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Isaksen's Research webpage
Below are old abstracts from the topology seminar of the Department of Mathematics at Wayne State University.
Current topology seminar schedule
Old topology seminar schedules
Tuesday April 21, 2009
Dan Isaksen, Wayne State University
The cohomology of motivic A(2)
Abstract: Working over an algebraically closed field of characteristic zero, I will compute the cohomology of the subalgebra A(2) of the motivic Steenrod algebra that is generated by Sq1, Sq2, and Sq4. The method of calculation is a motivic version of the May spectral sequence.
Speculatively assuming that there is a "motivic modular forms" spectrum with certain properties, we use an Adams-Novikov spectral sequence to compute the homotopy of such a spectrum at the prime 2.
Tuesday April 7, 2009
Peter Bubenik, Cleveland State University
Models and van Kampen theorems for directed homotopy theory
Abstract: Motivated by problems in concurrent computing, one is led to study spaces in which only certain paths are allowed. In particular, the execution paths in the state space are not reversible. Abstractly, we have a topological space with certain directed paths. Directed paths that are homotopic in a directed sense correspond to execution paths that are equivalent: for any input they will give the same output.
Directed homotopy theory has some immediate surprises. There are simple contractible partially-ordered spaces in which there are directed paths that are not (directed) homotopic. A main object of study is the fundamental category, which is the directed analog of the fundamental groupoid. I will construct some models and show some van Kampen theorems for this theory.
Thursday March 12, 2009
Pascal Lambrechts, Université de Louvain
Rational homology of spaces of smooth embeddings
Abstract: For a given compact smooth manifold M we consider the space Emb(M,Rk) of smooth embeddings of M into some large euclidean space Rk or rather some geometric variant of it which is a homotopy invariant of M. I will explain how Goodwillie's cutting method enables us to understand the homotopy type of this space of embeddings. I will then prove that the rational homology of that space is actually an invariant of th e rational homotopy type of M. The proof is based on Kontsevich's theorem on the formality of the little cubes operad and Arone's description of the layers of Weiss's orthogonal tower for the space of embeddings. This is joint work with Greg Arone and Ismar Volic.
February 10, 2009
John Klein, Wayne State University
Higher Reidemeister Torsion
Abstract: Let M be a compact manifold. Higher Reidemeister torsion can be thought of as cohomology classes on the moduli space of submanifolds N of Euclidean space that are diffeomorphic to M, equipped with a coordinatization of the rational homology of N.
Recently, Igusa gave axioms characterizing such torsions. The aim of my talk will be to explain this characterization and also vaguely to describe two (of the three known) ways to construct these theories.
February 3, 2009
Dan Isaksen, Wayne State University
Ostvaer's C*-homotopy theory
Abstract: The goal of the talk is to discuss the ideas in the preprint Homotopy theory of C*-algebras by Paul Arne Ostvaer (arXiv:0812.0154).
November 11, 2008
Mark Johnson, Penn State University Altoona
On modified Reedy structures
Abstract: In joint work with my colleague Wojciech Dorabiala, I have been looking at variations on the notion of Reedy model structures, first introduced by Bousfield and Kan in constructing their spectral sequence for a cosimplicial space. The usual Reedy structure gives a way to make (co)simplicial objects in a Quillen model category into a new model category, where the weak equivalences are the (co)simplicial maps whose entries are all weak equivalences. Our modification allows us to use appropriately chosen subsets of the entries to define weak equivalences, when defining model structures on more general types of diagram categories. Our intended application s come from recovering the algebraic K-theory of a model category by looking at such modified Reedy structures on grid-like diagram categories. We hope to eventually answer a variation on an open question recently posed by Dugger and Shipley about characterizing equivalences on K-theory via Quillen equivalences of various Reedy model structures on diagram categories.
November 4, 2008
Andrew Toms, York University
Dynamics, C*-algebras, and K-theoretic rigidity
Abstract: Dynamics provides some of the most interesting and ubiquitous examples in the theory of operator algebras. It is typically difficult, however, to understand their fine structure. A conjecture of Elliott (c. 1990) predicts that the C*-algebras associated to minimal dynamical systems on compact metric spaces (among others) will be classified up to isomorphism by their K-theory and tracial state spaces. In the case of uniquely ergodic systems, K-theory alone should suffice. In this talk I will explain how characteristic class obstructions can be used to prove that the conjecture can only hold for metric spaces of finite covering dimension. Modulo this necessary condition, I will present the solution of Elliott's conjecture in the uniquely ergodic case. (Joint work with Wilhelm Winter.)
September 30, 2008
Robert Bruner, Wayne State University
The homology and cohomology of cyclic groups
Abstract: (This is joint with Marcel Bokstedt.) Consider the permutation action of a cyclic group of order n on an n-fold tensor product of an algebra with itself. (The algebra may also have an action of the Steenrod algebra.) Cyclic invariants and coinvariants for this action fit into various exact sequences which generalize Hochschild homology. We will derive some of the elementary properties of these constructions. In the process, we conclude that the homology and cohomology of cyclic groups aren't as simple as they might seem at first sight.
September 23, 2008
Nguyen H. V. Hung, Wayne State University and Vietnam National University
The conjecture on spherical classes and the Lannes-Zarati homomorphism
Abstract: We study an algebraic version of the conjecture on spherical classes stating that the Lannes-Zarati homomorphism equals zero at any rank at least 3.
We show that the algebraic conjecture can equivalently be formulated in terms of Singer's invariant-theoretic description of the lambda algebra as follows: "any element in the Dickson algebra of at least 3 variables induces the zero-class in the homology of Singer's complex, which is dual to the lambda algebra".
