Algebra Seminar 

In this talk I will describe the construction of the category of non-commutative motives in Drinfeld-Kontsevich's non-commutative algebraic geometry program. In the process, I will present the first conceptual characterization of Quillen's higher K-theory since Quillen's foundational work in the 70's. As an application, I will show how these results allow us to obtain for free the higher Chern character from K-theory to cyclic homology.

We will analyse the relationship between Hochschild Homology and the manifold $S^1$. From this we can see how to associate naturally to manifolds (with certain geometric structures) operations generalizing Hochschild Homology. These operations are defined on certain algebraic structures related to spaces of embeddings.

I will survey Dominic Joyce's theory of quaternionic algebra, which provides the algebraic framework for studying quaternionic holomorphic functions on hyperkahler manifolds, as well as Quillen's description of the theory in terms of equivariant sheaves on the Riemann sphere.

Unprojection theory aims to analyze complicated commutative rings in terms of simpler ones. The talk will be about joint work in progress with Janko Boehm (Saarbruecken) that relates, on the algebraic level of Stanley--Reisner rings, stellar subdivisions of a certain class of simplicial complexes (which includes all sphere triangulations) with Kustin--Miller unprojection. I will also mention a related result about boundary complexes of cyclic polytopes.

Given an arbitrary field $k$ and a fan $F$, we study the classification of the various "toric $k$-forms" of the toric variety $X_{K,F}$, where $K$ is the algebraic closure of $K$ and $F$. This classification generalizes the work of Delaunay on "real toric varieties" and has a particularly simple description in the case of complete non-singular toric surfaces. We show how to use the Cox construction to perform explicit calculations and make a few applications. This is joint work with Javier Elizondo, Frank Sottile and Zach Teitler.

As higher analogues of group schemes, group stacks arise in several contexts in geometry. E.g., symmetries of stacks, structure group stacks of higher principal bundles, stacky abelianization of reductive groups (Deligne), and so on. Working with group stacks is, however, considerably more difficult than working with group schemes, especially when one needs to do explicit calculations. In these talks we introduce some general techniques for dealing with this problem. We discuss applications to 'group actions on stacks' and to 'classification of forms of stacks over a field'. Notes

We identify a class of local rings $(R,m,k)$ with $m4=0$ exhibiting the Koszul like property that $H_R(-t)P_M^R(t)$ is in $Z[t]$ for all finite $R$-modules $M$; where $H_R(t)$ denotes the Hilbert series of $R$ and $P_M^R(t)$ the Poincaré series of $M$ over $R$. This class includes generic graded Gorenstein algebras of socle degree $3$. We show the minimal free resolutions of finite modules over such rings admit Koszul syzygy modules.

Computing the topology of a real algebraic set given as the zero set of a list of polynomials remains a challenging problem, even for polynomials in 3 variables. Nevertheless, we can show that for certain systems of sparse polynomials, one can efficiently compute the topology in polynomial-time with high probability. This is recent joint work with Martin Avendano. We illustrate the algorithm through various examples, and see how a special case leads to the use of Diophantine approximation. We then show how, in more general cases, it is natural to expect a set of small set of inputs where the algorithm slows down. We assume no background in number theory or algorithms.

After the work of H. Miller and J. Lannes in the 80's we know that the homotopy classes of maps from BV (V some elementary abelian p-group) to some space are detected by ordinary mod p cohomology. I will review what happens when mod p cohomology is replaced by a complex oriented cohomology theory. As an interesting special case I will consider morphisms from elementary abelian p-groups to compact Lie groups.

We show that algebraic K-theory and periodic algebraic cobordism are localizations of motivic suspension spectra obtained by inverting the Bott element, generalizing theorems of V. Snaith in the topological case. This yields an easy proof of the motivic Conner-Floyd theorem and also implies that algebraic K-theory is E-infinity as a motivic spectrum.

The representations from a free group into $\mathrm{SL}(3,C)$ are an affine variety. The ring of invariants of the conjugation action is generated by traces of words in generic matrices. We have described minimal sets of these generators; providing global coordinates for the moduli of representations. In this talk, we describe maximal algebraically independent subsets of the minimal generators. In contrast, these sets should be thought of as local parameters for the moduli.

