. Recently, Matt Kahle has conjectured that for each prime p, the p-torsion in H_*(X) might be Cohen-Lenstra distributed. Loosely, the Cohen-Lenstra distribution on finite abelian p-groups predicts that a given group arises with probability inversely proportional to the size of its automorphism group. In this talk, we'll give evidence for this conjecture by proving a version of it for a suitable notion of random chain complexes (rather than simplicial complexes), using somewhat standard techniques of random p-adic matrices. Additionally, we'll give some number-theoretic, category-theoretic, and measure-theoretic background on the Cohen-Lenstra distribution.

• Nov 17

  • Tomer Schlank (Massachusetts Institute of Technology)

  • Stable étale obstruction to zero-cycles of degree 1

  Given a field K and an algebraic variety X/K, a zero-cycle on X is a formal sum of points on X(\bar{K}) which is Galois equivariant . The zero-cycles with total degree 1 can be considered as a abelianized version of a K-point on X. When K is a number field Colliot-Thélène, conjectured that the famous Brauer-Manin obstruction is the only one for the existence of zero-cycles of degree 1 on a smooth projective variety. From the viewpoint of algebraic geometry, solutions to a Diophantine equation are just sections of a corresponding map of schemes X -> S, where schemes are usually considered as a certain type of "Spaces" . When considering sections of maps of spaces f:X->S in the realm of algebraic topology, Bousfield and Kan developed an Obstruction-Classification Theory using the cohomology of the S with coefficients in the homotopy groups of the fiber of f. By using étale homotopy it is possible to transform this obstruction theory to the realm of algebraic geometry. This allows reinterpretation of classical obstructions to existences of rational points (like the Brauer-Manin Obstruction) and to define new obstructions. One example is using stable étale homotopy theory to get a new obstruction for zero-cycles of degree 1. In a work in progress we hope to use this new obstruction to attack Colliot-Thélène's conjecture. This is a joint project with Y.Harpaz and V.Stojanoska.

Talks in Spring 2014

• Mar 10

  • Tyler Lawson (University of Minnesota)

  • CANCELLED

  I'll be discussing the calculation of the Morava K-theory of Eilenberg-Mac Lane spaces, as part of a discussion of the Hopkins-Lurie paper on ambidexterity in $K(n)$-local homotopy theory.

• Mar 24

  • Yu Tsumura (Purdue University)

  • A 2-categorical extension of the Reshetikhin-Turaev theory

  I will discuss a concrete construction of a 2-categorical extended topological field theory that extends the Reshetikhin-Turaev TQFT. An extended TQFT is defined to be a 2-functor from a 2-category of cobordisms with corners to the Kapranov-Voevodsky 2-vector spaces. As an intermediate 2-category between these 2-categories, a 2-category of special ribbon graphs is introduced.

• Mar 31

  • Tyler Lawson (University of Minnesota)

  • Morava K-theory of Eilenberg-Mac Lane spaces

  I'll be discussing the calculation of the Morava K-theory of Eilenberg-Mac Lane spaces, as part of a discussion of the Hopkins-Lurie paper on ambidexterity in $K(n)$-local homotopy theory.

• Apr 07

  • Robert Lipshitz (Columbia University)

  • Morse flows on products and Khovanov homotopy

  We will start by recalling the definition of the Khovanov homology of a knot, and a refinement, a Khovanov homotopy type. We will then discuss the combinatorics of Morse flows on products of manifolds (this is surprisingly complicated), and a Kunneth theorem for the Khovanov homotopy type, and give an application to concordance. Along the way, we will present a number of as-yet unanswered questions, mostly of a homotopy-theoretic nature. This is joint work with Sucharit Sarkar.

• Apr 14

  • Amit Patel (University of Minnesota (IMA))

  • Persistent Sheaves for Stratified Maps

  Given a stratified map f : X -> Y, we may study the homology (relative homology) of its fibers by studying its Leray cosheaf (sheaf), one for each dimension. The Leray cosheaf assigns to each sufficiently small open set U of Y the ordinary homology of the pre-image f^{-1}(U). The Leray sheaf assigns to each sufficiently small open set U of Y the relative homology of the pair of spaces (X, X - f^{-1}(U)). For a stratified map g : X -> R to the real line, the indecomposable summands of the Leray cosheaf (sheaf) are in one-to-one correspondence with the appropriate (level-set zigzag) persistence diagram of g. It is therefore tempting to define the persistence diagram of an arbitrary stratified map f : X -> Y as the list of indecomposable summands associated to the Leray sheaf or Leray cosheaf of f. However, the Leray cosheaf (sheaf) is in general not stable to perturbations to f. For example, an arbitrarily small perturbation to f may introduce a hole (a point with a trivial co-stalk) well in the interior of the support of the Leray cosheaf (sheaf).

