For a group G, its dual appears via the Tannakian duality. In other words, the
group ^LG will not be given in terms of generators and relations, but rather by means of
the category of its finite dimensional representations.
It is this category that admits a geometric description. Namely, it can be realized by
perverse sheaves on a certain geometric object attached to G, called the
affine Grassmannian. Such a description of ^LG allows us to state the geometric Langlands
conjecture for an arbitrary group G (and not just GL(n)), which we
will do at the end of the lecture.
After a short review of the history of this problem, I will sketch its relationship with
some aspects of diophantine approximation and complexity theory.
Our treatement of the arithmetic Nullstellensatz is based on duality theory for Gorenstein
algebras (Tate trace formula). The trace formula allows one to perform division modulo
complete intersection ideals, with good control of the degree and height of the involved
The key ingredient is the notion of local height of a variety defined over a number field.
The local arithmetic intersection theory plays
– with respect to the height estimates – the role of the classical
intersection theory with respect to the degree bounds.
(Joint work with T. Krick (Univ. Buenos Aires, Argentina) and L. M. Pardo (Univ. Cantabria,
Nov. 15, 2009
David Smyth (Harvard)
Modular compactifications of Mg,n
A modular compactification of Mg,n is (roughly) a deformation-open class of singular curves with the property that every one-parameter family of smooth curves has a unique limit contained in that class. A modular compactification is stable if all the curves parametrized have the property that every rational component has three distinguished points. We will present a general classification of modular compactifications of Mg,n in terms of simple combinatorial data, which will include Schubert’s moduli space of pseudostable curves and Hassett’s spaces of weighted pointed stable curves as special cases.
Nov. 14, 2009
Hsian-Hua Tseng (Ohio State)
Gromov-Witten theory for root stacks
Let r be a positive integer. Given a line bundle L over a space X there is a stack over X which classifies r-th roots of L. This stack of r-th roots of L, which is a gerbe over X banded by the cyclic group of order r, appears naturally in various places such as the structure result of toric Deligne-Mumford stacks. The goal of this talk is to discuss recent progresses towards understanding Gromov-Witten theory of such stacks of roots (joint work with E. Andreini and Y. Jiang).
Nov. 14, 2009
Michael Zieve (Michigan)
Automorphism groups of curves
Hurwitz proved that a complex algebraic curve of genus g>1 has at most 84(g-1) automorphisms. In case equality holds, the automorphism group has a quite special structure. However, in a qualitative sense, all finite groups G behave the same way: the least g>1 for which G acts on a genus-g curve is on the order of (#G)·d(G), where d(G) is the minimal number of generators of G. I will present joint work with Bob Guralnick on the analogous question in positive characteristic. In this situation, certain special families of groups behave fundamentally differently from all others. If we restrict to G-actions on curves with ordinary Jacobians, we obtain a precise description of the exceptional groups and curves.
Nov. 14, 2009
Sabin Cautis (Columbia)
Equivalences via geometric sl(2) actions
I will explain how sl(2) actions can be used to construct equivalences between derived categories of coherent sheaves. The example we consider are the derived categories of coherent sheaves on cotangent bundles to Grassmannians. Our construction generalizes Seidel-Thomas twists.
Oct. 7, 2009
Daniel Erman (Berkeley)
Deformations of zero-dimensional schemes
A natural question in the study of zero-dimensional schemes is to determine when such a scheme deforms to a disjoint union of points. I will discuss a syzygetic invariant which yields sharp information about this question. I will also give applications to the study of Hilbert schemes of points. This is joint work with M. Velasco.
Sep. 23, 2009
Tommaso De Fernex (Utah)
Rigidity properties of Fano varieties
I will discuss some deformation properties of Fano varieties. The general methods rely on the investigation of the variation of the cone of effective curves and, more generally, of the Mori chamber decomposition, which, according to Mori theory, encode information on the geometry of the variety. The talk is based on joint work with C. Hacon.
