Mathematical Physics

Every toric variety is a GIT quotient of an affine space by an algebraic torus. In this talk, I will discuss a way to understand and compute the symplectic mirrors of toric varieties from this universal perspective using the concept of window subcategories. The talk is based on results from a work of myself and a joint work in progress with Peng Zhou.

The theory of quasimaps to Nakajima quiver varieties X has recently been used very effectively by Aganagic, Okounkov and others to study symplectic duality. For certain X, namely Hilbert schemes of ADE surfaces, it turns out quasimap theory is equivalent to a particular flavor of Donaldson-Thomas theory on a related threefold Y. I will explain this equivalence and how it intertwines concepts and tools from the two sides. For example, symplectic duality has something to say about the crepant resolution conjecture for Y.

A classical result of R. Courant gives an upper bound for the count of nodal domains (connected components of the complement of where a function vanishes) for Dirichlet eigenfunctions on compact planar domains.  This can be generalized to Laplace-Beltrami eigenfunctions on compact surfaces without boundary. When considering random linear combinations of eigenfunctions, one can make this count more precise and pose statistical questions on the geometries appearing amongst the nodal domains: what percentage have one hole? ten holes? what percentage have their boundary being tangent 100 times to a fixed non-zero vector field? The first 20-25 minutes will give a survey on some fundamental results of Nazarov-Sodin, Sarnak-Wigman, and Gayet-Welschinger before presenting some joint works with I. Wigman (King's College London) and Matthew de Courcy-Ireland (École Polytechnique Fédérale de Lausanne) answering these questions in the last 25-30 minutes.

In this talk I will describe a new link "invariant" (with certain wall-crossing properties) for links L in a three-manifold M, where M takes the form of a surface times the real line. This link "invariant" is constructed via a map, called the q-nonabelianization map, from the
gl(N) skein algebra of M to the gl(1) skein algebra of a covering three-manifold M'. In the special case of M=R^3, this map computes well-known link invariants in a new way. As a physical application, the q-nonabelianization map computes protected spin character counting BPS ground states with spin for line defects in 4d N=2 theories of class-S. I will also mention possible extension to more general three-manifolds, as well as further physical applications to class-S theories. This talk is based on joint work with Andrew Neitzke, and ongoing work with Gregory Moore and Andrew Neitzke.