Spring 2017

 

              FRAGMENT


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

 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
Feb 6
(monday 2-3pm)
 Ozlem Ejder    
 (USC)
 CSU Fermat Curves, Monodromy and Modular Curves
Feb 9
(12 noon)
 Jeff Achter
(CSU)    
 CSU Distinguished models of intermediate Jacobians
Feb 9    
 Travis Mandel
 (U. Utah)
 CSU Descendant log Gromov-Witten theory and tropical curves
Feb 16
 Thomas Goller
 (U.Utah)
 CSU Quantum cohomology, TQFTs, and quot schemes.
 Feb 21
3-4 PM
 Graeme Wilkin
(Nat. Univ. Singapore)
 Boulder The reverse Yang-Mills-Higgs flow in the neighborhood of a critical point
Feb 28
(tuesday 3-5pm)    
 Adam Boocher
 (U. Utah)    
 CSU Bounds for Betti Numbers of Graded Algebras
March 16    
 Spring break  at CSU - no seminar    
  
March 30

two talks!
12:00 Library 203
3:00 Weber 201

 Nathan Kaplan 
 (UC Irvine)
CSU
1. Counting Problems for Elliptic Curves over a Fixed Finite Field

2. Rational point count distributions for del Pezzo Surfaces over Finite Fields

April 4/6
 Druv Raganathan
 (MIT/IAS)
 Boulder/CSU The space of equations for an algebraic curve
Apr 13
  Nicola Tarasca
 (U. Georgia)
 CSU Du Val curves and the pointed Brill-Noether theorem
Apr 20

 Javier Carvajal
 (U. Utah)
CSU
Fundamental groups of $F$-regular schemes and singularities.

 April 27                       Keli Parker
(University of Colorado)   
 CSU An alternate compactification of marked genus one curves
 May 2 Philip Engel   
(Harvard)
 Boulder The Hurwitz theory of elliptic orbifolds
 May 4
 Luca Schaffler
 (U. Georgia)
  CSU The KSBA compactification of the moduli space of D_{1,6}-polarized Enriques surfaces

Have the dates show up in your calendar automatically: 

https://www.google.com/calendar/ical/4bddiire4eapkjuomee1p3l2k0%40group.calendar.google.com/public/basic.ics


Abstracts:

Ozlem Ejder: Fermat Curves, Monodromy and Modular Curves

Abstract: This talk will be about my mathematical explorations in
graduate school. Among the objects I will care about in this talk are
elliptic curves, their torsion subgroups, modular curves, Fermat
curves and monodromy. These terms are parts of two different projects
and I will explain how one work inspired the other.

Jeff Achter: Distinguished models of intermediate Jacobians

Abstract: Consider a smooth projective variety over a number field. The image of the associated (complex) Abel-Jacobi map inside the (transcendental) intermediate Jacobian is a complex abelian variety. We show that this abelian variety admits a distinguished model over the original number field, and use it to address a problem of Mazur on modeling the cohomology of an arbitrary smooth projective variety by that of an abelian variety.  
(This is joint work with Sebastian Casalaina-Martin and Charles Vial.)

Travis Mandel: Descendant log Gromov-Witten theory and tropical curves

Abstract: Gromov-Witten theory is concerned with enumerating the complex curves in a space which satisfy various conditions (e.g., two points determine one line).  I will give a brief introduction to this theory, including explaining some of the complications that arise and the advantages of using log structures.  I will then sketch how these log invariants can often be obtained by counting tropical curves satisfying tropical analogs of the conditions.  I will use the second half of my talk to give more details on this tropical correspondence, including a sketch of the proof and a brief discussion of applications to mirror symmetry.  This is based on joint work with H. Ruddat.

Thomas Goller: Quantum cohomology, TQFTs, and quot schemes.

Abstract: Deforming the cup product in the cohomology ring of the
Grassmannian yields the quantum product, which is the basis of several
amazing geometric structures (topological quantum field theories) that can
be used to study moduli spaces on curves via degeneration. In particular,
these structures can be used to compute the lengths of finite quot schemes
on curves as modified Verlinde numbers. This beautiful story on curves
leads to the question: are there analogous ways to study quot schemes on
surfaces?


