Algebra-Number Theory Archives for Academic Year 2013

Organizational Meeting

When: Wed, August 29, 2012 - 2:00pm
Where: Math 3206

Rank-level duality for conformal blocks of type so(2m+1)

When: Mon, September 17, 2012 - 2:00pm
Where: Math 1311
Speaker: Swarnava Mukhopadhyay (University of North Carolina)
Abstract: Classical invariants of tensor products of representations of one Lie group can often be related to invariants of some other Lie group. Physics suggests that the right objects to consider for these questions are certain refinements of these invariants known as conformal blocks. Conformal blocks appear in algebraic geometry as spaces of global sections of line bundles on the moduli stack of parabolic bundles on a smooth curve. Rank-level duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra. In this talk we will discuss a formulation of rank-level duality using conformal embeddings of Lie algebras. We will also give an outline of our proof of the rank-level duality for type so(2m+1) conjectured by T. Nakanishi and A. Tsuchiya.

Special divisors on hyperelliptic curves

When: Wed, October 10, 2012 - 2:00pm
Where: Math 1311
Speaker: Vivek Shende (MIT) -
Abstract: A divisor on a curve is called ``special'' if its linear
equivalence class is larger than expected. On a hyperelliptic curve,
all such come from pullbacks of points from the line. But one can ask
subtler questions. Fix a degree zero divisor Z; consider the space
parameterizing divisors D where D and D+Z are both special. In other
words, we wish to study the intersection of the theta divisor with a
translate; the main goal is to understand its singularities and its

The real motivation comes from number theory. Consider, in products
of the moduli space of elliptic curves, points whose coordinates all
correspond to curves with complex multiplication. The Andre-Oort
conjecture controls the Zariski closure of sequences of such points
(and in this case is a theorem of Pila) and a rather stronger
equidistribution statement was conjectured by Zhang. The locus
introduced above arises naturally in the consideration of a function
field analogue of this conjecture. This talk presents joint work with
Jacob Tsimerman.

Fall 2012 Algebra and Number Theory Day

When: Sat, October 27, 2012 - 9:15am
Where: Colloquium Room 3206
Speaker: (See Website for Speakers List) (Fall 2012 Algebra and Number Theory Day) -

Donaldson-Thomas invariants of 2-dimensional sheaves and modular forms

When: Wed, October 31, 2012 - 2:00pm
Where: MATH 1311
Speaker: Amin Gholampour (University of Maryland) -
Abstract: We define the Donaldson-Thomas invariants associated to the moduli space of stable 2-dimensional sheaves on a smooth threefold X. If X is a smooth K3 fibration over a curve, we express the DT invariants of X in terms of the Euler characteristics of the moduli spaces of stable torsion free sheaves on a K3 surface and the Noether-Lefschetz numbers of the fibration. From this we conclude that the generating functions of the DT invariants of X are modular. We extend this to the case that the K3 fibration has finitely many fibers with nodal singularities. Finally, we sketch a method to compute the DT invariants of the Calabi-Yau complete intersections such as Fermat quintic in P^4.

Curve neighborhoods of varieties in flag manifolds

When: Mon, November 5, 2012 - 2:00pm
Where: Math 1311
Speaker: Leonardo Mihalcea (Virginia Tech) -
Abstract: If X is a Schubert variety in a flag manifold, its curve neighborhood is defined to be the union of the rational curves of a fixed degree passing through X. It turns out that this is also a Schubert variety, and I will explain how to identify it explicitly in terms of the combinatorics of the Weyl group and of the associated (nil-)Hecke product. I will also show how the geometry and combinatorics of this and more general curve neighborhoods is reflected in computations in quantum cohomology of flag manifolds. This is part of several joint projects with A. Buch, P.E. Chaput, C. Li and N. Perrin.

Fluid-Particle Interaction and the Navier-Stokes-Smoluchowski Model & Orientation in Genome Assmebly and Dendrogram's: A novice's introduction

When: Wed, November 14, 2012 - 12:00pm
Where: CSS 4301
Speaker: Josh Ballew & Karl Schmitt (UMCP) -

Rigid curves on moduli spaces of stable rational curves and arithmetic breaks

When: Wed, November 14, 2012 - 2:00pm
Where: Math 1311
Speaker: Ana-Maria Castravet (Ohio State University) -
Abstract: The Mori cone of curves of the Grothendieck-Knudsen moduli space of
stable rational curves with n markings, is conjecturally generated by the one-dimensional
strata (the so-called F-curves). A result of Keel and McKernan states that a hypothetical
counterexample must come from rigid curves that intersect the interior. In this talk I will
show several ways of constructing rigid curves. In all the examples a reduction mod p
argument shows that the classes of the rigid curves that we construct can be decomposed
as sums of F-curves. This is joint work with Jenia Tevelev.

