On February 5, I discussed the problem of understanding the representation theory of Hecke correspondences between moduli spaces of stable sheaves on a smooth projective surface S (and presented reasons why one would care, stemming from mathematical physics and algebraic geometry). In this talk, I will give details on the proof, by studying the geometry and intesection theory of various nested moduli spaces. This will be essentially an algebraic geometry talk, but I will emphasize the big picture and future perspectives.

In this work in progress, I consider refined curve counting on local Pezzo surfaces and match to the physically derived formulas of Huang-Klemm-Poretschkin.  Due to the presence of a non-geometric global symmetry in the physics, generating functions for the invariants associated to dP_n (the blowup of n general points in P^2) are supposed to organize as representations of E_n, and as representations of affine E_8 for the rational elliptic surface.  By geometry I define and compute Weyl-invariant characters of a maximal torus associated with the refined BPS invariants.  In all cases to date where the calculation has concluded, these characters coincide with the characters of representations of E_n who dimensions match the predictions of HKP.

The Göttsche conjecture, now a theorem, states that the degree of the Severi locus of k-nodal curves in a linear system can be expressed universally as a polynomial in the four Chern numbers. Kleiman and Piene have formulated a family and cycle version of this conjecture. More precisely, it states that for a relative effective divisor on a family of surfaces, there exists a natural cycle on the base, enumerating the k-nodal fibres of the divisor. It's class can be expressed universally in the Chern data of the surface and the divisor. Using the BPS-calculus of Pandharipande and Thomas, we will prove the conjecture.

Kazhdan-Lusztig polynomials contain crucial data for the representation theory of reductive groups and the geometry of flag varieties.
In this talk I will review Braden-MacPherson-Fiebig's approach to KL polynomials using moment graphs. This provides an incarnation of the Hecke category where KL polynomials (or their positive characteristic analogue) can be computed using elementary algebra.
I will conclude by using moment graphs to give a purely combinatorial interpretation of the coefficient of q for KL polynomials in Type A.

We will discuss inverse and implicit function theorems in Scale Calculus. 
In the second hour, we will collaboratively make a list of existing, expected, and desirable local-local models towards describing more pseudoholomorphic curve moduli spaces as zero sets of scale-Fredholm sections.

I will describe recent results and conjectures relating knot invariants (such as HOMFLY-PT polynomial and Khovanov-Rozansky homology) to algebraic geometry of the Hilbert scheme of points on the plane. The talk is based on joint works with Andrei Negut, Jacob Rasmussen and Matthew Hogancamp.

I will introduce the concept of monoidal gerbes and show how to use them to twist suitable monoidal tensor categories. We will see interesting examples of this construction related to moduli stacks of objects in Calabi-Yau categories. For 3-Calabi-Yau categories we can go even further by constructing interesting monoidal functors. This allows us to define the Cohomological Hall (co)Algebra.

http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=117053

A K3 surface is a simply connected compact complex surface with trivial canonical bundle. Moduli space of K3 surfaces has been extensively studied in algebraic geometry and it can be characterized in terms of the period map by the Torelli theorem. The differential geometric significance is that every K3 surface admits a hyperkahler metric (a metric whose holonomy group is SU(2)), which is in particular Ricci-flat. The understanding of limiting behavior of a sequence of hyperkahler K3 surfaces gives prototype for more general questions concerning Ricci curvature in Riemannian geometry. In this talk I will survey what is known on this, and talk about a new glueing construction, joint with Hans-Joachim Hein, Jeff Viaclovsky and Ruobing Zhang, that shows a multi-scale collapsing phenomenon, and discuss the connection with the Kulikov classification in algebraic geometry.