Geometry Working Seminar 15

In Spring 2015 the working seminar will consist of two parts:

1.) There will be a seminar on buildings

2.) Members of the Geometry group will present their research.

The seminar is held weekly on Wednseday between 1-3 pm in MAA110.

References:

Books:

• Abramenko, Brown: Buildings
• Garret
• Weiss: The structure of spherical buildings, The structure of affine buildings
• Ronan, M.: Lectures on buildings

Review articles:

• Rousseau, G.: Euclidean buildings
• Everitt, B.: A (very short) introduction to buildings

​"Target" articles:

• Parreau, A.: Compactification d’espaces de représentations de groupes de type fini
• Kleiner, B., Leeb, B.: Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings
• Parreau, A.: Immeubles affines : construction par les normes et etude des isometries
• Maubon, J.: Symmetric spaces of the non-compact type: differential geometry

 

Seminars:

 

18 Februray 2015

• Overview of seminar

 

25 Februray 2015

• D. Jetchev: Coxeter Groups, Chamber and Coxeter Complexes, Buildings Axioms 

Abstract: I will first review some basic notions from Coxeter groups such as Coxeter groups, Coxeter diagrams, Tits' linear representation, roots, length functions and reflections. We will then discuss chamber and Coxeter complexes and prove some basic facts about those. I will define the gallery distance, foldings and half-apartments and prove the convexity of the latter. I may start discussing the building axioms (if time permits).  

 

4 March 2015

• D. Jetchev:Chamber complexes and Coxeter complexes 

Abstract: I will continue with the discussion of chamber and Coxeter complexes and prove some basic facts about those. I will define the gallery distance, foldings and half-apartments and prove the convexity of the latter. If time permits, I will present the axioms for buildings. 

 

11 March 2015

• D. Jetchev: Introductory lecture on buildings (III)

 

18 March 2015

• D. Jetchev: Introduction to buildings (IV)
  
   
• A. Lachowska: "The Kashiwara-Vergne and Grothendieck-Teichmuller Lie algebras agree in depth 2.
  The Grothendieck-Teichmuller Lie algebra encodes symmetries of the Drinfeld associators.Similarly, the Kashiwara-Vergne Lie algebra encodes symmetries of the solutions of the Kashiwara-Vergne problem on the properties of Cambpell-Haussdorf series. Both algebras admit presentation in terms of Lie words in two variables (x,y) satisfying certain relations, and both are filtered by depth - the total number of y’s in a Lie word. Conjecturally, the two Lie algebras are essentially isomorphic. We show that the conjecture holds for the associated depth-graded slices of depth 2. This is a joint work with A. Alekseev and E. Raphael (University of Geneva). “ 

 

25 March 2015

• H. Nakajima

 

1 April 2015

• W. Wong: Symmetric spaces and buildings
  Notes

 

15 April 2015

• N. Proudfoot: Category O and symplectic duality

  In the first lecture, I will define category O for a symplectic resolution, with special attention to the Springer resolution.  In the second, I will give an introduction to the symplectic duality program of Braden, Licata, Webster, and myself.  This is closely related to the relationship between Higgs and Coulomb branches of physical field theories (about which you may have heard recently).

• Johan Martens: Quantum representations and higher-rank Prym varieties.

  Riemann’s moduli space of curves can naturally be equipped with a range of bundles, whose fibres are spaces of non-abelian theta functions or, equivalently, spaces of conformal blocks.  These bundles come naturally equipped with flat projective connections, in many ways mirroring an old story for (abelian) theta functions, who were classically known to satisfy a heat-equation.  In some aspects however the non-abelian theta functions behave quite differently, most clearly exhibited when considering the projective representations of the mapping class group they give rise to.  For a few sporadic, low-level versions this difference brakes down though, a phenomenon best understood through strange duality.  In this talk we will describe the situation for rank 4, where the situation gets clarified by thinking about higher-rank Prym varieties.  This is joint ongoing work with T. Baier, M. Bolognesi and C. Pauly.

 

22 April 2015

 

• N. Proudfoot: Category O and symplectic duality

  In the first lecture, I will define category O for a symplectic resolution, with special attention to the Springer resolution.  In the second, I will give an introduction to the symplectic duality program of Braden, Licata, Webster, and myself.  This is closely related to the relationship between Higgs and Coulomb branches of physical field theories (about which you may have heard recently).

• A. Noll: R-Buildings

 

29 April 2015

 

• N. Proudfoot: Quantum cohomology at q=1

I will discuss ongoing work with Michael McBreen about the relationship between the quantum cohomology of a symplectic resolution and the intersection cohomology of the singular space that it resolves

• P. Samuelson: Character varieties, Hecke algebras, and knots

  The spherical double affine Hecke algebra SH(g; q,t) is an algebra depending on a Lie algebra g and two parameters q, t in C^*. When q=1 it is the ring of functions on a variety X(g;t), and the Dunkl embedding gives a rational map X(g;t=1) - - -> X(g; t). For g = sl_2 (and probably more), this gives a "wrong way" rational map from the

  SL_2 character variety of the torus T^2 to the character variety of the punctured torus T^2 \ p. We will briefly describe a conjecture with Berest that the restriction map from the character variety of a knot complement to X(sl_2, t=1) extends to X(sl_2, t), and will describe the Brumfiel-Hilden algebra, which is a useful tool for dealing with SL_2 character varieties in general.

 

13 May 2015

• Z. Chen: Teichmuller space and its compactification.

  In the first part of the talk, we will give several equivalent definitions of Teichmuller space and explain Teichmuller theorem from the hyperbolic point of view. In the second part, we’ll explain Thurston’s compactification of Teichmuller space, and the generalisation of Morgan-Shalen to certain character variety. If we have time, we’ll explain the later development due to Daskalopoulos, Dostoglou and Wentworth.

 

 

20 May 2015

• J. Kamnizer: Understand the link between quiver varieties and affine Grassmannians

  There are two geometric realizations of representations of simply-laced semisimple Lie algebras.  The first uses Nakajima quiver varieties and the second uses affine Grassmannians.  For a number of years, I have been trying to understand the relationship between these constructions.  I will explain the various progress to date and explain how we are getting closer to a solution to this problem, thanks to the recent work of Braverman, Finkelberg, Nakajima.