- September 23, 1:30 pm

Michael Schein, Harvard University

"Weights of Galois representations associated to Hilbert modular forms"Abstract: Let F be a totally real field, p > 3 a rational prime unramified in F, and P a place of F over p. Let

\rho: Gal(K/F) -> GL_2(F_q) be a two-dimensional mod p Galois representation which is assumed to be modular of some weight and whose restriction to a decomposition subgroup at P is irreducible. We specify a set of weights, determined by the restriction of \rho to the inertia subgroup at P, which contains all the weights for which \rho is modular (and, conjecturally, nothing else; we have partial results in this "converse" direction). This proves a special case of a conjecture of F. Diamond, which provides an analogue of Serre's epsilon conjecture for Hilbert modular forms mod p. - September 30, 1:30 pm

David Pollack, Wesleyan University and Harvard University

"Rigidity of p-adic cohomology classes for GL_3"Abstract: This is a report on joint work with Avner Ash and Glenn Stevens. Ash and Stevens have previously described p-adic analytic families of ordinary arithmetic cohomology classes for GL_n over Q, where the Hecke eigenvalues vary p-adic analytically as functions of the weight. In contrast to the GL_2 case, not all ordinary classes for GL_n deform in families. We will discuss a notion of p-adic rigidity for eigenclasses in the cohomology of GL_n, and give examples of non-selfdual cohomological cuspforms for GL_3 that are p-adically rigid.

- October 7, 1:30 pm

Richard Taylor, Harvard University

"Ihara's lemma, Calabi-Yau varieties and the Sato-Tate conjecture"Abstract: I will discuss a conjectural generalisation of Ihara's lemma in the theory of modular forms and sketch how it implies the Sato-Tate conjecture.

- October 14, 1:30 pm

Karen Acquista, Boston University

"A (down-to-earth) criterion for cohomological dimension"Abstract: In Serre's book "Galois Cohomology", he proves that a field k has cohomological dimension at most 1 if for all finite separable K/k and all finite Galois L/K, the norm map from L* to K* is surjective; the main ingredient of the proof is an argument from Tate cohomology. I'll talk about how to generalize this to get a criterion for fields of cohomological dimension at most n, involving norm maps on Milnor K-theory.

- October 21, 1:30 pm

Alireza Salehi Golsefidy, Yale University

"Lattices of minimum covolume in classical Chevalley groups over F_q((t))"Abstract: Studying the colvolume of lattices goes back to the work of Siegel in the forties where he shows that the (2,3,7)-triangular group is a lattice of minimum covolume in G = SL_2(R). The case of SL_2(C) has been open for a long time and just recently G. Martin announced that he has found the minimum covolume. Lubotzky considered the problem for non-Archimedean fields. First he showed that SL_2(F_q[1/t]) is a lattice of minimum covolume in SL_2(F_q((t))). Later in a joint work with Wiegel, he considered the characteristic zero case and calculated the minimum covolume. I studied lattices of simply connected classical Chevalley groups over a positive characteristic local field, and showed that any lattice of minimum covolume up to an automorphism of G(F_q((t))) is G(F_q[1/t]).

- October 28, 1:30 pm

Teruyoshi Yoshida, Harvard University

"On Shimura varieties of Harris-Taylor type"Abstract: Unitary Shimura varieties of Harris-Taylor type are (n-1)-dimensional proper smooth varieties over some CM number field, whose etale cohomology groups realize some cases of global Langlands correspondence for GL(n). I will discuss some results on the arithmetic geometry of its bad reduction: (i) a generalized semistable model for the tamely ramified case - which directly relates some cases of local Langlands correspondence with Deligne-Lusztig theory, (ii) weight spectral sequence for the semistable reduction case - this yields the compatibility of monodromy operator for global/local Langlands correspondence. If possible, I will talk about some intersection theory which computes the action of Hecke corresondences on the cohomology of semistable reduction case.

- November 4, 1:30 pm

Fred Diamond, Brandeis University

"Modularity conjectures and results"Abstract: I'll give an informal survey of conjectures and results on the modularity of 2-dimensional p-adic and mod p Galois representations, describing in particular recent progress due to Kisin, Khare, Wintenberger and Gee.

