Abstract: Given a Hida family of modular forms, a conjecture of Greenberg predicts that L-functions of forms in the family should generically vanish to order 0 or 1 at the center of the functional equation. Similarly the Selmer groups of forms in the family should generically be of rank 0 or 1. In this talk I will prove a generalization of the Gross-Zagier theorem, relating Neron-Tate heights of special points on the modular Jacobian J_1(N) to derivatives of L-functions, and explain how this generalization can be used to verify Greenberg's conjecture for any particular Hida family.

The usefulness of the various Riemann-Roch formulas as computational tools is well documented in literature. Grothendieck-Riemann-Roch is a commutative diagram relating pullback in K-theory to the pullback of associated Chow invariants for locally complete intersection (l.c.i.) morphisms. We extend this notion to quasi-smooth morphisms between derived schemes, this is the "derived" analog of l.c.i. morphisms and it encompasses relative perfect obstruction theories. We will concentrate on the naturality of the construction from the standpoint of pure intersection theory and how it interacts with the virtual Gysin homomorphism defined by Behrend-Fantechi. Time permitting we will discuss the relationship with existing formulas, i.e., Ciocan-Fonanine, Kapranov, Fantechi, and Goettsche.

I will explain some attempts to develop a theory of Schubert calculus on the affine Grassmannian. I will begin with the different descriptions of the (co)homology rings due to Bott, Kostant and Kumar, and Ginzburg. Then I will discuss the problems of finding polynomial representatives for Schubert classes and the explicit determination of structure constants in (co)homology.

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 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.

The cohomology rings of flag varieties have long been known to exhibit positivity properties. One such property is that the structure constants of the Schubert basis with respect to the cup product are non-negative. Brion (2002) and Anderson-Griffeth-Miller (2011) have shown that positivity extends to K-theory and T-equivariant K-theory, respectively. In this talk I will discuss recent work (joint with Shrawan Kumar) which generalizes these results to the case of Kac-Moody groups.

A numerical monoid is a subset of the nonnegative integers that is closed under addition. Given a numerical monoid S, consider the shifted monoid S_n obtained by adding n to each minimal generator of S. In this talk, we examine minimal relations between the generators of S_n when n is sufficiently large, culminating in a description that is periodic in the shift parameter n. We also explore several consequences, some old and some new, in the realm of factorization theory. No background in numerical monoids or factorization theory is assumed for this talk.

The fundamental group of a topological space is usually defined in terms of homotopy classes of based loops. The group structure is given by composition of loops. If X is a complex algebraic variety, one has an underlying topological space, and hence a fundamental group. Hain showed that the nilpotent completion of the group ring of this topological fundamental group carries a mixed Hodge structure. Grothendieck defined a fundamental group for schemes defined over any field. Applying this to a complex algebraic variety, one obtains the profinite completion of the topological fundamental group. This group comes with a natural action of the absolute Galois group of the field of definition. The above indicates that varieties over fields of characteristic zero have two notions of fundamental group, armed with either a Galois action or a mixed Hodge structure. This is similar to the situation with homology and cohomology groups, where one has both an etale and Betti version carrying Galois actions and Hodge structures. An important guiding principle is that both of these versions of homology and cohomology should come from an underlying ``motivic'' theory. This is a homology and cohomology theory for algebraic varieties over a field k taking values the abelian tensor category of mixed motives over k, denoted M_k. There should be functors from M_k to both the category of Galois modules and mixed Hodge structures. When applied to the motivic homology of a variety X, these functors should yield the etale homology or Betti homology of X. In this way, motives glue these two different theories together more strongly than just the comparison isomorphisms. Nori has recently given a definition of the category of mixed motives. In this talk, we will show how this category relates to the fundamental group. In fact, more generally there is a motivic version of paths between two different points x and y of X which is important for applications. We also show that the multiplication and comultiplication maps are motivic, and compare with Hain's theory.