## University of Colorado at Boulder Department of Mathematics Tuesdays 3-5pm, Math 350

This is a seminar series intended to involve people in the Front Range interested in Algebra, GeoMEtry and Number Theory. The seminar will meet either at CSU (green) or at CU Boulder (gold).

For more information, or to provide a speaker, contact Renzo, renzo AT math.colostate.edu, or Yano, casa AT math.colorado.edu.

## Schedule of Talks:

 Date Speaker Location Title Aug 31 Damiano Fulghesu(MSU  Moorhead/CSU) CSU From Classical to Equivariant Intersection Theory. Sep 7 Vance Blankers(CSU) CSU How to cope with a grumpy adviser +The recursive structure of hyperelliptic loci in moduli spaces of curves Sep 28 Fumitoshi Sato(U.Utah) CSU Talk 1. A construction of Moduli Spaces of Pointed Rational CurvesTalk 2. Topological recursion relations via degree 2 maps Oct 5 Linda Chen(Swarthmore U) CSU Brill-Noether theory and determinantal formulas for degeneracy loci Oct 12 Nicolas Addington     (U. of Oregon) CSU Complete intersections of unequal degrees Oct 19 Yunfeng Jiang(U. Kansas) CSU McPherson's index theorem and Donaldson-Thomas invariants Nov 2 Dan Edidin    (U. Missouri) CSU Towards an intersection Chow cohomology for GIT quotients and good moduli spaces Nov 9 Yuan Liu(U Wisconsin) CSU The realizability problem with inertia conditions Nov 16 Dhruv Ranganathan(MIT) CSU Tropological aspects of weighted stable curves Feb 13 David Zureick-Brown(Emory) CSU TBA Feb 15 Marc-Hubert Nicole(McGill/Marseille) CSU

## Abstracts:

Damiano Fulghesu:  From Classical to Equivariant Intersection Theory.

Abstract: The main goal of this expository talk is to give a short introduction to equivariant intersection theory as it has been developed by B. Totaro, D. Edidin, and W. Graham.

We will start from recalling some basic facts from classical intersection theory and explain how they generalize in the equivariant world.

We will also discuss some of the fundamental examples such as projective spaces, grassmannians, and moduli spaces of curves.

Graduate students are particularly welcome to participate.

Vance Blankers: The recursive structure of hyperelliptic loci in moduli spaces of curves

Abstract:  Moduli spaces of curves and their tautological rings enjoy a rich recursive combinatorial structure, which we describe in some detail. We then define a more generalized hyperelliptic locus \Hyp_{g,\ell,2m,n} in \Moduli_{g,\ell+2m+n}, requiring our hyperelliptic curves to have \ell marked Weierstrass points, m marked conjugate pairs, and n marked free points. Under the natural forgetful morphisms, these loci form a doubly-infinite family of tautological classes with its own recursive structure, which allows us to use an induction argument to show that any hyperelliptic class in genus two is extremal and rigid in the effective cone of codimension-(\ell+m) classes. Our main result extends work of Chen and Tarasca, who previously considered only Weierstrass point markings, and points towards a possible CohFT-like structure for hyperelliptic classes.

Fumitoshi Sato:  A construction of Moduli Spaces of Pointed Rational Curves/ Topological recursion relations via degree 2 maps

In the first hour I will explain a (new?) construction of moduli spaces of pointed rational curves as a wonderful compactification of a product of projective lines. In the second hour, the talk will turn to studying tautological classes in the moduli spaces of curves.

Mumford started to study enumerative geometry of moduli of curves.  I will give some generalizations of Mumford result.

Linda Chen: Brill-Noether theory and determinantal formulas for degeneracy loci

I will begin with an overview of degeneracy loci and applications to the Brill-Noether theory of special divisors, and then describe recent developments in Schubert calculus, including K-theoretic formulas for degeneracy loci and K-classes of Brill-Noether loci. These recover the formulas of Eisenbud-Harris, Pirola, and Chan-López-Pflueger-Teixidor for Brill-Noether curves. This is joint work with Dave Anderson and Nicola Tarasca.

