• Algebraic K-theory of group rings and topological cyclic homology.
  Wolfgang Lück, Holger Reich, John Rognes, and Marco Varisco.
  In preparation. Abstract

  We use topological cyclic homology and the cyclotomic trace to detect elements in the rationalized higher algebraic K-theory groups of integral group rings. Modulo a deep conjecture in number theory, the so-called Schneider conjecture, and under mild homological finiteness conditions on the group, we prove that the connective algebraic K-theory assembly map for the family of virtually cyclic subgroups is rationally injective. This generalizes a result of Bökstedt, Hsiang, and Madsen, and leads to a concrete description of a large direct summand inside the algebraic K-theory groups of an integral group ring in terms of group homology. Along the way we also prove integral splitting and isomorphism results for THH- and TC-assembly maps with coefficients in arbitrary connective ring spectra.

• Commuting homotopy limits and smash products.
  Wolfgang Lück, Holger Reich, and Marco Varisco.
  K-Theory, 30(2):137–165, 2003.
  [arXiv:math/0302116] [pdf] [djvu] [MR2064237]
  Abstract

  In general the processes of taking a homotopy inverse limit of a diagram of spectra and smashing spectra with a fixed space do not commute. In this paper we investigate under what additional assumptions these two processes do commute. In fact we deal with an equivariant generalization which involves spectra and smash products over the orbit category of a discrete group. Such a situation naturally occurs if one studies the equivariant homology theory associated to topological cyclic homology. The main theorem of this paper will play a role in the generalization of the results obtained by Bökstedt, Hsiang and Madsen about the algebraic K-theory Novikov Conjecture to the assembly map for the family of virtually cyclic subgroups.

• Algebraic L-theory and triangular Witt groups.
  Ph.D. thesis, Münster. Abstract

  The classical Witt groups of quadratic forms have been generalized in algebraic topology, with Wall’s and Ranicki’s L-groups for surgery theory, and in algebraic geometry, with Knebusch’s Witt groups of schemes and Balmer’s Witt groups of triangulated categories with duality.
  We introduce algebraic L-theory groups for so-called pre-triangulated differential graded categories with duality, generalizing and unifying the two aforementioned approaches.

“Mathematics, taught and learned appropriately,
improves the mind and implants good habits of thought.”
George Pólya

Fall 2009

Office hours: MW 3:30–5:00

• Math 461: Topology
  MWF 2:20–3:20 in LN-1406 and R 1:15–2:40 in LN-1408
• Math 590T: Topics in Topology — Graduate Topology Seminar
  R 10:00–11:00 in LN-2205

Summer 2009 (Session II)

Office hours: MTWRF 11:20–12:20

• Math 488/575/590: Topics in Higher Math & Special Topics for Teachers (MAT/MST) — Intro to Modern Geometry
  MTWRF 9:00–11:15 in LN-2205

Binghamton Geometry Topology Seminar

• Topology and Geometry at Münster 2009.
  Münster University (Germany), June 1–5, 2009.
• Topology Festival.
  Cornell University, Ithaca (NY), May 1–4, 2009.
• Visiting the University at Buffalo, and talk in the Colloquium.
  University at Buffalo, SUNY (NY), February 5–6, 2009.

Conferences and Seminars Organized

• Special Session on Geometric Topology at the AMS 2008 Spring Eastern Meeting (with David Rosenthal).
  Courant Institute of Mathematical Sciences, New York University, March 15–16, 2008.
• Geometry and Topology Seminar.
  Binghamton University, since Fall Semester 2007.
• Workshop on Trace Methods in Algebraic K-Theory (with Lars Hesselholt, Wolfgang Lück, and Holger Reich).
  Münster University (Germany), September 29–October 3, 2003.
• Seminar on trace maps (with Holger Reich).
  Münster University (Germany), Oberseminar Topologie, Sommersemester 2001 & Wintersemester 2001/2002.
• Fragestunde/Kleines Seminar & Rothenberge Workshops.
  Münster University (Germany), GK/SFB, 2001–2004.