Sep. 16, 2009
Mark DeCataldo (Stony Brook)
Topology of the Hitchin fibration and Hodge theory of character varieties
Given a compact Riemann surface X of genus at least two, there are two algebraic varieties attached to it: the character variety Ch, and the Hitchin moduli space M. The non Abelian Hodge theorem asserts that they are diffeomorphic (but have different complex structures). While the rational cohomology rings H
Apr. 5, 2000
Mihnea Popa (UMich)
Generalized theta linear series on moduli spaces of vector bundles on curves
The main theme will be some results on effective bounds for global generation and normal generation of multiples of generalized theta line bundles on moduli spaces of vector bundles on a smooth projective curve X. The techniques involved are of two different flavors: on one hand I will use a result (of independent interest) giving an optimal upper bound on the dimension of the Hilbert scheme of coherent quotients of a fixed vector bundle, while on the other hand I will appeal to general vector bundle techniques on abelian varieties via the notion of Verlinde bundle on the Jacobian of X.
Apr. 3, 2000
David Ben-Zvi (Chicago)
Moduli of curves and bundles via vertex algebras
We will examine some relations between the moduli space of bundles on a fixed algebraic curve and the moduli space of curves. It turns out that vertex algebras provide a powerful tool to examine the local geometry of these spaces. A basic element of the theory is the Sugawara construction, which relates the Virasoro and Kac-Moody algebras ("rotating the loop in a loop group"). We present a general geometric formulation of the Sugawara construction, which specializes to the well-known relations between curves and bundles (Hitchin system, isomonodromy equations, Beilinson-Drinfeld operators and heat equations for theta functions) as well as to new connections.
Mar. 15, 2000
Brendan Hassett (Chicago)
Ample divisors on holomorphic symplectic fourfolds
The Picard group of a polarized K3 surface S may be regarded as an integral quadratic form with respect to the intersection pairing. There is a dictionary between the geometry of S and the arithmetic properties of this form. For example, there are criteria for the existence of smooth rational curves in terms of the integers represented by the form. This yields a simple arithmetic description of the ample cone of S. Our goal is to extend this dictionary to certain higher dimensional analogs to K3 surfaces, known as holomorphic symplectic manifolds. These include punctual Hilbert schemes (i.e., desingularized symmetric products) of K3 surfaces. We give a conjectural framework generalizing the picture for K3 surfaces and prove the conjectures in certain cases. The variety parametrizing lines contained in a cubic fourfold X is a holomorphic symplectic manifold. Our conjectures predict the existence and nonexistence of certain ruled surfaces on X and these surfaces induce unirational parametrizations for X. Hence the rich projective geometry of cubic fourfolds provides a useful laboratory where we may test our ideas.
Jan. 26, 2000
Tom Nevins (Chicago)
Moduli spaces of framed sheaves on ruled surfaces
I will discuss some moduli spaces of framed torsion-free sheaves on ruled surfaces; these spaces arise in the study of algebras of Hecke operators coming from the geometric Langlands program for surfaces. I use a natural S1 action on these moduli spaces to investigate their topology, and in particular I obtain a complete characterization of their rational homology for a class of ruled surfaces over elliptic curves. These computations use ideas from the study of betti numbers of Hilbert schemes and the study of the topology of configuration spaces.
Nov. 22, 1999
Sasha Orlik (Cologne)
The cohomology of period domains over finite fields
Period domains are open subsets of generalized flag varieties, which are described by semistable conditions. In the case of a local ground field these period domains are open admissible subsets in the sense of rigid algebraic geometry, whereas in the finite ground field case one gets open subvarieties. The goal of the talk is to present the strategy of the compution of the l-adic cohomology of period domains. The result confirms a conjecture of Kottwitz and Rapoport.
Nov. 17, 1999
The appearance of the Langlands dual group
In this lecture we will be concerned with the (so far unique) geometric construction of the Langlands dual group. This construction lies in the core of the whole field.
Nov. 10, 1999
Martin Sombra (IAS)
Effective Nullstellensatz, division formulas, and all that
I will present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over the integers. This result improves previous work on this area of Berenstein-Yger and Krick-Pardo. I will also present an arithmetic Nullstellensatz for sparse polynomial systems.
Nov. 9, 1999
Michael Duff (UM Physics)
A layman's guide to M-theory
Superunification of the fundamental interactions underwent a major paradigm shift in 1984 when eleven-dimensional supergravity was knocked off its pedestal by ten-dimensional superstrings. 1995 witnessed another shift of equal proportions, however, when superstrings were themselves superseded by "M-theory", a non-perturbative theory which describes extended objects with two dimensions (supermembranes) and five dimensions (superfivebranes), which subsumes all five consistent string theories and whose low-energy limit is, ironically, eleven-dimensional supergravity.
