Topology Seminars Spring 2008

• 4 Feb
  2008 Miraculous Cancellation and Pick's Theorem
  Kostya Feldman (University of Cambridge)
  
  Time 4pm - Venue Frank Adams Room 2, Alan Turing
• 11 Feb
  2008 Topological aspects of motion planning
  Mark Grant (University of Durham)
  
  Time 4pm - Venue Frank Adams Room 2, Alan Turing
  Abstract (click to view)

  Inspired by the motion planning problem in robotics, M. Farber recently introduced a new numerical homotopy invariant, called the Topological Complexity, which provides a measure of the navigational complexity of a space when viewed as the configuration space of a mechanical system. As well as its practical motivation, computation of this invariant presents a challenge to Homotopy Theorists, which may be likened to computation of the Lusternik-Schnirelmann category. I will survey some results and examples intended to convey the difficulty and wider significance of the problem, as well as the methods used in effective computations. I will also describe joint work with M. Farber in which cohomology operations are utilised in finding effective lower bounds for the Topological Complexity, and discuss some open problems.

• 18 Feb
  2008 How to use algebraic cellular approximations to better understand the Eilenberg-Moore spectral sequence
  Shoham Shamir (University of Sheffield)
  
  Time 4pm - Venue Frank Adams Room 2, Alan Turing
  Abstract (click to view)

  Given chain-complexes k and M over a ring R, a k-cellular approximation to M is the "closest approximation" of M that can be glued together from copies of suspensions of k. I will discuss this concept, which is due to Dwyer, Greenlees and Iyengar, and how is can be used to study the Eilenberg-Moore cohomology spectral sequence for a fibration.

• 25 Feb
  2008 Rigidity theorems in stable homotopy
  Constanze Roitzheim (University of Sheffield)
  
  Time 4pm - Venue Frank Adams Room 2, Alan Turing
  Abstract (click to view)

  Is a stable model category determined by the triangulated structure of its homotopy category? We will investigate this question for the stable homotopy category localised at p-local complex topological K-theory. At the prime 2, the answer to the above question is yes. However, at odd primes there are algebraic models for this homotopy category that differ from the standard model on higher homotopy level. We will explain the set-up of this and present recent results.

• 3 Mar
  2008 Operations in K-theory, Completions and Discrete Modules
  Martin Crossley (Swansea University)
  
  Time 4pm - Venue Frank Adams Room 2, Alan Turing
  Abstract (click to view)

  The are known to be uncountably many stable operations in K-theory yet only two operations, and linear combinations thereof, are explicitly known. In this talk I will discuss work by Francis Clarke, Sarah Whitehouse and myself aimed at improving this poor state of affairs, at least in the p-local case. We have been able to construct a "basis" for the p-local operations, describing the ring as a certain completion. This facilitates a clean description of a category shown by Bousfield to accurately model K-local spectra. This category, in turn, is very similar to one used by Schwede to understand homotopy groups of symmetric spectra.

• 10 Mar
  2008 Models for the homotopy type of the Deligne-Mumford compactification of the moduli stack of curves
  Jeffrey Herschel Giansiracusa (University of Oxford)
  
  Time 4pm - Venue Frank Adams Room 2, Alan Turing
  Abstract (click to view)

  An old theorem of Charney and Lee says that the category of stable nodal surfaces and isotopy classes of degenerations has a classifying space with the same rational homology as the Deligne-Mumford compactification of the moduli space of curves. We show that this category actually has the same integral homotopy type as the moduli *stack* of stable curves. Along the way we clarify the structure of an atlas for the stack originally constructed by Bers.

• 7 Apr
  2008
  Lionel Schwartz (Paris 13)
  
  Time 4pm - Venue Frank Adams Room 2, Alan Turing
• 14 Apr
  2008 Polynomial maps of spheres
  Reg Wood (University of Manchester)
  
  Time 4pm - Venue Frank Adams Room 2, Alan Turing
• 21 Apr
  2008 CANCELLED
  
  
  Time 4pm - Venue Frank Adams Room 2, Alan Turing
• 28 Apr
  2008 The Face-Polynomial of Nestohedra
  Andrew Fenn
  
  Time 4pm - Venue Frank Adams Room 2, Alan Turing
  Abstract (click to view)

  I will present an introduction to polytopes and an overview of some of their uses. I will then move on to one of the most interesting aspects of polyhedra and present calculations of the face-polynomials of many important series of nestohedra. A nestohedron is a simple polytope which arises from a connected graph. A series of nestohedra is a set of nestohedra arising from a series of graphs which is defined so that each successive graph is obtained from the previous one by the addition of a node that is connected to some subgraph or by replacing a node with two nodes connected by an arc. This raises the problem of how the face polynomials are related to each other both within and between series. Examples of such series of nestohedron include the Permutohedron and the Associohedron or Stasheff polytope. Furthermore this raises the question of how natural operations on the graphs affect the polytopes anf their face-polynomials. The problem of how to describe the faces of a nestohedron in terms of its graph is well known. It is usually attacked using f-vectors. We show that an approach based on the face-polynomial of a polytope is a finer tool, since in the case of the face-polynomial we can transform operations on the graph into analytical techniques, which we cannot do with the f-vector.

• 15 May
  2008 Homotopy decompositions of loop spaces and modular representation theory
  Jie Wu
  
  Time 1pm - Venue Frank Adams Room 2, Alan Turing
  Abstract (click to view)

  In this talk, we will explain a connection between homotopy theory and modular representation theory. The question in homotopy theory is about the natural decompositions of loop spaces of co-H spaces. We will relate this question to the functorial co-algebra decompositions of tensor algebras and also to the modular representation theory of the symmetric group. More precisely, the problem is reduced to the study of the largest projective k[S_n]-submodule of Lie(n) and decompositions of Lie powers over general linear groups.

Further information

For further information please contact the Seminar organiser.