- November 11, 1:30 pm

Florian Herzig, Harvard University

"The weight in Serre's Conjecture for GL_3"Abstract: I will present a generalisation of the weight part of Serre's conjecture to (tame) three-dimensional mod p Galois representations. It relates predicted weights to the modular representation theory of GL_3(F_p). I will give theoretical and computational evidence. This improves on a conjecture of Ash, Doud, Pollack, and Sinnott. The same phenomenon also occurs in Serre's Conjecture for Hilbert modular forms, as formulated by Buzzard, Diamond, and Jarvis.

- November 18, 1:30 pm

Aaron Levin, Brown University

"Generalizing Siegel's and Picard's Theorems"Abstract: Let C be a complex affine curve with more than two points at infinity. Siegel's theorem states that, assuming C is defined over a number field, there are only finitely many integral points on C. On the other hand, it is a theorem of Picard that all holomorphic maps f:

**C**-> C are constant. I will briefly discuss the Nevanlinna theory-Diophantine approximation dictionary, in which the above two theorems correspond. I will then give new conjectures which generalize Siegel's and Picard's theorems to higher-dimensional varieties and describe progress I have made on proving these conjectures. - December 2, 1:30 pm

Philippe Cassou-Nogues, Universite Bordeaux I and Harvard University

"Galois module structure of torsors"Abstract: After recalling the main results in the theory of Galois module structures of rings of integers of finite extensions of Q, we will consider relative situations when the base field is an arbitrary number field. We will show how the classical question of the existence of a normal integral basis generalizes when dealing with torsors associated with the points of the Mordell-Weil group of an elliptic curve. Finally we will indicate how these arithmetic questions lead to problems on torsion line bundles on abelian schemes.

- December 16, 10:30 am

*** NOTE CHANGE IN DAY/TIME ***

Pierrette Cassou-Nogues, Universite Bordeaux I and Radcliffe Institute (Harvard)

"Birational morphisms from C^2 to C^2"Abstract: In this talk we want to explain some common work with Peter Russell (McGill, Montreal). We classify birational morphisms from C^2 to C^2 with k lines and one curve as missing curves, assuming a technical hypothesis. This is a step in the classification of birational morphisms from C^2 to C^2 and subsequently in the classification of generically rational polynomials.

- January 21, 1:30 pm

Manfred Einsiedler, University of Vienna

"Measure rigidity for Cartan subgroups"Abstract: Furstenberg conjectured that there are very few times 2, times 3-invariant probability measures on the unit circle. To date the best available theorem is Rudolph's result on measures that satisfy additionally that the entropy of times 2 (or equivalently times 3) is positive. Similar conjectures have been made by Margulis, Katok and Spatzier for actions on locally homogeneous spaces (which are in turn related to possible application in number theory). We will discuss the proof of measure rigidity under positive entropy for the Cartan action on SL(3,Z) \ SL(3,R) and similar actions [due to Lindenstrauss, Katok, and myself in various combinations]. Along the way, we will discuss entropy, conditional measures, and weight subgroups.

- January 28, 1:30 pm

Gabriele Mondello (M. I. T.)

"Combinatorial and tautological classes on the moduli space of curves"Abstract: Every presentation of the moduli space of curves gives rise to "natural" cohomology classes. The so-called tautological classes appear in the classical algebro-geometric setting. On the other side, the cellular decomposition of the moduli space via the ribbon graph complex allowed Witten to define some interesting cycles on it, which are called combinatorial. The aim of this talk is to illustrate the main ideas involved in the proof of the so-called Witten-Kontsevich conjecture: combinatorial classes are tautological.

- February 18, 1:30 pm

Robert Pollack (Boston University)

"mu-invariants of anticyclotomic p-adic L-functions"Abstract: "Main conjectures" of Iwasawa theory describe a precise relationship between algebraic and analytic invariants of various number theoretic objects such as class groups, elliptic curves and modular forms. In this talk, we will describe how to establish the "mu-part" of this conjecture for elliptic curves along the anticyclotomic Z_p-extension.

- March 4, 1:30 pm

Olga Plamenevskaya (M.I.T.)

"Contact structures, knots, and Khovanov vs. Heegaard Floer"Abstract: Khovanov homology is a combinatorially defined invariant of a link in S^3. A recent theorem of Ozsvath and Szabo relates Khovanov homology to Heegaard Floer homology of 3-manifolds. Both Khovanov's theory (an invariant of Rasmussen's) and the Ozsvath-Szabo theory can be used to give lower bounds for the slice genus of a knot. I will describe these results, and then discuss how the correspondence between the two theories can be extended to include invariants of contact 3-manifolds and transverse knots in S^3. I will also give bounds for the self-linking number of a transverse knot in terms of Ozsvath-Szabo and Rasmussen's invariants.

