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• October 19, 2009. Fred E.J. Linton, Wesleyan University

  
  

  Title: Something Old, Something New, Something Borrowed, ...

  
  Abstract: I talk about a peculiar topological space that Peter Johnstone and Bob Paré and I used some 30 years ago to solve a problem in topos theory.
  Remarks on selected words or phrases above:
  "Peculiar"? it has a discrete, dense, open subset, whose (closed) complement is a discrete subspace as well,
  yet ...
  "Problem in topos theory"? ... the whole space admits a double cover with no continuous global section;
  and
  "Some 30 years ago"? offered in the Proceedings of a 1978 Berlin categories get-together.
  Remarks on the choice of title:
  "old"? as said, the work cited dates back to 1978.
  "new"? the main proof offered here is a new cardinality-counting argument, rather than the Baire category reasoning of 30-odd years ago.
  "borrowed"? it's the base space that is borrowed: that's Heath's V-space, and is rather much older than 30 years.
  
  Seemed like the sort of mix of areas seminar participants might appreciate. And the pictures may be a pleasant diversion, too.
  
  
  

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• October 26, 2009. Louis Thrall, The Graduate Center, CUNY

  
  

  Title: Category theory, Combinatorial Homotopy Theory and Coverings of "Spaces".

  
  Abstract: I will begin by briefly reviewing the definition of Cech coverings of a space and the associated nerve. There is then a theorem that states that the geometric realization of the nerve and the original space are weakly equivalent. This may then be generalized to Grothendieck topologies, if one uses hypercovers. Some immediate applications to algebraic geometry can then be stated. In the course of this talk, we will flesh out what all of this means. We end with some concrete examples of this in practice: models of the S^2 and CP^n will be shown.
  
  
  

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• November 2, 2009. Louis Thrall, The Graduate Center, CUNY

  
  

  Title: Category theory, Combinatorial Homotopy Theory and Coverings of "Spaces" Part II.

  
  Abstract: I will begin by briefly reviewing the definition of Cech coverings of a space and the associated nerve. There is then a theorem that states that the geometric realization of the nerve and the original space are weakly equivalent. This may then be generalized to Grothendieck topologies, if one uses hypercovers. Some immediate applications to algebraic geometry can then be stated. In the course of this talk, we will flesh out what all of this means. We end with some concrete examples of this in practice: models of the S^2 and CP^n will be shown.
  
  
  

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• November 23, 2009. Alex Hoffnung, University of California, Riverside

  
  

  Title: The categorified Hecke algebra and the Zamolodchikov tetrahedron equation

  
  Abstract: We will look at a categorification of the Hecke algebra and its connections to incidence geometries and buildings. Further, we will see how this construction gives rise to solutions of the Zamolodchikov tetrahedron equation.
  
  
  

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• December 7, 2009. Dustin Mulcahey, The Graduate Center, CUNY

  
  

  Title: Towards a Change of Comonads Theorem

  
  Abstract: The issue of computing derived functors with respect to some comonad is a central one in unstable homotopy theory. In particular, the E2 term of any variant of the unstable Adams spectral sequence comes about this way. Here, I present the following question, "When do two comonads (over possibly different categories) yield the same derived functors?" I will go over some progress in answering this question. The discussion will be mostly categorical in nature, though the ultimate application is in unstable BP theory.
  
  
  

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• December 14, 2009. Rick Jardine, The University of Western Ontario

  
  

  Title: Cocycle categories

  
  Abstract: The set of maps between objects in a homotopy category can be identified with path components of a category of cocycles, in great generality. The applications to date have been predominantly consisted of homotopy classification results in non-abelian cohomology theory. Cocycle category methods lead to an expanded (and interesting) version of the Verdier hypercovering theorem. Cocycle categories might also be used to give a new description of etale homotopy theory, which would involve neither hypercovers nor pro-objects.
  
  Rick will also be speaking on Concurrency on Thursday December 10th, 2009 (4:15-5:30) at the The Computer Science Colloquium
  
  
  

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• Spring 2009

• Cambridge

• Montreal

• Sydney

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