Nov. 2, 1999
Eleny Ionel (Wisconsin)
Gromov-Witten invariants of symplectic sums and applications
The natural cut-and-paste operation for symplectic manifolds is the symplectic sum along a codimension 2 symplectic submanifold. In this talk we will describe a gluing formula for Gromov-Witten invariants of the symplectic sum in terms of the relative GW invariants of the two pieces. This degeneration formula (which is joint work with Tom Parker) describe s what happens to holomorphic curves as one pinches the neck and explains how to compute the GW invariant from the limiting curves. The degeneration formula has plenty of applications, ranging from the construction of infinitely many exotic symplectic structures on the same smooth manifold to solving enumerative questions in algebraic geometry. In the second part of the talk I will describe how to use the degeneration formula to get new relations in the cohomology of the moduli space of complex structures on a genus g Riemann surface with n marked points.
Oct. 20, 1999
Paul Feehan (Ohio State)
Witten's conjecture and gluing PU(2) monopoles
We will discuss new results on gluing theory for PU(2) monopoles and applications to the proof of Witten's conjecture on the relation between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds. The PU(2)-monopole moduli space provides a noncompact cobordism between links of compact moduli spaces of U(1) monopoles of Seiberg-Witten type and the (Donaldson) moduli space of anti-self-dual SO(3) connections, which appear as singularities in this larger moduli space. The purpose of the gluing theorem is to provide topological models for neighborhoods of ideal Seiberg-Witten moduli spaces appearing in lower levels of the Uhlenbeck compactification of the moduli space of PU(2) monopoles and thus permit calculations of their contributions to Donaldson invariants using the PU(2) monopole cobordism.
Oct. 11, 1999
Moshe Jarden (Tel Aviv)
Torsion on abelian varieties over large algebraic fields
A conjecture of Geyer and the speaker from 1978 asserts finiteness and infiniteness of the torsion part of abelian varieties over large algebraic fields. A theorem of Geyer and the speaker says the conjecture holds for elliptic curve. The aim of the talk is to prove several parts of the conjecture for arbitrary abelian varieties.
Oct. 11, 1999
Ching-Li Chai (Penn)
Congruences of Néron models: solution to a problem of Gross and Prasad
The formation of Néron models of a torus T over a local field K does not commute with change of base field; the discrepency gives an numerical invariant c(T). For an abelian variety A over K one can define this invariant c(A) similarly, and c(A) is the local contribution of the change of Faltings' height when A is defined over a global field. In joint work with J.-K. Yu, we give an affirmative answer to the question of Gross and Prasad, that c(T) is equal to one-half of the Artin conductor of the Galois representation on the character group X
Oct. 6, 1999
Gary Kennedy (UMich visitor)
Contact formulas for rational plane curves
Two algebraic plane curves are said to have a contact of order n at a point if both curves are smooth there and if the intersection number is n. If X is a family of such curves depending on s parameters, then one expects to find finitely many members of the family satisfying s specified conditions, where a "condition" means either passing through a specified point or having a contact of order n with a specified curve (this should be counted as n-1 conditions). A contact formula expresses this finite number in terms of simpler data for the family and for the specified individual curves, called characteristic numbers. One is then faced with the problem of determining the characteristic numbers. We will look in particular at the case of rational plane curves and where the specified order of contact is no more than three.
Sept. 29, 1999
Vangelis Mouroukos (UMich)
Arithmetic Hodge structures and higher Abel-Jacobi maps, after Green, Asakura and Saito
One of the traditional tools for the Hodge-theoretic study of algebraic cycles has been the Abel-Jacobi map, classically for divisors on Riemann surfaces and generalized by Weil and Griffiths to higher codimension cycles on varieties of arbitrary dimension. However, both the image and the kernel of the Abel-Jacobi map remain, in general, rather mysterious. Recent work by various authors associates to any complex variety a new Hodge-theoretic structure, called an arithmetic Hodge structure. Using these, one can define higher Abel-Jacobi maps from (higher) Chow groups to certain extension groups of arithmetic Hodge structures. These new invariants can be used to study the kernel and image of the Abel-Jacobi map in several interesting situations (for example, in the cases of zero-cycles on a surface and cycles on general hypersurfaces). We shall give an exposition of the above constructions and examples to illustrate that the new invariants can detect non-trivial algebraic cycles quite efficiently.