- March 11, 1:30 pm

Edward Lee (Harvard)

"A modular nonrigid Calabi-Yau threefold arising from the Horrocks-Mumford vector bundle"Abstract: We construct a determinantal quintic threefold by taking a one-parameter family of abelian surfaces coming from sections of the Horrocks-Mumford vector bundle on P^4. After passing to a big resolution, we show that its middle cohomology breaks up into a 4-dimensional piece arising from a pair of elliptic ruled surfaces and a 2-dimensional piece whose L-function is that of a modular form. An interesting feature is that the elliptic surfaces are a conjugate pair each defined over Q(i), and that only their union is defined over Q.

- March 18, 1:30 pm

Damiano Testa (M.I.T.)

"The Severi problem for rational curves on del Pezzo surfaces"Abstract: Let X be a smooth projective surface and choose a curve C on X. Let V_C be the set of all irreducible divisors on X linearly equivalent to C whose normalization is a rational curve. The Severi problem for rational curves on X with divisor class [C] consists of studying the irreducibility of the spaces V_C as C varies among all curves on X. In this talk, we will prove that all the spaces V_C are irreducible in the case where X is a del Pezzo surface, with a unique exception.

- April 1, 1:30 pm

Steve Miller (Brown University)

"Zeros Near the Central Point of Elliptic Curve L-Functions"Abstract: Based on the function field case, we expect zeros of families of L-functions to behave like eigenvalues of classical compact groups. In particular, zeros of elliptic curve L-functions near the central point should behave like eigenvalues of orthogonal groups near 1. After quickly reviewing random matrix theory we present some theoretical evidence supporting this conjecture and numerical evidence against it, and try to reconcile the two.

- April 8, 1:30 pm

Russell Mann (Brown University)

"Level-raising for automorphic forms on definite unitary groups" - April 15, 1:30 pm

Kiran Kedlaya (M.I.T.)

"Slope filtrations for Frobenius modules"Abstract: The classification of finite dimensional vector spaces over an algebraically closed field, equipped with a linear transformation, is well known. The theme of this talk will be to introduce analogues of this classification for semilinear transformations of finite free modules over a ring equipped with some sort of Frobenius endomorphism. I'll start with the original such analogue, the classification of rational Dieudonne modules over the Witt vectors of an algebraically closed field, attributed to Dieudonne and Manin. I'll then introduce a generalization of this classification, to rings resembling the ring of power series (over a p-adic field) convergent in an annulus; it bears a formal resemblance to the classification of vector bundles on the projective line, including the presence of a complete numerical invariant that looks like a Harder-Narasimhan polygon. Finally, I'll make some comments about how this classification shows up in the study of p-adic cohomology (via work of Crew, Tsuzuki) and/or p-adic Galois representations (via work of Colmez, Cherbonnier, Berger, Kisin).

- April 22, 1:30 pm

Gregory Soifer (Bar Ilan University, Israel)

"The Auslander conjecture:history, results and open problems"Abstract: Let Aff(R^n) be the group of affine transformations of the real affine space R^n. A subgroup G of Aff(R^n) is called properly discontinuous if {g in G; g \cap K nonempty} is finite for every compact subset K of R^n. And G is called crystallographic if G is properly discontinuous and the orbit space G\R^n is compact. A subgroup G of Aff(R^n) will also be called an affine group. A long standing conjecture of Auslander states that every affine crystallographic group G is virtually solvable. So far only special cases of this conjecture have been proved. One of the main goals of our talk is to give an updated survey on the Auslander conjecture. We will display new methods and ideas as well as new conjectures.

- September 10, 1:30 pm

Jennifer Johnson, Caltech

"On the Equivariant Bloch-Kato conjecture"Abstract: The Bloch-Kato conjecture, also called the Tamagawa number conjecture, is a generalization of the analytic class number formula to general motives. Focusing on the case of abelian extensions of imaginary quadratic fields, I will explain the conjectural framework, its relationship to other outstanding conjectures, and discuss aspects of the proof.

