Spring 2022
February 3 -- TBA
February 10 -- Ray Maresca
Title: Combinatorics of Exceptional Sequences of Type \tilde{A}_n
Abstract: It is known that there are infinitely many exceptional collections of representations of Euclidean quivers. In this talk we will study those of Euclidean type $\tilde{A}_n$ and completely describe them in terms of a more geometric combinatorial object known as strand diagrams. We will put every exceptional collection into a so-called family and show how infinitely families behave in terms of strand diagrams. Finally, we will show that there are only finitely many such families by giving a bijection between families and certain small strand diagrams.
February 17 -- Ian Montague
Title : Seiberg-Witten Floer K-Theory and Cyclic Group Actions on Spin 4-Manifolds with Boundary
Abstract: Over two decades ago, Furuta proved his celebrated "10/8ths" theorem, which gives a bound on the magnitude of the signature of a (smooth) spin 4-manifold in terms of its second Betti number. Soon afterwards, Bryan showed that if a spin 4-manifold admits a cyclic group action (e.g., a cyclic branched cover over a surface), then Furuta's bound can be strengthened. In another direction, Manolescu showed in 2013 that given a spin 4-manifold X with boundary a fixed rational homology sphere Y, the 4-manifold X satisfies a "10/8ths-type" inequality involving a correction term dubbed the kappa invariant of Y, which comes from analyzing the Pin(2)-equivariant complex K-theory of the Seiberg-Witten Floer spectrum associated to Y. As a common generalization of Bryan and Manolescu's work, I will describe work-in-progress which establishes a 10/8ths-type inequality for spin 4-manifolds with rational homology sphere boundary equipped with a Z/m action. In particular, the corresponding correction term in the inequality involves a family of "equivariant kappa invariants" associated to the bounding rational homology sphere, which take the form of local minima of a semi-infinite sub-lattice of Q^{m} or Q^{m-1} depending on the parity of m.
February 24 -- No GSS
March 3 -- No GSS
March 10 -- Zihao Liu
Title: Large Scale Homology
Abstract: Generally, large scale geometry considers properties of metric spaces that are visible to an observer at a vantage point preceding to infinity. Specifically, with large scale structures, all bounded metric spaces are equivalent to a point, and thus, the focus will be on unbounded spaces, such as the Cayley graph of a finitely generated infinite group. In this talk, I will introduce my recent work about the large scale homology theory by using the affine n-simplex to characterize topology of the boundary (points at infinity) of an unbounded metric space.
March 17 -- No GSS
March 24 -- No GSS
March 31 -- Shujian Chen
Title: Matrix Tree Theorem for reflection groups
Abstract: The famous Matrix Tree Theorem allows us to count the number of spanning trees of a graph by computing the determinant of the corresponding Laplacian. Guillaume Chapuy and Theo Douvropoulos extend the Laplacian to W-Laplacian for reflection groups W and thus generalize the Matrix Tree Theorem. In this talk, we will explore this generalization and some of its applications.
April 7 -- Tony Guo
Title: Frieze Varieties are Invariant under Coxeter Mutation
Abstract: One of the fundamental classification theorems in quiver representations is Gabriel's theorem (1978) which states that a connected quiver has finite representation type if and only if its underlying graph is a Dynkin diagram of type A, D, or E. Lee et al. (2018) introduced the concept of "frieze variety'' X(Q) for an acyclic quiver and showed that X(Q) detects the representation type of the quiver. Igusa and Schiffler (2019) generalized the definition of frieze variety and showed that the generalized frieze variety is invariant under "Coxeter mutation.'' They also introduced a new technique for computing frieze varieties more efficiently. In particular, this technique is easy to apply if the quiver has certain symmetries. I will also present plenty of examples and some open questions.
April 14 -- Alex Semendinger
Title: The Isomorphism Problem for Artin and Coxeter Groups
Abstract: TBA
April 21 -- TBA
April 28 -- Simon T. Huynh
Title: TBA
Abstract: TBA