See also the Brave New Rings seminar and Student seminar pages.
I will state and quickly outline Waldhausen's proof of the stable parametrized h-cobordism theorem, relating the space of h-cobordisms on a compact manifold M stabilized with respect to suspensions, to the loop space of the Whitehead space of M defined in terms of the algebraic K-theory A(M) of M.
I will focus on the manifold part of the proof of this theorem, reducing it to a "thickening theorem" about a space of PL manifold thickenings of a given polyhedron K.
I will give the inductive proof of the thickening theorem, using Hatcher's principle of global transversality in patches. This is joint work with Jahren and Waldhausen.
I will show how to classify free topological involutions on S^1xS^n up to conjugacy. This is done in three steps:
This is joint work with Slawomir Kwasik. A preprint is posted as http://arxiv.org/abs/0802.2035
I will present Ganter and Kapranov's character theory for 2-representations, their result that it produces 2-characters in the sense of Hopkins-Kuhn-Ravenel, and, if time permits, work an example or two.
(A continuation of Tuesday's talk.)
I will talk about two slightly different ways to exploit MU, and in particular the fact that it is a commutative S-algebra/E_\infty ring spectrum.
Bass defined an exotic Nil-summand of the algebraic K-theory of a polynomial extension. Later, Waldhausen extended the definition to tensor algebras and defined an exotic Nil-summand of the algebraic K-theory of an injective amalgam of groups.
The Nil-Nil theorem states, under a certain finiteness condition, that there is a natural isomorphism from the amalgam Nil to a tensor Nil. An important application is that the Farrell-Jones conjecture in algebraic K-theory can be sharpened from the family of virtually cyclic subgroups to the family of finite-by-cyclic subgroups of a discrete group G.
This is joint work with J.F. Davis and A.A. Ranicki. Time permitting, we may discuss current work on the L-theoretic version, generalizing the UNil-NL theorem of Connolly-Ranicki.
The Seiberg-Witten invariants are defined for closed, oriented, homology oriented, connected smooth 4-manifolds X with a spin-c structure, provided b_2^+ > 1. I will show that if in addition X contains a (smoothly embedded) non-separating rational homology 3-sphere Y then the Seiberg-Witten invariant of X can be expressed as the Lefschetz number of the endomorphism W_* of the reduced monopole Floer homology of Y, where W is the cobordism obtained by cutting X open along Y. I then use this Lefschetz number to extend the Seiberg-Witten invariant to 4-manifolds X with b_1=1 and b_2^+=0, assuming X contains some non-separating rational homology 3-sphere Y. If Y is an integral homology sphere, then the Floer homology h-invariant of Y is also an invariant of X (i.e. independent of Y).
Thursday August 28th at 14.15 - 16.00 in room B63
STEFFEN SAGAVE:
Characterizations of topological logarithmic structures
(Note the unusual weekday.)
Wednesday September 10th at 14.15 - 16.00 in room B63
JOHN ROGNES:
Half a magnetic monopole
(Joint work with Christian Ausoni and Bjørn Ian Dundas.) We show that when the gerbe μ representing a magnetic monopole is viewed as a virtual 2-vector bundle, then it decomposes, modulo torsion, as two times a virtual 2-vector bundle σ. We therefore interpret σ as representing half a magnetic monopole.
Wednesday September 17th at 14.15 - 16.00 in room B63
THOMAS KRAGH:
The Stable Homotopy Type of Floer Homology, I
In this first part we give an overview of the topics involved in understanding and defining Floer homology. We then give a description of Floer homology in trivial cases. If time permits we show how to construct a CW-spectrum instead of just the chain complex defining the Floer homology in these trivial cases.
Wednesday September 24th at 14.15 - 16.00 in room B63
PAUL ARNE ØSTVÆR:
The Bott inverted infinite projective space is homotopy algebraic K-theory
Joint work with Markus Spitzweck. We show a motivic stable weak equivalence between the Bott inverted infinite projective space and homotopy algebraic K-theory: http://www.math.uio.no/~paularne/bott.pdf
Wednesday October 1st at 14.15 - 16.00 in room B63
THOMAS KRAGH:
The Stable Homotopy Type of Floer Homology, II
Monday October 6th at 12.15 - 14.00 in room B91
TORE A. KRO (Sarpsborg):
Two-categorical bundles and their classifying spaces
Wednesday October 8th at 14.15 - 16.00 in room B63
STEFFEN SAGAVE (Münster):
Units of ring spectra and topological logarithmic structures
Wednesday October 15th at 14.15 - 16.00 in room B63
THOMAS KRAGH:
The Stable Homotopy Type of Floer Homology, III
Monday October 27th at 12.15 - 14.00 in room B91
CHRISTIAN SCHLICHTKRULL (Bergen):
Higher topological Hochschild homology of Thom spectra
Wednesday November 5th at 14.15 - 16.00 in room B63
JOHN ROGNES:
Topological logarithmic geometry 101
I'll discuss logarithmic topological André--Quillen homology, starting with logarithmic structures in algebra. Later talks will be about logarithmic topological Hochschild homology and logarithmic topological cyclic homology.
Wednesday November 12th at 14.15 - 16.00 in room B63
JOHN ROGNES:
Topological logarithmic geometry 102
I'll discuss logarithmic topological André--Quillen homology, continuing with logarithmic structures in topology.
Wednesday November 19th at 10.15 - 11.15 in auditorium 4, VB
Prof. ERNST HEINTZE (Augsburg):
Symmetric spaces and Kac-Moody algebras
Wednesday November 19th at 11.30 - 12.30 in auditorium 4, VB
HANS JAKOB RIVERTZ (Trondheim):
On real isometric immersions of CP^{2} into CP^{3}
Monday November 24th at 12.15 - 14.00 in room B91
JOHN ROGNES:
Topological logarithmic geometry 103
I'll discuss logarithmic topological André--Quillen homology, proceeding to logarithmic Kähler differentials.
Thursday November 27th and Friday November 28th
Wednesday December 10th at 14.15 - 16.00 in room B62
ANDREW DU PLESSIS (Århus):
Maps of finite codimension
A smooth map f between smooth manifolds is of finite codimension if its orbit under the natural action of diffeomorphisms of source and target is of finite codimension. Considerable effort needs to be made to see that this is meaningful - the upshot is that, at least when the source manifold is compact, there is a wide range of dimensions for the source and target manifolds where maps of finite codimension are ubiquitous, in the sense that any finite-dimensional family of maps can be arbitrarily closely approximated by a family all of whose elements are of finite codimension.
This makes the geometry of maps of finite codimension very interesting. It turns out that, topologically, there are locally only finitely many types of geometry for these maps, and that this geometry can be expressed in terms of well-behaved stratifications.