- October 1, 1:30 pm

Virgil Pierce, Brandeis University

"Asymptotic expansions of partition functions of random matrices"Abstract: The asymptotic expansions of a family of partition functions associated to random Gaussian Hermitian matrices enumerate a family of maps. These functions are also tau-functions of the Toda Lattice equations. We will show that the terms in the asymptotic expansion are computable in terms of an implicitly defined function. This function is shown to have a Taylor Series with coefficients given by the Higher Catalan Numbers. Our technique is to derive continuum limit equations for the Toda Lattice Equations, then derive a hierarchy of differential equations governing the terms in the asymptotic expansion of the partition function, then to integrate these equations. In conclusion we will show how these results produce a closed formula for the number of genus 0 maps with j vertices of degree k.

- October 8, 1:30 pm

David Fisher, Lehman College - CUNY

"Local rigidity and implicit functions theorems I: Hamilton's theorem and diffeomorphism groups" - October 15, 1:30 pm

Marc Levine, Northeastern University

"Twisting the Chow ring"Abstract: A new technique for constructing interesting new versions of the classical Chow ring has arisen from lookinng at algebraic geoemtry through the lens of algebraic homotopy theory, as constructed by Morel-Voevodsky. I will give some examples of these new Chow rings with applications to central simple algebras and equivariant K-theory.

- October 22, 1:30 pm

Sol Friedberg, Boston College

"Multiple Dirichlet series and the nonvanishing of cubic twists"Abstract: In this talk I will describe the construction of multiple Dirichlet series from a family of twisted L-functions, and I will compare and contrast the families of quadratic and of cubic twists. An application to the nonvanishing of cubic twists of GL(2) will be presented.

- October 29, 1:30 pm

Kiyoshi Igusa, Brandeis University

"Semi-invariants on quivers and cyclic posets"Abstract: The theory of semi-invariants on quivers was developed by Schofield Derksen and Weyman. We extended their theory to negative dimension vectors. The result for Dynkin type quivers is a well-known subcomplex of the Coxeter complex whose faces correspond to the ``tilting objects'' in the cluster category. There are several known interpretations of these tilting objects. For example, they correspond to the clusters in the corresponding cluster algebra of finite type. For the case of A_n the number of tilting objects is given by the Catalan number and the complex that we get is dual to the Stasheff associahedron. Much of this is already known. At the end I will talk about the case of A_n-tilde. Here we get a subcomplex of the affine Coxeter complex with vertices corresponding to preprojective and preinjective modules over circular quiver with n+1 vertices. Explaining the relationship of these with cyclic posets is the goal of this talk.

- November 5, 1:30 pm

Chiu-Chu Liu, Harvard University

"Gromov-Witten invariants of toric Calabi-Yau threefolds"Abstract: I will describe a mathematical theory of the topological vertex, an algorithm proposed by M. Aganagic, A. Klemm, M. Marino, and C. Vafa on effectively computing Gromov-Witten invariants of toric Calabi-Yau threefolds. This is a joint work with Jun Li, Kefeng Liu, and Jian Zhou.

- November 12, 1:30 pm

Adrian Ocneanu, Penn State University

"The internal symmetry of ADE graphs"Abstract: We discuss the construction of a finite dimensional semisimple Hopf algebra which describes the inner symmetry of the Coxeter ADE graphs. Its representations give the quantum symmetries of the ADE graph and lead to the quantum subgroup structure. We discuss the connections between this structure andgeneralizations of the mesh relations of the Auslander-Reiten quiver.

- November 19, 1:30 pm

Joseph Landsberg, Harvard University

"Vogel's universal Lie algebra and E_{7 1/2}"Abstract: Motivated by knot theory, Vogel proposed a tensor category that he regarded as a "Universal Lie algebra". Similarly, Deligne proposed a tensor category that should uniformly model the tensor powers of the adjoint representations of the exceptional series of Lie algebras. I will explain work with L. Manivel on the geometry of rational homogeneous varieties that sheds light on the conjectures of Deligne and Vogel. In particular Deligne's work indicated there should be an exceptional Lie algebra between E_7 and E_8. I will explain how Manivel and I were able to explicitly identify this algebra, which we call E_{7 1/2} and to what extent it can claim a place in the exceptional series.

