Talk_id  Date  Speaker  Title 
17506

Thursday 1/17 3:00 PM

Daping Weng, MSU

More on Scattering Diagram and Theta Functions
 Daping Weng, MSU
 More on Scattering Diagram and Theta Functions
 01/17/2019
 3:00 PM  4:00 PM
 C117 Wells Hall
I will continue the discussion on scattering diagram and theta functions and relate them to the classical cluster theories. I will sketch GrossHackingKeelKontsevich’s proofs of positive Laurent phenomenon, sign coherence, and a weak version of the cluster duality conjecture.

17512

Thursday 1/24 3:00 PM

Daping Weng, MSU

More on Scattering Diagram and Theta Functions
 Daping Weng, MSU
 More on Scattering Diagram and Theta Functions
 01/24/2019
 3:00 PM  4:00 PM
 C204A Wells Hall
I will continue the discussion on scattering diagram and theta functions and relate them to the classical cluster theories. I will sketch GrossHackingKeelKontsevich’s proofs of positive Laurent phenomenon, sign coherence, and a weak version of the cluster duality conjecture.

17528

Thursday 2/7 3:00 PM

Alek Vainshtein, University of Haifa

Exotic cluster structures on SL_n
 Alek Vainshtein, University of Haifa
 Exotic cluster structures on SL_n
 02/07/2019
 3:00 PM  4:00 PM
 C204A Wells Hall
Back in 2005, Berenstein, Fomin and Zelevinsky discovered a cluster
structure in the ring of regular functions on a double Bruhat cell in a
semisimple Lie group, in particular, SL_n. This structure can be easily
extended to the whole group. The compatible Poisson bracket is given by
the standard rmatrix PoissonLie structure on SL_n. The latter is a
particular case of PoissonLie structures corresponding to
quasitriangular Lie bialgebras. Such structures where classified in
1982 by Belavin and Drinfeld. In 2012, we have conjectured that each
PoissonLie structure on SL_n gives rise to a cluster structure, and
gave several examples of exotic cluster structures corresponding to
PoissonLie structures distinct from the standard one. In my talk I will
tell about the progress in the proof of this conjecture and its
modifications.
Joint with M.Gekhtman and M.Shapiro.

17507

Thursday 2/14 3:00 PM

Alexander Shapiro, The University of Edinburgh

Positive PeterWeyl theorem
 Alexander Shapiro, The University of Edinburgh
 Positive PeterWeyl theorem
 02/14/2019
 3:00 PM  4:00 PM
 C204A Wells Hall
The classical PeterWeyl theorem asserts that the regular representation of a compact Lie group $G$ on the space of squareintegrable functions $L^2(G)$ decomposes as the direct sum of all irreducible unitary representations of $G$. In the talk, I will use positive representations of cluster varieties, to obtain a "noncompact" quantum analogue of the PeterWeyl theorem. This is joint work with Ivan Ip and Gus Schrader.

17543

Thursday 2/28 3:00 PM

Linhui Shen, MSU

Cluster structure on moduli spaces of local systems for general groups
 Linhui Shen, MSU
 Cluster structure on moduli spaces of local systems for general groups
 02/28/2019
 3:00 PM  4:00 PM
 C204A Wells Hall
There have been several references in the literature devoted to the study of the cluster structures on moduli spaces of Glocal systems, all of which are based on case by case study. In this talk, we present a systematic construction that works for all groups at once. As an application, we will investigate the principal series representations of quantum groups from the perspective of cluster theory.

17550

Thursday 3/14 3:00 PM

Linhui Shen, MSU

Cluster structure on moduli spaces of local systems for general groups
 Linhui Shen, MSU
 Cluster structure on moduli spaces of local systems for general groups
 03/14/2019
 3:00 PM  4:00 PM
 C204A Wells Hall
There have been several references in the literature devoted to the study of the cluster structures on moduli spaces of Glocal systems, all of which are based on case by case study. In this talk, we present a systematic construction that works for all groups at once. As an application, we will investigate the principal series representations of quantum groups from the perspective of cluster theory.

17560

Thursday 3/21 3:00 PM

Daping Weng, MSU

Cluster Structures on Double BottSamelson Cells
 Daping Weng, MSU
 Cluster Structures on Double BottSamelson Cells
 03/21/2019
 3:00 PM  4:00 PM
 C204A Wells Hall
Let $G$ be a KacPeterson group associated to a symmetrizable generalized Cartan matrix. Let $(b, d)$ be a pair of positive
braids associated to the root system. We define the double BottSamelson cell associated to $G$ and $(b,d)$ to be the moduli space of configurations of flags satisfying certain relative position conditions. We prove that they are affine varieties and their coordinate rings are upper cluster algebras. We construct the DonaldsonThomas transformation on double BottSamelson cells and show that it is a cluster transformation. In the cases where $G$ is semisimple and the positive braid $(b,d)$ satisfies a certain condition, we prove a periodicity result of the DonaldsonThomas transformation, and as an application of our periodicity result, we obtain a new geometric proof of Zamolodchikov's periodicity conjecture in the cases of $D\otimes A_n$. This is joint work with Linhui Shen.

18564

Thursday 3/28 3:00 PM

Daping Weng, MSU

Cluster Structures on Double BottSamelson Cells
 Daping Weng, MSU
 Cluster Structures on Double BottSamelson Cells
 03/28/2019
 3:00 PM  4:00 PM
 C204A Wells Hall
Let $G$ be a KacPeterson group associated to a symmetrizable generalized Cartan matrix. Let $(b, d)$ be a pair of positive
braids associated to the root system. We define the double BottSamelson cell associated to $G$ and $(b,d)$ to be the moduli space of configurations of flags satisfying certain relative position conditions. We prove that they are affine varieties and their coordinate rings are upper cluster algebras. We construct the DonaldsonThomas transformation on double BottSamelson cells and show that it is a cluster transformation. In the cases where $G$ is semisimple and the positive braid $(b,d)$ satisfies a certain condition, we prove a periodicity result of the DonaldsonThomas transformation, and as an application of our periodicity result, we obtain a new geometric proof of Zamolodchikov's periodicity conjecture in the cases of $D\otimes A_n$. This is joint work with Linhui Shen.

17551

Thursday 4/4 3:00 PM

John Machacek, York University

Sign variation and boundary measurement in projective space
 John Machacek, York University
 Sign variation and boundary measurement in projective space
 04/04/2019
 3:00 PM  4:00 PM
 C204A Wells Hall
We are interested in the topology of some spaces obtained by relaxing total positivity in the real Grassmannian. We define two families of subsets of the Grassmannian each of which include both the totally nonnegative Grassmannian and the whole Grassmannian. In this initial study of such subsets of the Grassmannian we focus of subsets of real projective space where interesting topology already appears. We we are able to find a regular CW complex which can be leveraged to compute some invariants like the fundamental group and Euler characteristic. We also conjecture some "balllike" properties (e.g. CohenMacualayness).

18579

Thursday 4/18 3:00 PM

Michael Shapiro, MSU

Cluster algebras with Grassmann variables (joint with V. Ovsienko)
 Michael Shapiro, MSU
 Cluster algebras with Grassmann variables (joint with V. Ovsienko)
 04/18/2019
 3:00 PM  4:00 PM
 C117 Wells Hall
We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of “extended quivers” which are oriented hypergraphs. We describe mutations of such objects and deﬁne a corresponding commutative superalgebra. Our construction includes the notion of weighted quivers that has already appeared in diﬀerent contexts. This project is a step towards understanding the notion of cluster superalgebra.
