Spring 2014




Colordo State University
Department of Mathematics
Thursdays 3-5pm, Weber 201

 University of Colorado at Boulder
Department of Mathematics
Tuesdays 3-5pm, Math 220

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
 January 23
Tomoyuki Takenawa
(Tokyo University of Marine Science and Technology)

Anton Dzhamay

 Discrete Hamiltonian structure of Schlesinger systems

Isomonodromic transformations of Fuchsian systems and
Painlevé equations.

 January 30
 Renzo Cavalieri

 CSU The geometry of wall crossings in Hurwitz theory
 Februrary 3-5  Alex Lubotzky
(Hebrew University)
 CSU  Arne Magnus Lectures
 February 6
 Eric Miles
 CSU  Special FRAGMATHCLUB talk:
How to be a bouncer at a club (filled with vector spaces, dots, and arrows).
 February 13    

John Calabrese
(Rice University)

 Donaldson-Thomas invariants and flops of Calabi-Yau threefolds
 February 18  Samuel Grushevsky
 (Stony Brook University)
 Boulder  Extending the Prym map
 March 4  Dawei Chen
 (Boston College)
 Boulder  Effective cone of the moduli space of curves 
 March 13        
 Paul Johnson

 CSU     Topology and combinatorics of Hilbert schemes of points on orbifolds
 March 18  Maksym Fedorchuk
 (Boston College)
 Boulder  Toward GIT stability of syzygies of canonical curves
 March 27      
 Mark Shoemaker
 CSU A proof of the LG/CY correspondence via the crepant resolution conjecture
 Apr 10    
 Dusty Ross
(U. Michigan)    

 CSU Wall-crossing in genus zero Landau-Ginzburg theory
 April 12-13
WAGS  Boulder  WAGS is in Boulder this Spring
 April 14 10am
 (Eng E206)
 Douglas Ulmer Georgia Tech Jacobians with many points
 April 29 Samouil Molcho
 (Brown University)
 Boulder Logarithmic Stable Maps as Moduli of Broken Orbits
 May 1 Nicola Tarasca (U. Utah) CSU Double total ramifications for curves of genus 2
 May 8 Ekin Ozman (U. Texas Austin) CSU The p ranks of Prym varieties


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

Forming and studying spaces which parameterize geometric objects is a big part of algebraic geometry. We'll talk about some of the issues that arise in forming a parameter space, look at parameterizing representations of a quiver (these are vector spaces and linear maps attached to a directed graph), and see why Bridgeland stability conditions are a bit like bouncers at a club. These considerations intersect with my work on the Bridgeland stability of line bundles on surfaces, and illustrate some ideas and techniques from the theory.

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

The Torelli map associated to a genus g cover its Jacobian - a 
g-dimensional principally polarized abelian variety. It turns out, by
the works of Mumford and Namikawa in the 1970s (resp. Alexeev and
Brunyate in 2010s), that the Torelli map extends to a morphism from 
the Deligne-Mumford moduli of stable curves to the Voronoi (resp.
perfect cone) toroidal compactification of the moduli of abelian varieties.

The Prym map associates to an etale double cover of a genus g 
curve its Prym - a principally polarized (g-1)-dimensional abelian
variety. The indeterminacy locus of the extension of this map to 
a map to the Voronoi toroidal compactification was studied by
Alexeev, Birkenhake, and Hulek in 2000s, and in this talk we discuss
the extension of the Prym map to a map to the perfect cone
toroidal compactification, and a unified approach to all the 
results mentioned above.

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

Title: Topology and combinatorics of Hilbert schemes of points on orbifolds

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

Title: A proof of the LG/CY correspondence via the crepant resolution conjecture

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 Ross

Title: 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.

Samouil Molcho

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


Nicola Tarasca

Inside the moduli space of curves of genus 2 with 2 marked points, the loci of curves admitting a map to Pof 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.