- December 3, 1:30 pm

Karen Chandler, Harvard University

"Multiple directions on multiple points"Abstract: There is no mystery in finding the Hilbert function of a general collection of points in P^n, but there is much to be understood on that of a collection of multiple points. Given a point p in P^n a (k-uple) multiple point is the scheme given by a power (m_p)^k of the maximal ideal of p. We shall describe motivation on considering such collections, basic conjectures, and their geometric significance. We shall focus on our past, present, and future results on the conjectures of Froberg and Iarrobino. Much work on this problem has been done in P^2, in which the conjectures are not revealing. Thus we shall attend to the higher dimensional cases. Our past work has given an inductive technique of verification on these conjectures. We shall now illustrate how a modification of these methods gives a direct method of proof.

- December 10, 1:30 pm

Marie-France Vigneras, Harvard University

"Structure of the universal pro-p-Iwahori module over the Hecke algebra"Abstract: This structure is the key to construct p-integral models in representations of a p-adic group and to classify the mod p irreducible representations, two questions arising in the p-adic Langlands correspondence. We will explain this for the group G=GL(2,F).

- January 26, 2:00 pm

Mihnea Popa, Harvard University

"Asymptotic intersection theory and restricted volumes"Abstract: I will describe work (partly in progress) on defining asymptoti\c intersection numbers of big (or pseudoeffective) line bundles on smooth projective varieties. Intuition is provided by intersecting with the positive part of a Zariski decomposition, in case it exists. The technical tool is a notion of volume for restrictions of linear series. One shows that the asymptotic intersection numbers and the restricted volumes define continuous functions of the big cone of the ambient variety, and obtain an interesting decomposition of this cone given by their zero-locus. This is joint work with Ein, Lazarsfeld, Mustata and Nakamaye.

- February 2, 2:00 pm

Maryam Mirzakhani, Harvard University

"Simple geodesics on hyperbolic surfaces and intersection theory on the moduli space of curves"Abstract: In this talk we discuss a recursive formula for the Weil-Petersson volume of the moduli space of hyperbolic Riemann surfaces with geodesic boundary components. We use this result to obtain a recursive formula for the intersection numbers of tautological classes on the moduli space of curves with marked points.

- February 9, 2:00 pm

Izzet Coskun, Harvard University

"Scrolls and the quantum cohomology of Grassmannians"Abstract: Rational normal surface scrolls are simultaneously surfaces in projective space and curves in the Grassmannian. This dual perspective renders their geometry very attractive. In this talk I will discuss the limits of rational normal scrolls in one parameter families, then use this description to study their enumerative geometry. I will describe an efficient algorithm for computing some Gromov-Witten invariants of Grassmannians. If time permits, I will show how a similar analysis applies to Del Pezzo surfaces and the Hilbert scheme of conics.

- February 23, 2:00 pm

Stephanie Yang, Harvard University

"Linear systems of plane curves through base points of bounded multiplicity"Abstract: Let Gamma be a collection of multiple points in P^2. A well-known conjecture of Harbourne and Hirshowitz gives geometric meaning to when the multiple points in Gamma fail to impose independent conditions on plane curves of a given degree. I show that this conjecture is true if the multiplicities of the points of Gamma are not greater than 8.

- March 1, 2:00 pm

Igor Pak, M.I.T.

"Unfoldings of convex polyhedra in many dimensions"Abstract: We study several questions on combinatorics and geometry of surfaces of convex polyhedra. The most basic one is: can one compute the geodesic distance between two points on the surface? We present a general construction of a nonoverlapping unfolding of the surface of convex polyhedra\, which allows such computation. The construction is based on an intricate study of cut locus of the surface and uses ideas from Differential Geometry as well as from Discrete and Computational Geometry. The talk assumes no background whatsoever and should be accessible to a general audience. This is a joint work with Ezra Miller.

- March 15, 2:00 pm

Akshay Venkatesh, M.I.T.

"Quantum chaos on locally symmetric spaces"Abstract: It is a general question to understand how the behavior of Laplacian eigenfunctions on a manifold reflect the geometry of the geodesic flow. I'll discuss this type of question in the case of locally symmetric spaces, and in particular some cases where one can prove "quantum unique ergodicity." This was established first by Lindenstrauss in rank 1, and then in some higher rank cases by Silberman and the speaker.

