
FRAGMENT


Colorado
State University
Department of Mathematics
Thursdays 35pm, Weber 201

University
of Colorado at
Boulder
Department of Mathematics
Tuesdays 35pm, 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 Curves Talk 2. Topological recursion relations via degree 2 maps  Oct 5
 Linda Chen (Swarthmore U)
 CSU  BrillNoether 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 DonaldsonThomas 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 ZureickBrown (Emory)  CSU
 TBA  Feb 15  MarcHubert Nicole (McGill/Marseille)  CSU   Have the dates show up in your calendar automatically:
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 doublyinfinite 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 CohFTlike 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: BrillNoether theory and determinantal formulas for degeneracy loci
I will begin with an overview of degeneracy loci and applications to the BrillNoether theory of special divisors, and then describe recent developments in Schubert calculus, including Ktheoretic formulas for degeneracy loci and Kclasses of BrillNoether loci. These recover the formulas of EisenbudHarris, Pirola, and ChanLópezPfluegerTeixidor for BrillNoether 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^{k1}. What about a complete intersection of hypersurfaces of unequal degrees d_1...d_k? Do we get a similar bundle over weighted P^{k1}, 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 higherdimensional, singular hypersurface. The prototypical example is Kuznetsov's degree6 K3 surface resolving the category of matrix factorizations of a nodal cubic 4fold. We will discuss several other examples and state some general results. This is joint work with Paul Aspinwall.
Yunfeng Jiang: McPherson's index theorem and DonaldsonThomas Invariants Abstract: Let X be a smooth projective scheme. The GaussBonnetChern 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 blowups, and is called the ChernMather class or ChernSchwartzMacPherson class. The ChernSchwartzMacPherson 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 DonaldsonThomas on projective 3folds, which is now called the DonaldsonThomas theory. In the case of the CalabiYau threefolds, the DonaldsonThomas 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 DonaldsonThomas 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 DonaldsonThomas 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 FultonMacpherson 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 socalled 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 GrunwaldWang’s counterexample for which the localglobal 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 algebrogeometric 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 nonzero 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 ChanGalatiusPayne. 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.

