Talk_id  Date  Speaker  Title 
5130

Thursday 9/21 10:00 AM

Eric Bucher, MSU

What is a cluster algebra?
 What is a cluster algebra?
 09/21/2017
 10:00 AM  10:50 AM
 C304 Wells Hall
 Eric Bucher, MSU
This will be the first talk of the fall cluster algebra seminar. We have some new attendees this semester so we will start with the basics, discussing definitions and examples of cluster algebras.

5141

Thursday 9/28 10:00 AM

Alexander Shapiro, University of Toronto

Positive representations of quantum groups
 Positive representations of quantum groups
 09/28/2017
 10:00 AM  10:50 AM
 C304 Wells Hall
 Alexander Shapiro, University of Toronto
Positive representations are certain bimodules for a quantum group and its modular dual. In 2001, Ponsot and Teschner constructed these representations for U_q(\mathfrak{sl}_2) and proved that they form a continuous braided monoidal category, where the word "continuous" means that a tensor product of two representations decomposes into a direct integral rather than a direct sum. Ten years later, their construction was generalized to all other types by Frenkel and Ip. Although the corresponding categories were braided more or less by construction, it remained a conjecture that they are monoidal. Following a joint work with Gus Schrader, I will discuss the proof of this conjecture for U_q(\mathfrak{sl}_n). The proof is based on our previous result where the quantum group is realized as a quantum cluster \mathcal Xvariety. If time permits, I will outline a relation between this story and the modular functor conjecture in higher Teichmüller theory along with several other applications.

5150

Thursday 10/5 10:00 AM

Leonid Chekov, Michigan State University

Cluster algebras of geometric type I
 Cluster algebras of geometric type I
 10/05/2017
 10:00 AM  10:50 AM
 C304 Wells Hall
 Leonid Chekov, Michigan State University
I will describe how cluster algebras arise in hyperbolic geometry of Riemann surfaces \Sigma_{g,s} with s>0 holes with the identification of cluster variables with Penner's lambda lengths, Xvariables with shear coordinates, and terms of exchange matrices  with coefficients of Poisson relations between the shear coordinates. I also describe sets of geodesic functions and their algebras induced by semiclassical/quantum relations for the shear coordinates.

5151

Thursday 10/12 10:00 AM

Leonid Chekov, Michigan State University

Cluster algebras of geometric type II
 Cluster algebras of geometric type II
 10/12/2017
 10:00 AM  10:50 AM
 C304 Wells Hall
 Leonid Chekov, Michigan State University
In the second part, I generalize the above structures to the case of Riemann surfaces \Sigma_{g,s,n}  R.s. with holes and decorated boundary cusps on hole boundaries. There, an interesting phenomenon occurs: (i) we have to consider generalized cluster transformations; (ii) we can establish 11 correspondence between (extended) shear coordinates and lambdalengths, so we can investigate both Poisson and symplectic structures on the both sets of variables.

5163

Thursday 10/19 10:00 AM

Eric Bucher, MSU

Component Preserving Mutations and Maximal Green Sequences
 Component Preserving Mutations and Maximal Green Sequences
 10/19/2017
 10:00 AM  10:50 AM
 C304 Wells Hall
 Eric Bucher, MSU
In this talk we will outline the study of maximal green sequences for cluster algebras. We will discuss new results which allow one to define component preserving mutations for quivers, and utilize them to create maximal green sequences by considering maximal green sequences for induced subquivers. This is ongoing work with John Machacek and three undergraduate researchers here at Michigan State University, Ethan Zewde, Evan Runberg, and Abe Yeck

6168

Tuesday 10/24 10:00 AM

Gregg Musiker, University of Minnesota

Beyond Aztec Castles and from Dungeons to Dragons
 Beyond Aztec Castles and from Dungeons to Dragons
 10/24/2017
 10:00 AM  10:50 AM
 C304 Wells Hall
 Gregg Musiker, University of Minnesota
In this talk I will discuss certain tessellations of the torus, known as brane tilings, and highlight how cluster algebras give rise to connections between combinatorics and physics. In particular, I will focus on algebraic and combinatorial formulas arising from quivers associated to del Pezzo surfaces and how such formulas allow us to investigate anew perfect matching problems of Propp from the turn of the century.

6174

Thursday 11/2 10:00 AM

Leonid Chekov, MSU

Cluster Algebras of Geometric Type Part III
 Cluster Algebras of Geometric Type Part III
 11/02/2017
 10:00 AM  10:50 AM
 C304 Wells Hall
 Leonid Chekov, MSU
This is a continuation of the previous 2 talks.

6183

Thursday 11/9 10:00 AM

Nick Ovenhouse, Michigan State University

Logcanonical Poisson brackets
 Logcanonical Poisson brackets
 11/09/2017
 10:00 AM  10:50 AM
 C304 Wells Hall
 Nick Ovenhouse, Michigan State University
TBA

6191

Thursday 11/16 10:00 AM

Linhui Shen, Michigan State University

Cluster Algebras and Representation Theory Part I
 Cluster Algebras and Representation Theory Part I
 11/16/2017
 10:00 AM  10:50 AM
 C304 Wells Hall
 Linhui Shen, Michigan State University
One of the main motivations for the study of cluster algebras is to provide an algebraicgeometric framework for studying problems arising from representation theory. In the first talk, we will focus on the cluster theory of configuration spaces of decorated flags. The tropical sets of the latter spaces parametrize top components of the affine Grassmannian convolution varieties. By the geometric Satake Correspondence, they parametrize bases in the tensor invariants of representations of the Langlands dual groups. If time permits, I will talk about their connections with FockGoncharov Duality Conjecture and Mirror Symmetry.

7213

Thursday 11/30 10:00 AM

Linhui Shen, Michigan State University

Cluster algebras and representation theory Part II
 Cluster algebras and representation theory Part II
 11/30/2017
 10:00 AM  10:50 AM
 C304 Wells Hall
 Linhui Shen, Michigan State University
In the second part, we will further study the configuration spaces of decorated flags and their connections with the geometric Satake Correspondence, FockGoncharov Duality Conjecture, and Mirror Symmetry. We will present a natural construction of the cluster coordinates of configuration spaces and generalize them to the moduli space of decorated Glocal systems.

7218

Thursday 12/7 10:00 AM

Linhui Shen, Michigan State University

Cluster algebras and representation theory Part III
 Cluster algebras and representation theory Part III
 12/07/2017
 10:00 AM  10:50 AM
 C304 Wells Hall
 Linhui Shen, Michigan State University
This is a continuation of the previous 2 talks.