- March 29, 2:00 pm

Ozgur Ceyhan, Max Planck Institute (Bonn)

"On moduli of pointed real curves of genus zero"Abstract: We introduce the moduli space of pointed real curves of genus 0 and the moduli space of admissible curves as its orientation double cover. Their strata correspond to real curves of different types and may be encoded by trees with certain decorations. By using the stratification of moduli of admissible curves, we identify its homology with graph homology of trees with corresponding decoration. We determine some relations in homology analogous to the relations in the homology of the moduli of stable pointed complex curves.

- April 19, 2:00 pm

David Dummit, University of Vermont

"Introduction to the abelian Stark Conjecture"Abstract: The Stark Conjectures, first formulated in a series of papers in the 1970s, postulate some deep connections between the leading terms at s=0 of the L-series of number fields and the arithmetic of corresponding extensions. These conjectures are much more precise in the case of abelian extensions, and over the years a number of generalizations and/or refinements have been made. This talk will try to explain what the abelian Stark Conjecture is (particularly in the so-called rank one case), mention how the Conjecture can be used computationally, describe the "local" conjecture of Gross, and indicate the recent unifying version of Burns.

- April 26, 2:00 pm

Antony Iarrobino, Northeastern University

"Hilbert function of Artinian Gorenstein algebras"Abstract: What are the possible sequences H of vector space dimensions for the first, second ...., j-th higher order partial derivatives of a degree-j homogeneous form f in a polynomial ring in r variables? Consider the family PGOR(H) of such forms f up to k*-multiple having a given sequence H=(1,r,...,r,1). Is this family an irreducible algebraic subset of the projective space paremetrizing all forms of degree j? What is its closure? Equivalently, what are the possible Hilbert functions H for graded Artinian Gorenstein algebra in codimension r? Is the family of such algebras irreducible? We first describe briefly the idyllic situation in three variables, consequence of the Buchsbaum-Eisenbud Pfaffian structure theorem, and work of R. Stanley, S.J. Diesel, J.O. Kleppe and many others. We then explore the case of r=4, reporting on joint work with Hema Srinivasan. The issue of characterizing H when r=4 remains open: we give an opinion.

- September 8, 2:00 pm

Uri Bader, Technion - Israel Institute of Technology

"Normal subgroup theorem for lattices in products of groups" (joint with Y. Shalom)Abstract: We will discuss the following analogue of the celebrated normal subgroup theorem of Margulis: Theorem: Let G = G_1 x G_2 be a locally compact group. Assume any proper quotient group of each G_i is compact. Let H < G be an irreducible lattice. Then any proper quotient of H is finite. The proof goes (like Margulis' proof) by showing that any quotient of H has both property (T) and amenability. These two parts of the proof are independent, yet there is a considerable formal similarity between them. It is this similarity that we will try to emphasize.

- September 22, 2:00 pm

Takashi Kimura, Boston University

"Admissible covers, equivariant topological field theories, and orbifolding"Abstract: Let G be a finite group. An admissible G-cover is essentially a principal G-bundle over a stable curve except that the G action need no longer be free over the marked points and nodes. A suitably pointed version of such data forms a moduli space which is responsible for G-equivariant (1+1)-dimensional topological field theories. We prove that by taking "G invariants," one obtains a (1+1)-dimensional TFT. This procedure can be promoted to a G equivariant version of a cohomological field theory, in the sense of Kontsevich-Manin. Such theories arise when considering the Chen-Ruan orbifold cohomology of a global quotient by G.

- September 29, 2:00 pm

Harry Tamvakis, Brandeis University

"Topological symmetry groups of graphs embedded in the 3-sphere"Abstract: The topological symmetry group of a finite, connected graph embedded in S^3 is the subgroup of the automorphism group of the graph consisting of those automorphisms which can be induced by a homeomorphism of the ambient sphere. We obtain strong restrictions on the finite groups which can arise as topological symmetry groups for some embedded graph. This is joint work with Erica Flapan, Ramin Naimi, and James Pommersheim.