Sept. 22, 1999
Ana Bravo (UMich visitor)
Smoothness and arithmetical schemes
We study the notion of quasi-smoothness introduced by O. Villamayor in some recent works. This notion makes possible to associate a "canonical tangent bundle" to certain arithmetical schemes. In this context we can associate a "jacobian ideal" to any irreducible scheme embedded in a quasi-smooth scheme. We describe the class of centers so that quasi-smoothness is preserved by blowing-ups. Finally we characterize this class of centers in terms of the jacobian ideal.
Apr. 21, 1999
Alexander Barvinok (UMich)
Let f : Rn→R2 be a map given by a pair of quadratic forms. A (strong) version of the Toeplitz-Hausdorff Theorem asserts that the image f (S) of the unit sphere S in Rn is a convex set in R2 provided n >2. A similar result holds for a map Cn→R3 defined by three Hermitian forms. In this talk, I discuss convexity properties of the image for more than two real (three Hermitian) forms. New convexity results will be presented as well as connections with distance geometry and molecular conformation.
Apr. 7, 1999
Vangelis Mouroukos (Chicago)
Cohomological connectivity and algebraic cycles
We give a systematic way to produce interesting algebraic cycles on general complete intersections of sufficiently high multidegree in a non-singular projective variety, that contain a fixed non-singular subvariety. This gives rise to new non-torsion elements in the Griffiths group of homologically trivial cycles modulo algebraic equivalence of such a complete intersection. Moreover, we formulate injectivity and surjectivity results for the corresponding Chow groups; they are consistent with Beilinson's conjectures on mixed motives and filtrations on Chow groups.
Apr. 6, 1999
Geometric construction of central elements in the affine Hecke algebra
Let G be a p-adic group and let HI denote the Iwahori Hecke algebra of G, i.e., HI consists of compactly supported functions on G, which are bi-invariant with respect to the Iwahori subgroup of G. It is well-known that the center of HI is isomorphic to the spherical Hecke algebra Hs of G, and the latter can be identified by means of the Satake isomorphism with the Grothendieck ring of finite-dimensional representations of the corresponding Langlands' dual group. Moreover, D. Kazhdan and G. Lusztig gave a geometric interpretation of HI. They showed that elements of HI can be viewed as perverse sheaves on a certain infinite-dimensional algebraic variety, called the affine flag manifold. Therefore, it becomes a natural question to ask if one can construct explicitly the center of HI using algebraic geometry. (In this form the question was first formulated by R. Kottwitz). In this talk we shall present such a construction, using Drinfeld's idea of fusion of Hecke operators. Moreover, we shall see that it has many favorable properties. In particular, it has been used by R. Bezrukavnikov for the proof of Lusztig's conjecture about the relation between two-sided cells in the affine Weyl group and centralizers of unipotent elements in the Langlands' dual group. Results which are going to be mentioned in this talk were obtained in a joint work with A. Beilinson.
Mar. 31, 1999
Matthew Emerton (UMich)
The geometric Langlands construction
In this talk we will explain Beilinson and Drinfeld's construction of the geometric Langlands correspondence between certain D-modules on the moduli of G-bundles on a curve and opers for the dual group on the curve itself.
Mar. 24, 1999
Ravi Vakil (MIT)
Twelve points on the projective line
There are many ways to define a (PGL(2)-invariant) divisor in the variety of "12 points in P1", Sym12(P1), isomorphic to P12. For example: the 12 nodal cubics in a pencil of plane cubics; the branch points of a degree 4 map of a genus 3 hyperelliptic curve to P1; the branch points of a genus 3 curve mapped to P1 by the canonical divisor; the branch points of a genus 4 curve mapped to P1 by a theta-divisor. We'll see that these divisors (and others) are all the same, by showing that the corresponding moduli spaces are covers of others. The links involve various beautiful classical constructions.