Graeme Wilkin: The reverse Yang-Mills-Higgs flow in the neighborhood of a critical point


Abstract: The Yang-Mills-Higgs flow was originally studied by Simpson in the

context of constructing Hermitian-Einstein metrics on Higgs bundles. Methods of

Donaldson and Simpson show that the Yang-Mills-Higgs flow resembles a nonlinear

heat equation on the space of Hermitian metrics on the bundle and therefore the

downwards flow is well-behaved with respect to existence and uniqueness, and it

also has nice smoothing properties. On the other hand, the reverse flow is ill-

posed and therefore even existence of solutions is problematic.


In this talk I will describe a new method to construct long-time solutions to

the reverse Yang-Mills-Higgs flow that converge to a given critical point. The

methods naturally lead to an algebraic criterion for two critical points to be

connected by a flow line and a geometric condition to distinguish between broken

and unbroken flow lines in terms of secant varieties of the underlying curve. If

time permits I will talk about work in progress to describe the compactification

of the space of flow lines in terms of the geometry of secant varieties.



Adam Boocher: Bounds for Betti Numbers of Graded Algebras

Let R be a standard graded algebra over a field.  The set of graded Betti numbers of R provide some measure of the complexity of the defining equations for R and their syzygies.  Recent breakthroughs (e.g. Boij-Soederberg theory, structure of asymptotic syzygies, Stillman's Conjecture)  have provided new insights about these numbers and we have made good progress toward understanding many homological properties of R.  However, many basic questions remain.  In this talk I'll talk about some conjectured upper and lower bounds for the total Betti numbers for different classes of rings. Surprisingly, little is known in even the simplest cases.


Dhruv Ranganathan (MIT / IAS) The space of equations for an algebraic curve


The Brill-Noether varieties of Riemann surface C are objects of classical interest. They parametrize embeddings of C of prescribed degree into a projective space of prescribed dimension, i.e. equations for the curve. When C is general, these varieties are well understood: they are smooth, irreducible, and have the "expected" dimension. As one ventures deeper into the moduli space, past the general locus, these varieties exhibit intricate, even pathological, behaviour: they can be highly singular and their dimensions are unknown. A first measure of the failure of a curve to be general is its gonality. I will present a generalization of the Brill-- Noether theorem, which determines the dimensions of the Brill-Noether varieties on a general curve of fixed gonality, i.e. "general inside a chosen special locus”.


In the first hour, I will give an introduction to the tropical linear series techniques that allow combinatorial tools to be used to attack such problems. In the second hour, I will explain some of the new ideas used, which introduce logarithmic geometry as a new tool in Brill-Noether theory. This is joint work with Dave Jensen.


Luca Schaffler: The KSBA compactification of the moduli space of D_{1,6}-polarized Enriques surfaces.

Abstract:
In the first hour of the talk, we introduce the concept of secondary polytope. Given a polytope Q, the secondary polytope of Q is an interesting combinatorial object which keeps track of certain polyhedral subdivisions of Q. After defining it, we take a close look at some concrete examples of secondary polytopes. Then we discuss how all this convex geometry interacts with algebraic geometry and the study of moduli spaces.

In the second part, we describe the moduli compactification by stable pairs (also known as KSBA compactification) of a 4-dimensional family of Enriques surfaces, which arise as the bidouble covers of the blow up of the projective plane at three general points branched along a configuration of three pairs of lines. The chosen divisor is an appropriate multiple of the ramification locus. Using the theory of stable toric pairs we are able to study the degenerations parametrized by the boundary and its stratification. We relate this compactification to the Baily-Borel compactification of the same family of Enriques surfaces. Part of the boundary of this stable pairs compactification has a toroidal behavior, another part is isomorphic to the Baily-Borel compactification, and what remains is a mixture of these two. To conclude, we construct an explicit Looijenga semitoric compactification of this 4-dimensional family which we conjecture is isomorphic to the KSBA compactification studied.


