Topology Seminars Spring 2013

Seminars will begin this Semester on Monday 28 January. They will be held at 3.00pm in the Frank Adams 2 seminar room, preceded by high-class tea and biscuits from 2.30pm on the atrium bridge. There will be a vist to Sandbar after each meeting, for further discussion and refreshments.

• Monday 28 January
  2013 Cobordism 1; basic ideas and examples
  Nigel Ray (University of Manchester)
  
  3pm FRANK ADAMS 2
  Abstract (click to view)

  This is the first of two talks that are designed to introduce a geometrically-curious audience to cobordism. Since the days of Pontryagin and Thom in the 1940s and 50s, cobordism theory has developed as an amalgam of homotopy theory and differential topology. Over the last 20 years or so, it has spread into algebraic and symplectic geometry, and further afield, via topological quantum field theory into theoretical physics. By way of introduction, I shall focus mainly on examples of smooth manifolds without additional structure, and related cobordism invariants such as Stiefel-Whitney numbers. The aim is to outline Thom's original computation of the cobordism ring (which led to his 1958 Fields medal) in down-to-earth terms. As usual, I shall give preference to philosophy and examples, over details and precise truth!

• Monday 04 February
  2013 Cobordism 2; beyond the complex case
  Nigel Ray (University of Manchester)
  
  3pm FRANK ADAMS 2
  Abstract (click to view)

  This is the second of two talks that are designed to introduce a geometrically-curious audience to cobordism. In the 1960s, Milnor and Novikov (both Fields medallists!) independently generalised Thom's work to complex cobordism, which currently underlies some of the most powerful tools of homotopy theory. Beyond the complex case lie quaternionic and framed cobordism, which, to this day, host many unsolved problems - and bring the wheel full circle, and back to Pontryagin. The aim is to provide an accessible introduction to a selection of these delights, such as the twisted quaternionic structure on the 5-sphere and the twisted framing on the circle. As usual, I shall give preference to philosophy and examples, over details and precise truth!

• Monday 11 February
  2013 Configuration spaces and homological stability
  Martin Palmer (Oxford)
  
  3pm FRANK ADAMS 2
  Abstract (click to view)

  First of all, I will give an overview of what the phenomenon of homological stability is and why it's useful, with plenty of examples. I will then specialise to the case of configuration spaces -- of various different kinds -- and explain some of the key ideas involved in proving that these sequences of spaces are homologically stable. A configuration here can be more than just a finite collection of points in a background space: in particular, the points may be equipped with a certain non-local structure (an orientation), or one can consider unlinked embedded copies of a fixed manifold instead of just points. If time permits, I may also say something about homological stability with local coefficients, in general and in particular for configuration spaces.

• Monday 18 February
  2013 Geometry and topology of infinite-dimensional bundles and the caloron correspondence
  Raymond Vozzo (Glasgow and Adelaide)
  
  3pm FRANK ADAMS 2
  Abstract (click to view)

  The caloron correspondence is a method for translating between principal bundles with structure group an infinite-dimensional Lie group and bundles with finite-dimensional structure group. Crucially, it also allows us to describe the geometry of these infinite-dimensional bundles in terms of finite dimensional data. In this talk I will describe this correspondence and give an indication of some of the results that have so far been obtained using it.

• Monday 25 February
  2013 Towards the algebraic model of rational equivariant cohomology theories
  Magdalena Kedziorek (Sheffield)
  
  3pm FRANK ADAMS 2
  Abstract (click to view)

  Cohomology theories are represented by spectra. However the category of spectra is quite complicated. The machinery of model categories allows us to look for different (easier, algebraic) models with the same homotopy information as the category of spectra. It is well known that rational chain complexes give an algebraic model for the rational cohomology theories. However, for G-equivariant cohomology theories, no general result of this form is known when G is a compact Lie group.
  In this talk I will assume only basic knowledge of algebraic topology and I will remind the audience of the ideas of spectra and model structures. I will describe some earlier work and introduce a framework for constructing algebraic models in general. It is conjectured that the models will take the form of sheaves of modules over a topological category of subgroups of G.

• Monday 04 March
  2013 G-topologies explained from a different perspective 1
  Harold Simmons (Manchester)
  
  3pm FRANK ADAMS 2
  Abstract (click to view)

  There are two kinds of G-topologies each of which is used in a certain setting. (1) Each Gabriel topology produces a quotient of the category ModR of modules over a given ring R; the result can also be viewed as a sub-category. (2) Each Grothendieck topology produces a quotient of the category C^ of the pre-sheaves on a given category C; the "experts" view this as a producing a sub-category. The two kinds of G-gadgets and the associated constructions are very similar, but this is rarely mentioned in the literature.
  In the talks I will explain (i) what each G-topology is (ii) why "topology" is the wrong word, since the gadgets have almost no connections with point-set topologies (iii) a more general notion, of each kind, which is easier to understand and use to generate a G-gadget (iv) an equivalent notion which is even easier to understand and use. I will try to emphasize the module and the pre-sheaf parallels.
  The talks should interest algebraist, topologists, and geometers, and I will try to use simple examples from all three areas. They can be seen as companions to the talks by Amit Kuber, but you need not have attended those talks.

• Monday 11 March
  2013 G-topologies explained from a different perspective 2
  Harold Simmons (Manchester)
  
  3pm FRANK ADAMS 2
  Abstract (click to view)

  See 04 March

• Monday 18 March
  2013 Enzymes that unknot DNA; a mathematical study
  Julian Gibbons (Imperial)
  
  3pm FRANK ADAMS 2
  Abstract (click to view)

  In certain organisms, such as bacteria, the axis of the DNA double helix is closed and may therefore be knotted. If this occurs, it is generally harmful to the organism, so enzymes have evolved to unknot the DNA. In this talk, I shall discuss the mathematical methods used to study this sort of phenomenon, mentioning in particular how these techniques generalise to a new Heegaard Floer obstruction to Dehn surgeries in S^3. If time permits, I shall also extend this discussion to a broader class of enzymes.

• Monday 15 April
  2013 Characteristics of commutative S-algebras
  Andrew Baker (Glasgow)
  
  3pm FRANK ADAMS 2
  Abstract (click to view)

  The notion of characteristic of a (commutative) ring is a basic one. In derived settings it is less clear what a correct definition should be. Following work by Markus Szymik, I'll give a definition in the setting of p-local commutative S-algebras (aka E-infinity ring spectra) that satisfies reasonable properties such as homotopy invariance. Then I will discuss some examples which involve very classical stable homotopy theory and an extension of Joel Cohen's old work on matric Toda brackets/Massey products. At the prime 2, ku, ko, tmf and MSp will make appearences.

• Monday 22 April
  2013 An overview of Dehn surgery
  Fionntan Roukema (Sheffield)
  
  3pm FRANK ADAMS 2
  Abstract (click to view)

  This talk will give an overview of some classical results from Dehn surgery. We will introduce Dehn surgery and use the mapping class group of an orientable genus g surface to indicate why every closed connected orientable 3-manifold is obtained by surgery on a link. We then restrict our attention to surgeries on links whose exteriors are simple but whose surgeries are not simple; this will bring us to the statement of some very important results due to Thurston and Perelman. The talk will be informal and contain many pictures that will hopefully describe a classical and beautiful story.

• Monday 29 April
  2013 TBA
  TBA
  
  3pm FRANK ADAMS 2
  Abstract (click to view)

Further information

For further information please contact the seminar organiser.