The SGGTC seminar meets on Fridays in Math 407 at 1pm, unless noted otherwise (in red).
Previous semesters: Fall 2016, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007.
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|Perverse sheaves, knot contact homology and relative Calabi-Yau completions|
|Jan. 27-29||FRG Workshop: Crossing the Walls in Enumerative Geometry|
(University of Bonn)
|Representation theory of Coxeter groups:Some first steps|
(University of Notre Dame)
(University of Georgia)
January 20, 2017: Wai-kit Yeung " Perverse sheaves, knot contact homology and relative Calabi-Yau completions"
Abstract: In this talk, I will present joint work [arXiv:1610.02438] with Yu. Berest and A. Eshmatov, where we give a universal construction, called homotopy braid closure, that produces invariants of links in R^3 starting with a braid group action on objects of a (model) category. Applying this construction to the natural action of the braid group B_n on the category of perverse sheaves on the two-dimensional disk with singularities at n marked points, we obtain a differential graded (DG) category that gives knot contact homology in the sense of L. Ng. As an application, we show that the category of finite-dimensional modules over the 0-th homology of this DG category is equivalent to the category of perverse sheaves on R^3 with singularities at most along the link. If time allows, I will also discuss my recent work [arXiv:1612.06352] that generalizes this picture using non-commutative algebraic geometry.
Abstract: This talk is an introduction as well as a survey about 2-representation theory of Coxeter groups. The motivation to study such 'higher' representations is as follows: In groundbreaking work Chuang-Rouquier and, independently, Khovanov-Lauda introduced 2-representation theory of Lie algebras and their quantum analogs. As a 'higher' version of classical representation theory of Lie groups, their ideas have already led to many successive works as well as applications throughout mathematics (and physics). Historically speaking, what `should have come first' is 2-representation theory of finite groups. Sadly the story of `higher' representations of finite groups is not understood at all at the moment. But, quite recently, Mazorchuk-Miemietz made very good progress towards a 'higher' analog of representations of finite-dimensional algebras (as e.g. groups rings of finite groups), and applied it quite successful to one of the most well-behaved family of finite groups: finite Coxeter groups. Already in this case a lot of interesting new phenomena show up, most of which are neither present in classical representation theory nor in the story of Chuang-Rouquier, Khovanov-Lauda, and which might lead to interesting connections and applications in the years to come. However, in this talk we will focus on one (completely explicit) example, i.e. the first non-trivial family of examples given by the dihedral groups, where already several new phenomena are visible. Based on joint work with Marco Mackaay, Volodymyr Mazorchuk and Vanessa Miemiet