Nicolas Addington: Complete intersections of unequal degrees

Abstract
For a Fano hypersurface in P^n, the derived category decomposes into an exceptional collection and a category of matrix factorizations. For a complete intersection of k hypersurfaces of degree d, it decomposes into an exceptional collection and a sort of bundle of categories of matrix factorizations over P^{k-1}. What about a complete intersection of hypersurfaces of unequal degrees d_1...d_k? Do we get a similar bundle over weighted P^{k-1}, with weights d_1...d_k? Not really: it is better to view it as a categorical resolution of the category of matrix factorizations of some higher-dimensional, singular hypersurface. The prototypical example is Kuznetsov's degree-6 K3 surface resolving the category of matrix factorizations of a nodal cubic 4-fold. We will discuss several other examples and state some general results. This is joint work with Paul Aspinwall.

Yunfeng Jiang: Mc-Pherson's index theorem and Donaldson-Thomas Invariants

Abstract:
Let X be a smooth projective scheme. The Gauss-Bonnet-Chern theorem states that the integration of the top Chern class of X over X is the topological Euler characteristic of X. In order to study Chern class for singular schemes, R. MacPherson introduced the notion of local Euler obstruction of singular varieties. The local Euler obstruction is an integer value constructible function on X, and the constant function 1_X can be written down as the linear combination of local Euler obstructions. A characteristic class for a local Euler obstruction was defined by using Nash blow-ups, and is called the Chern-Mather class or Chern-Schwartz-MacPherson class. The Chern-Schwartz-MacPherson class of the constant function 1_X is defined as the Chern class for X.  I will focus on these basic materials in the first hour of the talk.

Inspired by gauge theory in higher dimension and string theory, the curve counting theory via stable coherent sheaves was constructed by Donaldson-Thomas on projective 3-folds, which is now called the Donaldson-Thomas theory. In the case of the Calabi-Yau threefolds, the Donaldson-Thomas invariants are proved by Behrend to be "weighted Euler characteristic" of the moduli space X, where the weights come from the local Euler obstruction of the moduli space X. In this talk I will survey some results of the Donaldson-Thomas invariants along this line, and talk about one case that how the Behrend weighted Euler characteristic is related the Y. Kiem and J. Li's cosection localization invariants.  I will also talk about the motivic version of Donaldson-Thomas invariants. This is the main topic of the second part of the talk.

Dan Edidin: Towards an intersection Chow cohomology for GIT quotients and good moduli spaces

Abstract. We study the Fulton-Macpherson operational Chow rings of good moduli spaces of properly stable, smooth, Artin stacks. Such spaces are \'etale locally isomorphic to geometric invariant theory quotients of affine schemes, and are therefore natural extensions of GIT quotients. Our main result is that, with rational coefficients, every operational class can be represented by a so-called topologically strong cycle on the corresponding stack. Moreover, this cycle is unique modulo rational equivalence on the stack. This is based on joint work with Matt Satriano.

Yuan Liu: The realizability problem with inertia conditions

Abstract: We consider the inverse Galois problem with described inertia behavior. For a finite group $G$, one of its subgroups $I$ and a prime $p$, we ask whether or not $G$ and $I$ can be realized as the Galois group and inertia subgroup at $p$ of an extension of $\mathbb{Q}$. We discuss the results when $|G|$ is odd and when $G=GL_2(\mathbb{F}_p)$. Finally, we provide an example arising from Grunwald-Wang’s counterexample for which the local-global principle of our realizability problem fails.

Dhruv Ranganathan: Tropological aspects of weighted stable curves

Abstract: The moduli spaces of weighted stable curves, introduced by Hassett in 2003, reveals a rich network of algebro-geometric parameter spaces with beautiful properties. I will discuss new results that reveal that the dual complexes of these spaces can be topologically intricate. By using arguments in tropical moduli theory, one finds that the boundary complexes of spaces of weighted stable curves can be disconnected, support torsion in the fundamental group, and exhibit non-zero homology groups in a large range of degrees. Nonetheless, we identify natural hypotheses on the moduli problems that tame the chaos. I will explain how these results contrast starkly with the unweighted variant of the problem, studied by Vogtmann, and recently by Chan and Chan-Galatius-Payne. The story I will present started in work with Cavalieri, Hampe, and Markwig (2014), while more recent progress was made in joint work with Cerbu, Marcus, Peilen, and Salmon (2017).

In previous semesters the seminar page was maintained by Renzo Cavalieri, Rachel Pries, Jeff Achter and Yano Casalaina. You can find the Fall 09 page here, Spring 10 here, Fall 10 here , Spring 11 here, Fall 11 here, Spring 12 here,  Fall 12 here,  Spring 2013.  For more recent semesters, see the side bar above.