Where: Math 3206

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.

Where: Math 1311

Speaker: Vivek Shende (MIT) - http://math.mit.edu/~vivek/

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

cohomology.

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.

Where: Colloquium Room 3206

Speaker: (See Website for Speakers List) (Fall 2012 Algebra and Number Theory Day) - www.math.jhu.edu/antd

Where: MATH 1311

Speaker: Amin Gholampour (University of Maryland) - http://www2.math.umd.edu/~amingh/

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.

Where: Math 1311

Speaker: Leonardo Mihalcea (Virginia Tech) - http://www.math.vt.edu/people/lmihalce/

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.

Where: CSS 4301

Speaker: Josh Ballew & Karl Schmitt (UMCP) -

Where: Math 1311

Speaker: Ana-Maria Castravet (Ohio State University) - http://www.math.osu.edu/~castravet.1/

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.

Where: Math 1311

Speaker: Kiran S. Kedlaya (University of California at San Diego) - http://math.ucsd.edu/~kedlaya/

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).

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.

Where: Math 3206

Speaker: Harry Tamvakis (University of Maryland) - http://www2.math.umd.edu/~harryt/

Where: Math 3206

Speaker: Gregory Pearlstein (Michigan State University) - http://www.math.msu.edu/~gpearl/

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

Where: Math 3206

Speaker: Marc Levine (Universität Duisburg-Essen) - http://www.esaga.uni-due.de/marc.levine/

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].

Where: MTH 0303

Speaker: Larry Washington (UMCP) - http://www.math.umd.edu/~lcw

Abstract: Note time and place. This is an organizational meeting for an RIT on class field theory, to go through Milne's book http://www.jmilne.org/math/CourseNotes/cft.html

Eventually the RIT should have its own website.

Where:

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.

Where: Math 1311

Speaker: Stephen Lichtenbaum (Brown University) - http://www.math.brown.edu/faculty/lichtenbaum.html

Where: Math 1311

Speaker: Zhiyuan Li (Stanford University) - http://stanford.edu/~zli2/

Where: Math 1311

Speaker: Eric Katz (University of Waterloo) - http://www.math.uwaterloo.ca/~eekatz/

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.