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, ## Schedule of Talks:
Tomoyuki Takenawa and Anton Dzhamay
Part I: Tomoyuki Takenawa
Abstract: Consider, modulo simultaneous adjoint action by a regular matrix, a space of n x n matrices whose Jordan forms are matrices Theta_i with an additional condition that their traces add up to zero. Each element of this space corresponds to a Schlesinger system, i.e. a system of linear ordinary differential equations whose singularities are at most order one. Recently, Schlesinger (or almost equivalently Fuchsian) systems were classified by Katz, Kostov and Oshima using the notion of the spectral type, modulo two types of transformations: additions and middle convolution. After their theory, many related problems, such as isomonodromy deformations and Deline-Simpson’s problem have been studied extensively. In this talk, I will briefly review such theories, which is the background of second part. Part II: Anton Dzhamay Abstract: In this part I will briefly explain the relationship between isomonodromic transformations and Painlevé equations. I will first briefly review the differential case and then focus on the discrete case (and in particular I will explain what a discrete Hamiltonian dynamic is). Then I will explain how to obtain the equations for discrete Schlesinger transformations of the system, write them in the Hamiltonian form, and then look at the reduction to the difference Painlevé equations and the related birational geometry of the space of initial conditions. Renzo Cavalieri
It has been observed in the early part of the millennium (Goulden, Jackson, Vakil) that double Hurwitz numbers exhibit piecewise polynomial behavior with inductive wall crossing formulae. After an overview of the problem, I want to discuss some of the geometry that explains such phenomena- specifically deriving a tautological intersection theoretic formula for the doube Hurwitz numbers (joint with Marcus) and discussing, in genus zero, how tropical geometry determines spaces of relative stable maps as birational models of M_{0,n} (joint with Markwig and Ranganathan)
Eric Miles
John Calabrese We'll have a look at how DT invariants (which are similar in flavour to Gromov-Witten invariants) change under birational modifications, when the starting variety is a Calabi-Yau threefold (i.e. a smooth three-dimensional projective complex manifold with trivial canonical bundle). There is an explicit formula for flops and a similar formula when orbifolds are also allowed (this would be the setting of the McKay correspondence). I would like to sketch the idea of the proof which goes via motivic Hall algebras. Samuel Grushevsky Extending the Prym map Based on joint work with Casalaina-Martin, Hulek, Laza.
Dawei Chen Title: Effective cone of the moduli space of curves Abstract: The cone of effective divisors plays a central role in the study of birational geometry of a variety X. In this talk I will give a short survey on this subject, with a focus on the case when X is the moduli space of curves with marked points. Moreover, I will report some recent progress in low genera. If time allows, I will also discuss some work in progress towards understanding higher codimensional geometry of the moduli space of curves. Part of the talk is based on joint work with Izzet Coskun.
Paul Johnson The Hilbert scheme of n points on C^2 is a smooth manifold of dimension 2n. The topology and geometry of Hilbert schemes have important connections to physics, representation theory, and combinatorics. Hilbert schemes of points on C^2/G, for G a finite group, are also smooth, and when G is abelian their topology is encoded in the combinatorics of partitions. When G is a subgroup of SL_2, the topology and combinatorics of the situation are well understood, but much less is known for general G. After outlining the well-understood situation, I will discuss some conjectures in the general case, and a combinatorial proof that their homology stabilizes. Maksym Fedorchuk Title: Toward GIT stability of syzygies of canonical curves
Abstract: I will discuss a GIT problem concerning stability of syzygy points of canonical curves, and explain its solution for the first syzygy point in the generic case (joint with Deopurkar and Swinarski). I will also explain connections with Green's conjecture and applications related to the minimal model program for the moduli space of curves. Mark Shoemaker Given a homogeneous degree five polynomial W in the variables X_1, . . . , X_5, we may view W as defining a quintic hypersurface in P^4 or alternatively, as defining a singularity in [C^5/Z_5], where the group action is diagonal. In the first case, one may consider the Gromov-Witten invariants of {W=0}. In the second case, there is a way to construct analogous invariants, called FJRW invariants, of the singularity. The LG/CY correspondence conjectures that these two sets of invariants are related. In this talk I will explain this correspondence, and its relation to a much older conjecture, the crepant resolution conjecture (CRC). I will sketch a proof that the CRC is equivalent to the LG/CY correspondence in certain cases using a generalization of the ``quantum Serre-duality'' of Coates-Givental. This work is joint with Y.-P. Lee and Nathan Priddis. Dusty RossTitle: Wall-crossing in genus zero Landau-Ginzburg theory Abstract: Given a quasi-homogeneous polynomial of degree d, Landau-Ginzburg theory studies certain intersection numbers on the moduli space of d-spin curves (parametrizing curves with d-th roots of the canonical bundle). I will describe a generalization of these intersection numbers obtained by allowing some of the points on the curves to be weighted in the sense of Hassett. As one changes the weights, the invariants thus obtained can be related by a wall-crossing formula. I will explain how the wall-crossing formula generalizes the mirror theorem of Chiodo-Iritani-Ruan, and in particular how it gives a completely enumerative (A-model) interpretation of the mirror phenomenon.
The space of stable log maps of Abramovich-Chen and Gross-Siebert is a central object of study in singular Gromov-Witten theory. In this talk, I will give an explicit combinatorial description of the space of genus 0, 2 pointed stable maps to a toric variety: it is an appropriate enrichment of the Chow Quotient of Kapranov-Sturmfels-Zelevinsky into a toric stack. The idea for this description comes from Morse Theory. This is joint work with William
Gillam.
Inside the moduli space of curves of genus 2 with 2 marked points, the loci of curves admitting a map to P1 of degree d totally ramified at the two marked points have codimension two. In this talk I will show how to compute the classes of the compactifications of such loci in the moduli space of stable curves. I will also discuss the relation with the related work of Hain, Grushevsky-Zakharov, Chen-Coskun, Cavalieri-Marcus-Wise. 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 and Fall 12 here. For more recent semesters, see the side bar above. |