I will survey some applications of a special kind of stratification of an algebraic stack called a theta-stratification. The goal is to eventually be able to study semistability and wall-crossing

in a large array of moduli problems beyond the well-known examples. The most general application is to studying the derived category of coherent sheaves on the stack, but one can use this to understand the topology (K-theory, Hodge-structures, etc.) of the semistable locus and how it changes as one varies the stability condition. I will also describe a ``virtual non-abelian localization theorem'' which computes the virtual index of certain classes in the K-theory of a stack with perfect obstruction theory. This generalizes the virtual localization theorem of Pandharipande-Graber and the K-theoretic localization formulas of Teleman and Woodward.

# Past Algebraic and Symplectic Geometry Seminar

The moduli space of tropical curves (and its variants) is one of the most-studied objects in tropical geometry. So far this moduli space has only been considered as an essentially set-theoretic coarse moduli space (sometimes with additional structure). As a consequence of this restriction, the tropical forgetful map does not define a universal curve

(at least in the positive genus case). The classical work of Knudsen has resolved a similar issue for the algebraic moduli space of curves by considering the fine moduli stacks instead of the coarse moduli spaces. In this talk I am going to give an introduction to these fascinating tropical moduli spaces and report on ongoing work with R. Cavalieri, M. Chan, and J. Wise, where we propose the notion of a moduli stack of tropical curves as a geometric stack over the category of rational polyhedral cones. Using this framework one can give a natural interpretation of the forgetful morphism as a universal curve. The coarse moduli space arises as the set of $\mathbb{R}_{\geq 0}$-valued points of the moduli stack. Given time, I will also explain how the process of tropicalization for these moduli stacks can be phrased in a more fundamental way using the language of logarithmic algebraic stacks.

It has been observed long time ago (by many people) that singular affine symplectic varieties come in pairs; that is often to an affine singular symplectic variety $X$ one can associate a dual variety $X^!$; the geometries of $X$ and $X^!$ (and their quantizations) are related in a non-trivial way. The purpose of the talk will be 3-fold:

1) Explain a set of conjectures of Braden, Licata, Proudfoot and Webster which provide an exact formulation of the relationship between $X$ and $X^!$

2) Present a list of examples of symplectically dual pairs (some of them are very recent); in particular, we shall explain how the symplectic duals to Nakajima quiver varieties look like.

3) Give a new approach to the construction of $X^!$ and a proof of the conjectures from part 1).

The talk is based on a work in progress with Finkelberg and Nakajima.

Let F be a global field and let A denote its adele ring. The usual Tamagawa number formula computes the (suitably normalized) volume of the quotient G(A)/G(F) in terms of values of the zeta-function of F at the exponents of G; here G is simply connected semi-simple group. When F is functional field, this computation is closely related to the Atiyah-Bott computation of the cohomology of the moduli space of G-bundles on a smooth projective curve.

I am going to present a (somewhat indirect) generalization of the Tamagawa formula to the case when G is an affine Kac-Moody group and F is a functional fiend. Surprisingly, the proof heavily uses the so called Macdonald constant term identity. We are going to discuss possible (conjectural) geometric interpretations of this formula (related to moduli spaces of bundles on surfaces).

This is joint work with D.Kazhdan.

Let p>0 be a prime number. We shall describe a short Frobenius-theoretic proof of the Adams-Riemann-Roch theorem for the p-th Adams operation, when the involved schemes live in characteristic p and the morphism is smooth. This result implies the Grothendieck-Riemann-Roch theorem for smooth morphisms in positive characteristic and the Hirzebruch-Riemann-Roch theorem in any characteristic. This is joint work with R. Pink.

Following an idea of Bridgeland, we study the operator on the K-group of algebraic stacks, which takes a stack to its inertia stack. We prove that the inertia operator is diagonalizable when restricted to nice enough stacks, including those with algebra stabilizers. We use these results to prove a structure theorem for the motivic Hall algebra of a projective variety, and give a more conceptual definition of virtually indecomposable stack function. This is joint work with Pooya Ronagh.

Igusa's p-adic zeta function $Z(s)$ attached to a polynomial $f$ in $N$ variables is a meromorphic function on the complex plane that encodes the numbers of solutions of the equation $f=0$ modulo powers of a prime $p$. It is expressed as a $p$-adic integral, and Igusa proved that it is rational in $p^{-s}$ using resolution of singularities and the change of variables formula. From this computation it is immediately clear that the order of a pole of $Z(s)$ is at most $N$, the number of variables in $f$. In 1999, Wim Veys conjectured that the only possible pole of order $N$ of the so-called topological zeta function of $f$ is minus the log canonical threshold of $f$. I will explain a proof of this conjecture, which also applies to the $p$-adic and motivic zeta functions. The proof is inspired by non-archimedean geometry and Mirror Symmetry, but the main technique that is used is the Minimal Model program in birational geometry. This talk is based on joint work with Chenyang Xu.

The question that the Crepant Resolution Conjecture (CRC) wants to address is: given an orbifold X that admits a repant resolution Y, can we systematically compare the Gromov-Witten theories of the two spaces? That this should happen was first observed by physicists and the question was imported into mathematics by Y.Ruan, who posited it as the search for an isomorphism in the quantum cohomologies of the two spaces. In the last fifteen years this question has evolved and found different formulations which various degree of generality and validity. Perhaps the most powerful approach to the CRC is through Givental's formalism. In this case, Coates, Corti, Iritani and Tseng propose that the CRC should consist of the natural comparison of geometric objects constructed from the GW potential fo the space. We explore this approach in the setting of open GW invariants. We formulate an open version of the CRC using this formalism, and make some verifications. Our approach is well tuned with Iritani's approach to the CRC via integral structures, and it seems to suggest that open invariants should play a prominent role in mirror symmetry.

Over a decade ago Welschinger defined invariants of real symplectic manifolds of complex dimension 2 and 3, which count $J$-holomorphic disks with boundary and interior point constraints. Since then, the problem of extending the definition to higher dimensions has attracted much attention.

We generalize Welschinger's invariants with boundary and interior constraints to higher odd dimensions using the language of $A_\infty$-algebras and bounding chains. The bounding chains play the role of boundary point constraints. The geometric structure of our invariants is expressed algebraically in a version of the open WDVV equations. These equations give rise to recursive formulae which allow the computation of all invariants for $\mathbb{CP}^n$.

This is joint work with Jake Solomon.

Let $G$ be a compact Lie group and $k$ be a field of characteristic $p\ge 0$ such that $H^*(G)$ does not have $p$-torsion. We show that a free Lagrangian orbit of a Hamiltonian $G$-action on a compact, monotone, symplectic manifold $X$ split-generates an idempotent summand of the monotone Fukaya category over $k$ if and only if it represents a non-zero object of that summand. Our result is based on: an explicit understanding of the wrapped Fukaya category through Koszul twisted complexes involving the zero-section and a cotangent fibre; and a functor canonically associated to the Hamiltonian $G$-action on $X$. Several examples can be studied in a uniform manner including toric Fano varieties and certain Grassmannians.