The algebraic conjecture has been proved in several special cases:
(1) The Lannes-Zarati homomorphism vanishes at ranks 3 and 4;
(2) It vanishes on the image of the algebraic transfer at rank at least 3,
or equivalently every element in the Dickson algebra of at least 3 variables
is hit by the Steenrod algebra acting on the corresponding polynomial
algebra;
(3) It vanishes on any decomposable element of homological degree at
least 3 in the cohomology of the Steenrod algebra;
(4) It vanishes on elements of any finite Sq0-families,
except possibly on the initial element.
Some expositions of the existence of the Hopf invariant one and the
Kervaire invariant one classes are also investigated:
(1) What is the smallest subgroup of the general linear group whose
invariants in the polynomial algebra are all hit by the
Steenrod algebra A?
(2) The restriction homomorphism from A-generators for the
cohomology of either the symmetric or the alternating group to that of
its maximal elementary abelian subgroups are always zero,
except when the elementary abelian subgroup is of order 2 or 4.
September 16, 2008
Vigleik Angeltveit, University of Chicago
Uniqueness of Morava K-theory
Abstract: Classical obstruction theory seemingly produces uncountably many A∞ structures on the Morava K-theory spectrum K(n). We show that these A∞ structures are all equivalent, using a Bousfield-Kan spectral sequence converging to the homotopy groups of the moduli space of A∞ ring spectra equivalent to K(n). This spectral sequence has infinitely many differentials, and to show that all the relevant classes die we study the connective Morava K-theory spectrum k(n) and use the theory of Postnikov towers and S-algebra k-invariants developed by Dugger and Shipley.
April 15, 2008
John Harper, University of Notre Dame
Quillen homology of modules over operads
Abstract:
Even in the case of a simple algebraic structure such as commutative
algebras, homology in the derived sense of Quillen provides interesting
invariants; in Haynes Miller's proof of the Sullivan conjecture on maps
from classifying spaces, derived homology of commutative algebras is a
critical ingredient. This suggests that homology for the larger
class
of algebraic structures parametrized by an operad will also provide
interesting and useful invariants. In recent work, for the two contexts
of symmetric spectra and unbounded chain complexes, I have established
a homotopy theory for studying Quillen homology of modules and algebras
over operads, and have shown that homology can be calculated using
simplicial bar constructions. A larger goal is to determine the extra
structure that appears on the derived homology and the extent to which
the original object can be recovered from its homology when this extra
structure is taken into account. This talk is an introduction to these
results with an emphasis on several of the motivating ideas.
April 1, 2008
Aravind Asok, University of Washington
Highly A1-connected hypersurfaces
Abstract: This talk has its goals: (i) to study the A1-homotopy types of smooth affine quadric hypersurfaces, (ii) to study smooth hypersurfaces that are ``highly connected" from the standpoint of A1-homotopy theory, and (iii) to explain why algebraic geometers and homotopy theorists might be interested in these notions.
I'll begin by explaining how to view certain explicit models of smooth affine quadrics (odd- or even-dimensional) as motivic spheres. There are various ``classical" geometric applications of this observation (e.g., to construct elements of motivic homotopy groups of spheres). Then, I'll give one explanation of what the expression ``highly connected from the standpoint of A1-homotopy theory" means. Algebro-geometric implications of this condition for the hypersurfaces in question will be discussed, and concrete examples will be used to guide the discussion throughout. Time permitting, I'll give a more detailed explanation of the context of these results (e.g., classification problems for algebraic varieties). This talk is based on joint work with Brent Doran (IAS).
March 25, 2008
Son Nguyen, Wayne State University
Atiyah-Segal completion theorem
Abstract: This is an expository talk about the classical Atiyah-Segal completion theorem and one of its generalizations, namely, the work of Hopkins-Kuhn-Ravenel on generalized characters and complex cohomology theories.
February 26, 2008
Ian Putnam, University of Victoria
A homology theory for chaotic systems
Abstract: In the 1960's, Smale introduced the notion of an Axiom A system. These are diffeomorphisms of manifolds which display certain chaotic behaviour. The chaotic part has a natural decomposition into irreducible parts called basic sets. One of Smale's great insights was that these sets usually failed to be submanifolds; they are now called fractals. For zero-dimensional basic sets, Krieger introduced an invariant called the dimension group. I will describe this and an extension of it to general basic sets using results of Bowen and some standard ideas from topology.
February 12, 2008
Dan Isaksen, Wayne State University
Motivic homological algebra
Abstract: I will describe some preliminary explicit computations in motivic homotopy theory. Over arbitrary ground fields, we just don't know enough to compute much. But over the complex numbers, we have explicit descriptions (due to Voevodsky) of the cohomology of a point and of the Steenrod algebra of all cohomology operations. I will describe some computations of Ext groups over the motivic Steenrod algebra (over the complex numbers). Via the motivic Adams spectral sequence, these computations say something about motivic stable homotopy groups.
Over the real numbers, the cohomology of a point and the Steenrod algebra are also explicitly known (again due to Voevodsky). Similar Ext computations are possible over the real numbers, but the homological algebra is trickier.
I believe that these calculations will be an important guide for
further research in motivic homotopy theory.
January 29, 2008
José Manuel
Gómez, University of Michigan
Stringy bundles and ring spectra
Abstract: In
their work, P. Hu and I. Kriz considered Stringy bundles to give a
geometric description of Elliptic cohomology. These are geometric
objects over a fixed elliptic curve. After some inverting process, they
were able to obtain a multiplicative infinite loop space out of these
objects.
In this talk, I will present machinery that can be used directly on a slight modification of stringy bundles, that produces a strictly commutative ring spectrum out of these objects.
This construction relies on the concept of Multicategories and the recent work of Elemdorf and Mandell.
No prior knowledge of the theory of motives will be assumed!