  In this talk, I will present the persistent sheaf for any stratified map f : X -> M to an oriented manifold M. Roughly speaking, the persistent sheaf is a 2-functor that assigns to each sufficiently small open set U of M, a subgroup of the homology of the pre-image f^{-1}(U). I will give a rough construction of the persistent sheaf and show how it is stable to sufficiently small perturbations to f.

  This is joint work with Robert MacPherson (IAS).

• Apr 21

  • Craig Westerland (University of Minnesota)

  • The Dieudonné module approach to the Morava K-theory of Eilenberg-MacLane spaces.

  We'll introduce Dieudonné modules and p-divisible groups in the context of homology of H-spaces, and talk about the reformulation of Ravenel-Wilson's computation of the Morava K-theory of Eilenberg-MacLane spaces in this language.

• Apr 28

  • Craig Westerland (University of Minnesota)

  • The Dieudonné module approach to the Morava K-theory of Eilenberg-MacLane spaces, II

  Continuing from last week, we will compute the Dieudonne module of K(n)_* K(Z/p^j, 1)), explain some duality results due to Buchstaber-Lazarev, and discuss the p-divisible groups associated to these Hopf algebras.

• May 05

  • Donald Kahn (University of Minnesota)

  • Coincidence sets and Kervaire invariants, d'apres Koschorke and Randall

  Abstract not available

Talks in Fall 2013

• Sep 09

  • Craig Westerland (University of Minnesota)

  • p-compact groups and étale homotopy types

  Connected, compact Lie groups are quite rich in geometric structure. However, they have a classification in terms of the rather simple algebraic data of an integral reflection group, coming from the action of the Weyl group on the character lattice. From the point of view of homotopy theory, the geometric structure is largely invisible, and compact Lie groups are simply loop spaces with finite rank homology. Dwyer and Wilkerson introduced a p-complete analogue of this notion called "p-compact groups," and established some remarkable structure theorems for them. This culminated in a classification due to Andersen et al which looks remarkably similar to the classification of compact Lie groups, only employing p-adic reflection groups. Nonetheless, p-compact groups are decidedly lacking in geometry, and it is the purpose of this talk to try to equip them with some coming from the étale homotopy theory of algebraic groups.

• Sep 16

  • Alexander Voronov (University of Minnesota)

  • Dijkgraaf-Witten theory using cohomology with coefficients in Picard groupoids

  This is a report on work in progress with Amit Sharma in which we define cap product between homology and cohomology with coefficients in Picard groupoids and use it to construct Dijkgraaf-Witten theory, a gauge theory with a finite gauge group.

• Sep 23

  • Tyler Lawson (University of Minnesota)

  • Cohomology operations and power operations

  In the singular cohomology of spaces, one can define both cohomology operations and power operations, and the former turn out to be special instances of the latter. In different situations, however, these start to deviate from each other. I'll discuss some work in progress on understanding how these two structures interact with the specific goals of understanding Morava K-theory and E-theory.

• Sep 30

  • Michael Mandell (Indiana University, Bloomington)

  • The homotopy theory of cyclotomic spectra

  In joint work with Andrew Blumberg, we construct a category of cyclotomic spectra that is (something like) a closed model category and which has well-behaved mapping spectra. We show that topological cyclic homology (TC) is the corepresentable functor on this category given by maps out of the sphere spectrum, verifying a conjecture of Kaledin.

• Oct 14

  • Pavle Blagojevic (Freie Universität Berlin and Mathematical Institute of the Serbian Academy)

  • On k-regular maps

  The question about the existence of a continuous $k$-regular map from a topological space $X$ to an $N$-dimensional Euclidean space $R^N$, which would map any $k$ distinct points in $X$ to linearly independent vectors in $R^N$, was first considered by Borsuk in 1957. In this talk we present a proof of the following theorem, which extends results by F. Cohen and Handel 1978 (for $d=2$) and Chisholm 1979 (for $d$ power of 2): For integers $k$ and $d$ greater then zero, there is no $k$-regular map $R^d \to R^N$ for $N \lt d(k-a(k))+a(k)$, where $a(k)$ is the number of ones in the dyadic expansion of $k$.

  Joint work with G. M. Ziegler and W. Lück.

