2010

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Talk Summaries

Day 1

  • Survey 1 -- Constantin Teleman, UC Berkeley. ~60 min. 2d TQFTs, string topology, finite group gauge theory, K_G(G), relation to holomorphic bundles
  • Introduction to K-theory -- Jesse Wolfson, Northwestern. ~75 min. Generalized cohomology. K-theory: Vector bundles, BU, Bott periodicity, Thom isomorphism, Gysin maps. Chern character, topological Riemann-Roch. Notes posted here
  • More K-theory -- Chris Kottke, MIT. ~75 min. Fredholm and GL(res) models of BU. C* algebra K theory and K-homology (quickly). Clifford algebras and odd K-theory. Dirac operators, Gysin map as the index. Hopefully, relation between Dirac and Dolbeault, and Spin^c vs complex orientations. Notes posted here
  • Twisted K-theory -- Mehdi Sarikhani Khorami, Wesleyan. ~60 min. Units and twistings in cohomology theories. Special twistings of K from line bundles and gradings. Twisted K (by graded lines) with examples. C* algebra model for twisted K via Azumaya algebras. Twisted Chern isomorphism (statement only). Notes by Daniel Berwick-Evans posted here
  • More twistings -- Braxton Collier, University of Texas. ~45+ min. Distinction between twistings and their characteristic class (Example in computations). Category of twistings, Projective bundle model, gerbes. Gradings? Chern-Weil construction of Chern character? (informal) Could extend this to an optional discussion.

Day 2

  • Equivariant K-theory and its twisted versions -- Mio Alter, University of Texas. ~75 min. Vector bundle and C* algebra models in untwisted case. Completion theorem. Atiyah-Segal construction of twisted equivariant K-theory. C* (Azumaya) algebra construction. Notes posted here
  • Twisted equivariant Chern character -- Owen Gwilliam, Northwestern. ~75 min. Generalities. Delocalised Chern character. Examples. Twisted version. Computation of twisted complex K_G(G) \otimes C. (Time permitting but unlikely: the graded case, computation for G=SO(3)) Notes by Daniel Berwick-Evans posted here
  • Twisted K_G(G) -- Daniel Halpern-Leistner, UC Berkeley. ~75 min. Computation in simply connected case (or free \pi_1) case. General (additive) description in terms of affine Weyl group. Beginnings of TFT structure: transgression of twistings from H^4 and the Pontrjagin product.
  • K-theory of Topological Stacks -- Ryan Grady, Notre Dame. ~75 min. K-theory of topological stacks. Example of local-but-not global quotient stacks: gerbes as topological stacks. Stack description of twisted K. Computation of S^1-equivariant K_G(G), induction from Rep(G\times S^1) and appearance of the Kac numerator. Notes posted here

Optional discussion/example session?

Day 3

  • Loop groups and their PER's -- Harold Williams, UC Berkeley. ~75 min. Case of a torus. Affine Weyl group, flag variety. Borel-Weil construction of irreps, for compact groups and for loop groups by holomorphic induction on co-adjoint orbits. Parametrization of irreps by affine Weyl action on weights.
  • Character formulae -- Dario Beraldo, UC Berkeley. ~60 min. Weyl-Kac and Kirillov formulae for compact groups and for loop groups. [Mention: Character of the Spin representation.] Notes posted here
  • Dirac family construction of K-classes -- Sander Kupers, Utrecht. ~70 min. [Could be split in 2 short lects] Construction of K_T(T) classes by spectral flow, description in terms of Heisenberg-like groups based on a torus. Dirac operator on a compact group; algebraic description via Peter-Weyl, Dirac family on g and K-class. Outline of the construction for loop groups. Notes posted here and here

Day 4 (morning is tentative)

  • 2-tier TQFT structure -- AJ Tolland, SUNY Stony Brook. ~75 min. Construction of TQFT operations via correspondence spaces. Trace on twisted K_G(G) and topological Peter-Weyl. Verlinde formula as index formula in twisted K-theory. Higher twistings of K and more general index formula. Notes by Daniel Berwick-Evans posted here
  • Survey 2: Known and unfinished business -- Constantin Teleman, UC Berkeley. ~75 min.
  • Open-closed 2d field theories -- Matt Young, SUNY Stony Brook. ~75 min. After Kontsevich, Costello, Hopkins-Lurie, Discuss Hochschild homology, etc Notes posted here
  • B-model of a singularity as open-closed theory -- Kevin Lin, UC Berkeley. ~60 min. Landau-Ginzburg potentials, category of branes, Frobenius property
  • Twisted K_G(G) as open closed theory -- Constantin Teleman, UC Berkeley. ~60 min.

Day 5

  • Chern-Simons theory as 3-2-1 theory -- Hiro Tanaka, Northwestern. ~75 min. (after Witten, Reshetikhin-Turaev etc). Discussion of braided tensor categories. Notes by Daniel Berwick-Evans posted here
  • Chern-Simons for a torus via the categorified group ring -- Konrad Waldorf, UC Berkeley. ~60 min. Notes posted here
  • Chern-Simons for nonabelian G via conformal nets -- Chris Douglas, UC Berkeley (unconfirmed). ~60 min.
  • Elliptic cohomology -- Nick Rozenblyum, MIT. ~50 min. Informal introduction, restriction to the Tate curve and K-theory of loop space. Twistings of elliptic cohomology from H^4.
  • Equivariant elliptic cohomology and loop group reps -- Toly Preygel, MIT. ~50 min.

(The organizers would like to thank Daniel Berwick Evans and Jesse Wolfson for taking notes and drawing pictures for all the talks, and to all the speakers for editing these notes.)

References

Pre-Talbot Seminars

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