Mar. 23, 1999
Bernd Sturmfels (Berkeley)
Resonant hypergeometric series
We present a basis of logarithmic series solutions at a point of maximal degeneracy to the Gelfand-Kapranov-Zelevinsky hypergeometric equations. These differential equations were studied by Batyrev, Hosono-Lian-Yau and Stienstra in the context of toric mirror symmetry. Our new construction is combinatorial and gives an explicit formula for the terms of such a series. Main ingredients are volumes of convex polytopes and shellings of triangulations. This talk is based on the material in Section 3.6 of the forthcoming book "Gröbner Deformations of Hypergeometric Differential Equations" (with M. Saito and N. Takayama). The current draft of our book is available at my homepage.
Mar. 17, 1999
Vladimir Baranovsky (Chicago)
Moduli of sheaves on surfaces and the action of the oscillator algebra
This talk will be devoted to a generalization of the work by Nakajima and Grojnowski who proved that the cohomology groups of Hilbert schemes of points on an algebraic surface X possess an action of a certain Heisenberg/Clifford algebra constructed from the cohomology of X. We will show how to construct a similar action on the cohomology of the moduli spaces of torsion-free sheaves on X. Related open questions will be mentioned. The technical machinery required for understanding the talk will be kept to minimum.
Mar. 10, 1999
Kalle Karu (Boston U.)
Semistable reduction in characteristic zero
Let f: X→B be a family of varieties. We consider the problem of replacing the family f with a new family f': X'→B' such that all fibers of f' are as nice as possible. The correct definition of "as nice as possible" is given in terms of toric geometry, and a morphism f' satisfying it is called semistable. The semistable reduction problem then asks to find a generically finite proper base change B'→B and a proper birational morphism X'→ X xB B' such that the induced morphism f': X'→B' is semsitable. We discuss the proof of a slightly weaker version of the problem and some interesting (unsolved) questions in the combinatorics of polyhedral complexes that the strong semistable reduction problem gives rise to.
Feb. 23 and Mar. 9, 1999
Anders Buch (Chicago)
Formulas for degeneracy loci
I will describe a new class of polynomials which describe the cohomology class of a very general type of degeneracy loci, associated to a sequence of vector bundle maps and arbitrary rank conditions on these maps and their compositions. These polynomials generalize and give new formulas for all known types of Schubert polynomials. Our polynomials are expressed as a linear combination of products of Schur polynomials. We conjecture that all coefficients are non-negative, and given by a generalized Littlewood-Richardson rule. This is joint work with William Fulton.
Feb. 22, 1999
David Ben-Zvi (Harvard)
Spectral curves, opers and integrable systems
We describe some relations between loop groups, soliton equations, and algebraic geometry. Soliton equations (such as the Korteweg-deVries hierarchy) are naturally expressed as flows on spaces of flat connections (differential data). On the other hand, integrable systems in algebraic geometry are often expressed, using the theory of spectral curves, as linear flows on Jacobians (spectral data). We present a simple and general equivalence between moduli of differential data and moduli of spectral data, which provides a comprehensive geometric picture for a wide range of systems. An interesting feature is the role played by formal versions of spectral curves, which are also responsible for some interesting phenomena in loop groups.
Feb. 10, 1999
Sándor Kovács (Chicago)
A new singularity class is introduced in arbitrary characteristic. In characteristic 0 these singularities are shown to be equivalent to rational singularities. As a corollary we obtain a very simple amd natural proof of Elkik's Theorem: log terminal singularities are rational. Connections with other flavors of rationality will also be discussed.
For a group G, its dual appears via the Tannakian duality. In other words, the group ^LG will not be given in terms of generators and relations, but rather by means of the category of its finite dimensional representations.
It is this category that admits a geometric description. Namely, it can be realized by perverse sheaves on a certain geometric object attached to G, called the affine Grassmannian. Such a description of ^LG allows us to state the geometric Langlands conjecture for an arbitrary group G (and not just GL(n)), which we will do at the end of the lecture.
After a short review of the history of this problem, I will sketch its relationship with some aspects of diophantine approximation and complexity theory.
Our treatement of the arithmetic Nullstellensatz is based on duality theory for Gorenstein algebras (Tate trace formula). The trace formula allows one to perform division modulo complete intersection ideals, with good control of the degree and height of the involved polynomials.
The key ingredient is the notion of local height of a variety defined over a number field. The local arithmetic intersection theory plays – with respect to the height estimates – the role of the classical intersection theory with respect to the degree bounds.
(Joint work with T. Krick (Univ. Buenos Aires, Argentina) and L. M. Pardo (Univ. Cantabria, Spain).)