• Oct 21

  • F.R. Cohen (University of Rochester)

  • A leisurely excursion into polyhedral products

  Polyhedral products are natural subspaces of a product space "indexed by a simplicial complex". They are natural extensions of "moment-angle complexes" of Buchstaber-Panov dating back to Poincaré, Coxeter, as well as many others.

  Definitions, previous developments as well as a survey of applications of polyhedral products from toric manifolds, analogues of Stanley-Reisner ring of a simplicial complex to packing questions will be described. This talk is based on joint work with A. Bahri, M. Bendersky, S. Gitler, and the speaker.

• Oct 28

  • Michael Hill (University of Virginia)

  • Equivariant Operadic Actions and the Transfer

  Classically, loop spaces are algebras over little disks operads for some inner product space, and this is true equivariantly. Here, however, the indexing universe for spectra plays a role, as $G$-spectra on a universe $U$ have certain additional structure maps: transfers. In this talk, I'll describe how the transfer arises from the operadic structure directly. This is joint with Andrew Blumberg.

• Nov 04

  • Gunnar Carlsson (Stanford University)

  • Derived completion and K-theory of fields

  I will discuss a formulation of the descent problem for the algebraic $K$-theory of fields in terms of derived functors of completion applied to the complex representation ring of the absolute Galois group of the field, together with a discussion of how this conjecture can be proved.

• Nov 18

  • Donald Kahn (University of Minnesota)

  • Homotopy, schemes, and Weibel's road map

  Abstract not available

• Nov 25

  • Daniel Schaeppi (University of Western Ontario)

  • A Tannakian characterization of categories of coherent sheaves

  Classical Tannaka duality is a duality between groups and their categories of representations. The two basic questions it answers are the reconstruction problem (when can a group be reconstructed from its category of representations?) and the recognition problem (can we characterize categories of representations abstractly?).

  I will outline how the notion of a Tannakian category can be weakened in order to solve the recognition problem for categories of coherent sheaves of algebraic stacks (if you are an algebraic geometer), respectively categories of comodules of Adams Hopf algebroids (if you are an algebraic topologist). As an application I will outline how this notion can be used to describe coherent sheaves on fiber products in terms of Deligne's tensor product of categories.

• Dec 02

  • Andrew Blumberg (University of Texas, Austin)

  • The higher-categorical perspective on algebraic K-theory

  This talk will give an overview of the recent work interpreting the formal properties of algebraic K-theory in the context of new formalisms for handling the category of homotopical categories.

• Dec 09

  • Tarje Bargheer (Australian National University)

  • Stretching String Topology

  Higher dimensional string topology generalizes the Chas-Sullivan product on the free loop space on M to the space of maps from higher-dimensional manifolds into M. I will present a geometric operad that describes structures on the space of maps from n-dimensional spheres into M, generalizing the Chas-Sullivan structure and shows they are $E_{n+1}$-algebras. I will use this introduction of the operad to show how the geometric construction of the operad allows us to stretch string topology in various directions, suggesting new ways of enriching string topology.

Talks in Spring 2013

• Jan 28

  • Tyler Lawson (University of Minnesota)

  • Secondary operations

  Secondary operations featured prominently in Adams' proof of the Hopf invariant one conjecture, which implies there are no division algebra structures on $\mathbb{R}^n$ when $n \neq 1,2,4,8$. I'll discuss how secondary products and secondary operations appear in different contexts in algebra and topology.

• Feb 04

  • Alexander Voronov (University of Minnesota)

  • Secondary invariants from Dijkgraaf-Witten theory

  As Tyler Lawson explained to us in his last seminar talk, secondary operations arise when a primary operation vanishes or two primary operations are equal. If you have a bordism between two closed manifolds, their homology classes will be equal and, by definition, their cobordism classes will be equal. A Topological Field Theory (TFT) creates a secondary invariant in this case: a linear map between certain vector spaces associated to each manifold. Dijkgraaf-Witten (DW) theory is an example of a TFT associated to a finite group. Recently, there has been a push towards describing an Extended TFT, and that generated deeper understanding of DW theory. I will discuss new insights on DW theory as per Jacob Lurie and others.

• Feb 18

  • Tyler Lawson (University of Minnesota)

  • Forms of K-theory and elliptic cohomology

  I'll discuss K-theory, and how its theory of Chern classes differs from that for ordinary cohomology. I'll then discuss "forms" of K-theory, which provide some twist on the multiplicative rules of ordinary K-theory, and how real K-theory is universal among those. Elliptic cohomology theories will then be introduced, along with the analogous properties a universal object should have.