We introduce a version of Deligne cohomology for smooth proper real varieties which is related to bigraded Bredon cohomology in the same fashion that the usual version of Deligne cohomology is related to singular cohomology. For complex manifolds, the Deligne cohomology group $H_{D/C}^2(X;Z(2))$ can be identified with the group of equivalence classes of pairs $(L,\nabla )$, where $L$ is a holomorphic line bundle and $\nabla$ is a holomorphic connection on $L$. However, when $X$ is a Real manifold, the straightforward generalization of this result does not work due to a certain obstruction related to the set of real points of the variety, and one needs an additional geometric piece which would be a certain real quadratic form on the line bundle. We will provide a gentle introduction to Deligne cohomology and some examples.

I will discuss connections between the calculus of functors and the Whitehead Conjecture, both for the classical theorem of Kuhn and Priddy for symmetric powers of spheres and for the analogous conjecture in topological K-theory. It turns out that key constructions in Kuhn and Priddy's proof have bu-analogues, and there is a surprising connection to the stable rank filtration of algebraic K-theory.

Given a ring $A$ there is a geometric construction of the space that classifies the cohomology functor $H^n(-;A)$; it is just the the free $A$-module $S^n\otimes A$ generated by the space $S^n$. For spaces with an action of finite group $G$, the role of cohomology with coefficients in a ring is played by equivariant cohomology with coefficients in an appropriate algebraic object $M$ -- called a Mackey functor. In this talk we will describe a geometric construction for the classifying spaces of equivariant cohomology with coefficients in a Mackey functor $M$. This is joint work with Zhaohu Nie.

The original McKay correspondence, due to John McKay in the late 1970s, relates for a finite subgroup $G$ of $\mathrm{SL}(2,C)$ the geometry of the minimal resolution of singularities of the quotient $C^2/G$ with the representation theory of $G$. The talk will be introductory, and will try to discuss both classical and modern approaches to the topic.

The concept of smallness in homotopy theory generalizes the concept of compactness from classical topology. However, there are two possible generalizations of this notion: one is used in model category theory, while the other one is used in the realm of triangulated categories. The relation between these two concepts remained mysterious for a long time. Mark Hovey has shown in his book on Model categories that smallness in a stable finitely generated model category implies smallness in its homotopy category. Recently Rosicky generalized this result to combinatorial model categories. In this talk we will exhibit an example of a model category Quillen equivalent to the category of spaces with the following property: every homotopy type has a countably small representative. In particular, smallness in this model category does not translate into smallness in the homotopy category. Our example stems from work on enriched Brown representability. Connections with homotopy calculus and orthogonal calculus will also be discussed.

We will give an overview of the relevant definitions, including the notions of critical and $F$-essential subgroups of a given 2-group $S$. Then we will present a systematic procedure to find those subgroups of $S$ and how to determined all nonconstrained centerfree fusion systems over $S$, up to isomorphism, using that information. We will finish the talk by applying those methods to some examples.

To quantify something means to compare it with something else which is presumed to be more fundamental. In my talk I will compare taking extensions by fibrations to the operations of homotopy push-out. I will reformulate the nilpotence theorem of Devinats-Hopkins-Smith in these terms. The aim is to give an overview of what is known and what is not about Dror Farjoun's cellularity of topological spaces.

Let X be the moduli of SL(3,C) representations of a free group; that is the character variety. We determine minimal generators of the coordinate ring of X for any rank free group. This at once gives explicit global coordinates for X and determines the dimension of the moduli's minimal affine embedding. In this talk we present the minimal generators and discuss the constructive methods employed to establish the minimal generating set.

For a pointed cosimplicial space $X$, Bousfield and Kan constructed a pointed space $\mathrm{Tot}X$, which is analogous to the geometric realization of a simplicial space, and developed a spectral sequence abutting to the homotopy groups of $\mathrm{Tot}X$. In addition they showed that this spectral sequence supports a multiplicative pairing. In this talk I wish to present an analogous property for pointed simplicial spaces as well as discussing a cap product pairing for cosimplicial and simplicial pointed spaces and their respective homotopy spectral sequences.