Nathan Kaplan

Counting Problems for Elliptic Curves over a Fixed Finite Field

Let E be an elliptic curve defined over a finite field with q elements.  Hasse’s theorem says that #E(F_q) = q + 1 - t_E where |t_E| is at most twice the square root of q.  Deuring uses the theory of complex multiplication to express the number of isomorphism classes of curves with a fixed value of t_E in terms of sums of ideal class numbers of orders in quadratic imaginary fields.  Birch shows that as q goes to infinity the normalized values of these point counts converge to the Sato-Tate distribution by applying the Selberg Trace Formula.

In this talk we discuss finer counting questions for elliptic curves over a fixed finite field.  We study the distribution of rational point counts for elliptic curves containing a specified subgroup, giving exact formulas for moments in terms of traces of Hecke operators.  This leads to formulas for the expected value of the exponent of the group of rational points of an elliptic curve over F_q and for the probability that this group is cyclic.  This is joint with work Ian Petrow (ETH Zurich).

Rational point count distributions for del Pezzo Surfaces over Finite Fields

A del Pezzo surface of degree d over a finite field of size q has at most q^2+(10-d)q+1 F_q-rational points. A surface attaining this maximum is called 'split', and if all of these rational points lie on the exceptional curves of the surface, then it is called 'full'. Can we count and classify these extremal surfaces? We focus on del Pezzo surfaces of degree 3, cubic surfaces, and of degree 2, double covers of the projective plane branched over a quartic curve. We will see connections to the geometry of bitangents of plane quartics, counting formulas for points in general position, and error-correcting codes. 

Nicola Tarasca: Du Val curves and the pointed Brill-Noether theorem

The pointed Brill-Noether theorem describes under which condition a general pointed curve admits a linear series with prescribed vanishing sequence at the marked point. While the statement holds for a general pointed curve, no examples was known of smooth pointed curves satisfying the theorem. In recent joint work with Gavril Farkas, we show that a general element in an explicit class of what we called pointed Du Val curves satisfies the theorem. In particular, we give explicit examples of smooth Brill-Noether general pointed curves of arbitrary genus defined over Q.


Javier Carvajal: Fundamental groups of $F$-regular schemes and singularities.

 I will present recent work on the \'etale fundamental group of $F$-regular schemes (joint work with subsets of B. Bhatt, P. Graf, K. Schwede and K. Tucker). I will give applications of this work to the Picard group of $F$-regular singularities. Time permitting, I will discuss some limitations of these applications and thus the necessity of considering the Nori's fundamental group-scheme, which is well suited to the study of positive characteristic phenomena. As we will see, it gives a better understanding of the aforementioned results on the Picard group. For example, we will see that the torsion, and not just the prime-to-$p$ torsion, of the Picard group of $F$-regular singularities is bounded.

Keli Santos-Parker: An alternate compactification of marked genus one curves

In this talk I will introduce my thesis work completed at CU Boulder. The central object of interest is the notion of an "m-stable partially aligned log curve," and I will give a brief tour of the intuition that led to this pleasingly simple moduli problem, as well as the logarithmic geometry needed to make this work. These moduli spaces serve as birational models that resolve the indeterminacies of maps from stable n-marked genus one curves to the alternate compactifications of Smyth that allow elliptic m-fold singularities.

Philip Engel: The Hurwitz theory of elliptic orbifolds

Consider a tiling of a compact, oriented surface by triangles, squares, or hexagons. Enumeration of such tilings can be phrased in the language of Hurwitz theory of an elliptic orbifold. There is a combinatorial curvature associated to each vertex. We will outline a proof that the generating series for the number of tilings with a given number of tiles and specified nonzero curvatures lies in a ring of quasi-modular forms. The asymptotics as the number of tiles goes to infinity allows one to derive formulas for the volumes of strata of cubic, quartic, and sextic differentials---in particular, these volumes are a power of pi times a rational number.  This generalizes work of Eskin and Okounkov in the case of abelian and quadratic differentials.







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.



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