• Feb 25

  • Tyler Lawson (University of Minnesota)

  • Elliptic curves and elliptic cohomology theories

  I'll be continuing last week's talk by introducing elliptic cohomology theories, and universal objects among these.

• Mar 04

  • Erin Manlove (University of Minnesota)

  • Morava Stabilizer Groups

  Morava stabilizer groups arise in the study of homotopy groups of spheres as objects that encode topological information. Though Morava stabilizer groups themselves remain relatively mysterious, we may study them by looking at their maximal finite subgroups. In my talk, I will give two characterizations of Morava stabilizer groups, describe the type of subgroups that are of interest, and discuss how we connect these subgroups to geometric objects by identifying them with automorphism groups of points in an algebraic stack.

• Mar 11

  • Niles Johnson (Ohio State)

  • Obstruction theory for homotopical algebra maps

  We take an obstruction-theoretic approach to the question of algebraic structure in homotopical settings. At its heart, this is an application of the Bousfield-Kan spectral sequence adapted for the action of a monad $T$ on a topological model category. This yields an obstruction theory for lifting maps from $T$-algebras in a homotopy category to the homotopy category of $T$-algebras. Maps in this latter category are required to commute with a coherent system of higher homotopies encoding the $T$-action. We give general conditions under which the $E_2$ page of the resulting spectral sequence may be identified with André-Quillen cohomology.

  This talk will give an overview of the general theory, but focus mainly on the breadth of applications. These include categories of algebraic theories in the sense of Lawvere and categories of algebras over operads. We will describe specific examples in $G$-spaces, $G$-spectra, and rational $E_\infty$ algebras. Furthermore, we will outline calculations connected to rational unstable homotopy theory which distinguish between $E_\infty$ and $H_\infty$.

• Apr 01

  • Eric Peterson (University of California, Berkeley)

  • Cotangent spaces of certain spectra

  We outline a construction in derived algebraic geometry which produces elements of the $K(n)$-local Picard group. As an example, we produce a new model of a spectrum called the determinantal sphere, and as time permits we discuss newly indicated patterns in $K(n)$-local homotopy theory.

• Apr 08

  • Robert Hank (University of Minnesota)

  • Massey products in $A_\infty$ algebras

  During this talk, I'll develop a relationship between Massey products and multiplicative structure in such a way that the generalization of a Massey product from a strictly associative to an $A_\infty$ structure arises naturally.

• Apr 15

  • William Abram (University of Michigan)

  • The equivariant complex cobordism ring of a finite abelian group

  The calculation of the non-equivariant cobordism ring due to Milnor and Quillen was one of the great successes of algebraic topology. Equivariantly, Kriz described tom Dieck's stable equivariant complex cobordism ring $(MU_G)_*$, in the case $G=\mathbb Z/p$, as the pullback of a diagram of rings arising from the Tate diagram for $MU_{\mathbb Z/p}$. We extend this work to the case of $G$ a finite abelian group, where we describe $(MU_G)_*$ as the inverse limit over certain $G$-spectra $F(S)_*$ indexed over chains of subgroups of $G$. In the case $G=\mathbb Z/p^n$, we get a simple description of $(MU_G)_*$ as the n-fold pullback of a diagram of rings. In the general case, we are still able to compute the algebraic structure of $F(S)_*$ explicitly. I will discuss this computation in some detail.

• Apr 22

  • Yifei Zhu (University of Minnesota)

  • Power operations in an elliptic cohomology theory

  As a result of the geometry of vector bundles underlying K-theory, power operations in K-theory correspond to linear representations of finite symmetric groups. For elliptic cohomology theories, lacking analogous geometric interpretations, we can study their power operations via a correspondence to cyclic isogenies of elliptic curves, in hope of understanding their conjectural geometry. This approach is valid for Morava E-theories in general, including versions of K-theory and elliptic cohomology as special cases where computations can be carried out explicitly.

• Apr 29

  • Donald Kahn (University of Minnesota)

  • Schemes and homotopy after Fabian Morel

  I will attempt to survey the works of Morel on his construction of the homotopy category of schemes and the relation to motivic homotopy with applications to to the Grothendieck-witt ring of quadratic forms and questions about when relations in classic homotopy are actually algebraic. This will have to be a sketch.

• May 06

  • Nathaniel Stapleton (M.I.T.)

  • The E-theory of centralizers in symmetric groups

  Using cohomology theories closely related to Morava E-theory, we will provide an algebro-geometric description of the cohomology of particular centralizers in symmetric groups in terms of the scheme classifying subgroups of a certain p-divisible group.