Talk_id  Date  Speaker  Title 
19636

Tuesday 9/17 3:00 PM

Honghao Gao, MSU

Legendrian knots and augmentation varieties
 Honghao Gao, MSU
 Legendrian knots and augmentation varieties
 09/17/2019
 3:00 PM  4:00 PM
 C204A Wells Hall
We begin with a gentle introduction to Legendrian knot and its invariant theory. We will define the ChekanovEliashberg different graded algebra and augmentations associated to the dga. We also present an example where the augmentation variety is a cluster variety.

19647

Tuesday 9/24 3:00 PM

Honghao Gao, MSU

Legendrian knots and augmentation varieties
 Honghao Gao, MSU
 Legendrian knots and augmentation varieties
 09/24/2019
 3:00 PM  4:00 PM
 C204A Wells Hall
We begin with a gentle introduction to Legendrian knot and its invariant theory. We will define the ChekanovEliashberg different graded algebra and augmentations associated to the dga. We also present an example where the augmentation variety is a cluster variety.

19667

Tuesday 10/1 3:00 PM

Daping Weng, MSU

Double BottSamelson cell and the moduli space of microlocal rank1 sheaves
 Daping Weng, MSU
 Double BottSamelson cell and the moduli space of microlocal rank1 sheaves
 10/01/2019
 3:00 PM  4:00 PM
 C204A Wells Hall
Shende, Treumann, and Zaslow gave a combinatorial description of the moduli space of microlocal rank1 sheaves in their paper “Legendrian Knots and Constructible sheaves”. Following a result of Guillermou, Kashiwara, and Schapira, this moduli space is an invariant of Legendrian links. In this talk, I will review the definition and the cluster structure on the (undecorated) double BottSamelson cells, and show that in the cases of positive braids of Dynkin type A_r, the undecorated double BottSamelson cells are isomorphic to moduli spaces of microlocal rank1 sheaves associated to the corresponding braid closures. As a corollary, the undecorated double BottSamelson cells of Dynkin type A_r are also Legendrian link invariants for positive braid closures. If time allows, I will also talk about how to count F_q points on the undecorated double BottSamelson cells.

19679

Tuesday 10/8 3:00 PM

Daping Weng, MSU

Double BottSamelson cell and the moduli space of microlocal rank1 sheaves
 Daping Weng, MSU
 Double BottSamelson cell and the moduli space of microlocal rank1 sheaves
 10/08/2019
 3:00 PM  4:00 PM
 C204A Wells Hall
Shende, Treumann, and Zaslow gave a combinatorial description of the moduli space of microlocal rank1 sheaves in their paper “Legendrian Knots and Constructible sheaves”. Following a result of Guillermou, Kashiwara, and Schapira, this moduli space is an invariant of Legendrian links. In this talk, I will review the definition and the cluster structure on the (undecorated) double BottSamelson cells, and show that in the cases of positive braids of Dynkin type A_r, the undecorated double BottSamelson cells are isomorphic to moduli spaces of microlocal rank1 sheaves associated to the corresponding braid closures. As a corollary, the undecorated double BottSamelson cells of Dynkin type A_r are also Legendrian link invariants for positive braid closures. If time allows, I will also talk about how to count F_q points on the undecorated double BottSamelson cells.

20682

Tuesday 10/15 1:05 PM

Leonid Chekhov, MSU

FenchelNielsen coordinates on Riemann surfaces and cluster algebras
 Leonid Chekhov, MSU
 FenchelNielsen coordinates on Riemann surfaces and cluster algebras
 10/15/2019
 1:05 PM  2:05 PM
 C304 Wells Hall
It is a 30(at least)year old subject: it is known since long that both the standard FenchelNIelsen (lengthstwists) coordinates and (Y)cluster coordinates (if we have holes) result in the same Goldman bracket on the set of geodesic functions on Riemann surfaces. The proof (of "local" nature in the first case and of "global" in the second) implies that these two sets of coordinates realise the same Poisson algebra. Nevertheless, constructing a direct transition between these two sets was elusive mainly due to complexity of the transition. For a sphere with 4 holes and torus with one hole, the corresponding formulas were obtained by Nekrasov, Rosly and Shatashvili in 2011. I present some preliminary results on the corresponding algebras in the general case and discuss possible relations to objects called YangYang functionals.

