Fall 2013 Abstracts
- Eduardo González, University of Massachusetts Boston
- Mirror Symmetry, potential functions and Seidel's construction
- Let $X$ be a non-singular projective toric variety whose anti-canonical class is semipositive (nef). I will present work with Iritani regarding invertible elements in the quantum cohomology $QH(X)$ introduced by Seidel and mirror symmetry. I will discuss extensions of this when we extend the construction to discs with lagrangian boundary conditions. Then we will discuss consequences and expected relations, first with potential functions in lagrangian Floer theory introduced by Fukaya-Oh-Ohta-Ono and then with work in progress by Woodward et al. on gauged Floer theory.
- Lisa Jeffrey, University of Toronto
- A Hamiltonian circle action on the triple reduced product of coadjoint orbits of $\mathbf{SU}(3)$
- (Joint work in progress with Gouri Seal, Paul Selick and Jonathan Weitsman.)
- The fundamental group of the three-punctured sphere is the free group on two generators—or, more symmetrically, the group on three generators with one relation (that the product of the generators equal the identity). Representations of this group in compact Lie groups have been much studied (as a building block in the theory of flat connections on 2-manifolds).
- Analogously one may study the symplectic quotient at 0 of the product of three coadjoint orbits of a Lie group (the triple reduced product). For regular orbits of $G=\mathbf{SU}(3)$ this symplectic quotient is a 2-sphere. We exhibit a function whose Hamiltonian flow gives an $S^1$ action on it, and study the period of the $S^1$ action.
- Sema Salur, University of Rochester
- Geometric Structures on Manifolds with $\mathbf{G}_2$ Holonomy
- A $7$-dimensional Riemannian manifold $(M,g)$ is called a $\mathbf{G}_2$ manifold if the holonomy group of the Levi-Civita connection of $g$ lies inside $\mathbf{G}_2$. In this talk, I will first give brief introductions to $\mathbf{G}_2$ manifolds, and then discuss relations between $\mathbf{G}_2$ and contact structures. If time permits, I will also show that techniques from symplectic geometry can be adapted to the $\mathbf{G}_2$ setting. These are joint projects with Hyunjoo Cho, Firat Arikan and Albert Todd.
- Milen Yakimov, Louisiana State University
- Poisson unique factorization domains and cluster algebras
- Cluster algebras were defined by Fomin and Zelevinsky for the purposes of the axiomatic study of canonical bases and total positivity. In connection to those applications, in 2003 Berenstein, Fomin and Zelevinsky constructed explicit upper cluster algebra structures on the coordinate rings of all double Bruhat cells in complex simple Lie groups. An important open problem was to prove or disprove that those upper cluster algebras coincide with the corresponding cluster algebras. In this talk we will define a notion of Poisson Unique Factorization Domain, describe some of its properties, and use them to settle (positively) the above problem.