The Robbins phenomenon: unexpected numerical stability in p-adic arithmetic

When: Mon, December 3, 2012 - 2:00pm
Where: Math 1311
Speaker: Kiran S. Kedlaya (University of California at San Diego) -
Abstract: Since one cannot represent an arbitrary real number on a computer, it is
standard to approximate real-number arithmetic using floating-point
approximations. The situation is similar for p-adic numbers; we begin by
introducing the analogue of floating-point arithmetic for p-adics. We
then describe some known and conjectural examples of p-adic numerical
stability in which algebraic structures (e.g., cluster algebras) work
behind the scenes to keep the loss of numerical precision much lower
than one might initially expect. A key example is the Dodgson (Lewis
Carroll) condensation algorithm for computing determinants, for which we
obtain a partial result towards a conjecture of Robbins. Joint work with
Joe Buhler (CCR La Jolla).

Semi perfect obstruction theories and higher rank Donaldson-Thomas type invariants

When: Mon, December 10, 2012 - 4:00pm
Where: Math 1311
Speaker: Artan Sheshmani (Max Planck Institute) -
Abstract: We introduce a higher rank analog of the Pandharipande-Thomas theory of stable pairs on a Calabi-Yau threefold X. More precisely, we develop a moduli theory for frozen triples
given by the data O^r---->F where "F" is a sheaf of pure dimension 1. The moduli space
of such objects does not naturally determine an enumerative theory: that is, it does not naturally
possess a perfect symmetric obstruction theory. Instead, we show how to use the technology
of semi perfect obstruction theories and the luxury of infinity stacks in obtaining a well behaved
truncation of an obstruction theory coming from the moduli of objects in the derived category.
After building a suitable zero-dimensional virtual fundamental class by hand, we obtain the
first deformation-theoretic construction of a higher-rank enumerative theory for Calabi-Yau
threefolds. Finally If time permits we explain how to use virtual localization techniques to
compute the invariants using equivariant intersection theory.

Schubert polynomials and degeneracy loci for the classical Lie groups

When: Wed, December 12, 2012 - 1:00pm
Where: Math 3206
Speaker: Harry Tamvakis (University of Maryland) -

Normal Functions and the Hodge Conjecture

When: Thu, January 17, 2013 - 2:00pm
Where: Math 3206
Speaker: Gregory Pearlstein (Michigan State University) -
Abstract: The theory of normal functions and the
Hodge conjecture have their origin in the study
of algebraic cycles by Lefschetz and Poincare.
I will sketch the history of the subject and
discuss some of my recent work on singularities
of normal functions to the Hodge conjecture
and the zero locus of a normal function to
a conjectural filtration of Bloch and Beilinson

Connective algebraic K-theory

When: Thu, January 17, 2013 - 3:30pm
Where: Math 3206
Speaker: Marc Levine (Universit├Ąt Duisburg-Essen) -
Abstract: Connective topological K-theory is defined as a connected cover of
usual Bott periodic topological K-theory, and is useful in that it
mediates between K-theory and singular cohomology. In algebraic
geometry, one can perform a similar truncation of algebraic K-theory,
giving the theory of connective algebraic K-theory, which mediates
between K-theory and motivic cohomology (e.g., Chow groups). One has
as well the G-theory version. We give a general discussion of these
theories and present two theorems on connective G-theory. The first
relates the "geometric part" of connective G-theory to the
Grothendieck group of coherent sheaves supported in varying
codimensions, the second gives a completely different description via
algebraic cobordism, as the universal quotient formed by imposing the
multiplicative group law with coefficients in \Z[t].

Organizational meeting: RIT on class field theory

When: Mon, January 28, 2013 - 12:00pm
Where: MTH 0303
Speaker: Larry Washington (UMCP) -
Abstract: Note time and place. This is an organizational meeting for an RIT on class field theory, to go through Milne's book

Eventually the RIT should have its own website.

When is a variety a quotient of a smooth variety by a finite group?

When: Wed, February 20, 2013 - 2:00pm
Speaker: Matt Satriano (Michigan)
Abstract: In this talk we explore the following local-global question: if X is locally a quotient of a smooth variety by a finite group, then is it globally of this form? We show that the answer is "yes" whenever X is quasi-projective and already known to be a quotient by a torus. In particular, this applies to all quasi-projective simplicial toric varieties. We discuss the proof and show how it can be made explicit in the case of toric varieties. This is joint work with Anton Geraschenko.

Derived Exterior Powers

When: Mon, April 8, 2013 - 2:00pm
Where: Math 1311
Speaker: Stephen Lichtenbaum (Brown University) -

Picard Groups of Shimura Varieties

When: Mon, April 22, 2013 - 2:00pm
Where: Math 1311
Speaker: Zhiyuan Li (Stanford University) -

Log concavity of characteristic polynomials and toric intersection theory

When: Mon, May 6, 2013 - 2:00pm
Where: Math 1311
Speaker: Eric Katz (University of Waterloo) -
Abstract: In a recent joint work with June Huh, we proved
the log concavity of the characteristic polynomial of a realizable
matroid by relating its coefficients to intersection numbers on an
algebraic variety and applying an algebraic geometric inequality.
This extended earlier work of Huh which resolved a conjecture in graph
theory. In this talk, we rephrase the problem in terms of more
familiar algebraic geometry, outline the proof, and discuss an
approach to extending this proof to all matroids. Our approach
suggests a general theory of positivity in tropical geometry.