20686

Tuesday 10/22 3:00 PM

Leonid Chekhov, MSU

FenchelNielsen coordinates on Riemann surfaces and cluster algebras
 Leonid Chekhov, MSU
 FenchelNielsen coordinates on Riemann surfaces and cluster algebras
 10/22/2019
 3:00 PM  4:00 PM
 C204A Wells Hall
It is a 30(at least)year old subject: it is known since long that both the standard FenchelNIelsen (lengthstwists) coordinates and (Y)cluster coordinates (if we have holes) result in the same Goldman bracket on the set of geodesic functions on Riemann surfaces. The proof (of "local" nature in the first case and of "global" in the second) implies that these two sets of coordinates realise the same Poisson algebra. Nevertheless, constructing a direct transition between these two sets was elusive mainly due to complexity of the transition. For a sphere with 4 holes and torus with one hole, the corresponding formulas were obtained by Nekrasov, Rosly and Shatashvili in 2011. I present some preliminary results on the corresponding algebras in the general case and discuss possible relations to objects called YangYang functionals.

20692

Tuesday 10/29 3:00 PM

Alexander Shapiro, UC Berkeley

Character varieties, Coulomb branches, and clusters.
 Alexander Shapiro, UC Berkeley
 Character varieties, Coulomb branches, and clusters.
 10/29/2019
 3:00 PM  4:00 PM
 C204A Wells Hall
Quantum groups admit two different geometric realizations: as quantized character varieties and as quantized Coulomb branches of certain gauge theories. These realizations endow a quantum group with two, a priori different, cluster structures. In this talk I will show these structures, explain why they coincide, and say what they have to do with GelfandTsetlin subalgebras, higher rank Fenchel–Nielsen coordinates, and modular functor from higher Teichmüller theory. This talk will be based on joint works with Gus Schrader.

19648

Tuesday 11/5 3:00 PM

Dan Rutherford, Ball State University

Normal rulings and augmentation varieties
 Dan Rutherford, Ball State University
 Normal rulings and augmentation varieties
 11/05/2019
 3:00 PM  4:00 PM
 C204A Wells Hall
Normal rulings are combinatorial structures associated to the front diagrams of 1dimensional Legendrian knots in R^3. They were introduced independently by Fuchs and ChekanovPushkar in the context of augmentations of the Legendrian DGalgebra and generating families. In this talk I will present joint work with B. Henry in which we construct a decomposition of the augmentation variety into disjoint pieces indexed by normal rulings. The pieces of the decomposition are products of algebraic tori and affine spaces with dimensions determined by the combinatorics of the ruling. As a consequence, the ruling polynomial invariants of ChekanovPushkar are seen to be equivalent to augmentation number invariants defined by counting augmentations to finite fields. The construction of the decomposition is based on considering Morse complex sequences which are combinatorial analogs of generating families.

20708

Tuesday 11/19 3:00 PM

Honghao Gao, MSU

Applications of augmentations in contact topology
 Honghao Gao, MSU
 Applications of augmentations in contact topology
 11/19/2019
 3:00 PM  4:00 PM
 C204A Wells Hall
Chekanov introduced a differential graded algebra as an invariant for Legendrian knots in standard contact manifold R^3. An augmentation is a rank 1 representation of the dga. Augmentations are accessible invariants, and the moduli space of augmentations carries important properties from both algebraic and geometric perspectives. In this talk, I will review some problems in contact topology and discuss the applications of augmentations.