- October 13, 2:00 pm

Willaim Stein, Harvard University

"The arithmetic of J_1(p)"Abstract: Let p be a prime. The modular curve X_0(p) (almost) parametrizes isomorphism classes of pairs (E,C), where E is an elliptic curve and C is a subgroup of order p. In his paper, "Modular Curves and The Eisenstein Ideal", Mazur proved many results about the arithmetic of the Jacobian J_0(p) of X_0(p). For example, he proved that the torsion subgroup is generated by cusps and has order the numerator of (p-1)/12. He showed that the component group of the special fiber of J_0(p) at p is also cyclic of order (p-1)/12. Let X_1(p) be the modular curve that (almost) parametrizes isomorphism classes of pairs (E,P), where E is an elliptic curve and P is a point on E of order p. In this talk, I will talk about analogous questions about the arithmetic of the Jacobian J_1(p) of X_1(p). In particular, in this talk, I will discuss a paper, due to Edixhoven, Conrad, and myself, in which we prove that J_1(p) has trivial componenent group at p. I will also discuss explicit computations with J_1(p) and its factors, which suggest questions and conjectures about, e.g., the structure of the rational torsion subgroup of J_1(p).

- October 20, 2:00 pm

Ilia Zharkov, Harvard University

"Metric degenerations of Calabi-Yau families."Abstract: Recent conjecture of Kontsevich-Soibelman and Gross-Wilson asserts that the Gromov-Hausdorff limit of a maximally degenerate Calabi-Yau family is a sphere with Monge-Ampere Kahler affine structure with singularities in codimension 2. I will describe some examples of such structures and state an analog of the Calabi conjecture in the affine setting.

- October 27, 2:00 pm

Samuel Vandervelde, Wellesley College and University of Chicago

"Computing Mahler measure with a cyclic quadrilateral""Abstract: To begin, we will briefly trace the development of Mahler measure over the past century and provide some unexpected identities which help to explain the current interest in this area of mathematics. We will then develop the tools needed to evaluate the Mahler measure of certain two-variable polynomials, including an introduction to the Bloch-Wigner dilogarithm function. Finally, we provide an appealing formula which gives a value for Mahler measure in terms of the shape, side lengths, and angles of an associated cyclic quadrilateral. If time permits, we will also explore the connection between Mahler measure, special values of L-functions, and the Bloch group.

- November 3, 2:00 pm

V. Lakshmibai, Northeastern University

"A geometric approach to Standard Monomial Theory"Abstract: The talk will be on our recent work jointly with M. Brion, on a geometric construction of a "standard monomial basis" for the homogeneous co-ordinate ring associated with any ample line bundle on the generalized flag variety. This basis is compatible with Schubert varieties and opposite Schubert varieties. The construction is done using a degeneration of the diagonal.

- November 10, 2:00 pm

Paul Monsky, Brandeis University

"Calculation of some Hilbert-Kunz functions via ideal classes and dynamical systems"Abstract: If char(k)=p and g is in k[[x,y,z]], let e_n be the degree of the ideal (x^q,y^q,z^q,g) where q=p^n. The function n-->e_n is difficult to compute. I'll discuss what happens when g=z^D-H(x,y) where H is a smooth form of degree 4. There is an elegant complete description of all these e_n that involves iteration of various rational functions (modulo a conjecture that must be true, but that still defies a complete proof). We make heavy use of ideal classes in k[[x,y]]/H and "magnification operators" acting on these ideal classes, as in Teixeira's thesis.

- November 17, 2:00 pm

David Helm, Harvard University

"Analogues of character groups for Hilbert modular varieties"Abstract: The character group of the maximal subtorus of the Jacobian of a modular curve at a prime of semistable reduction has been a useful tool for the study of classical modular forms. We present an interpretation of this character group in terms of vanishing cycles that generalizes to the higher-dimensional Hilbert modular varieties. Moreover, we show how the Jacquet-Langlands correspondence can be understood as a relationship between the character groups for a Hilbert modular variety and a corresponding Shimura curve.

- November 24, 2:00 pm

Hui June Zhu, Harvard and McMaster University

"Constructing crystalline families"Abstract: A few years ago Breuil conjectured a complete classification of mod-p reduction types of 2-dimensional p-adic crystalline representations of Gal(Q_pbar/Q_p) with Hodge-Tate weights 0 and k where k < 2p+1. This naturally requires a satisfactory theory about the "integral models" (over Z_p) of such Galois representations, which exists previously only for k < p by the Fontaine-Laffaille theory. A general theory (without restriction on weight) is important in the study of Galois deformation. In this talk, I will introduce our objects of study, describe our approach of Galois deformation via explicitly constructing families of etale (phi,Gamma)-modules, then I will discuss applications of our results in partially solving Breuil's conjecture and some further questions. This is a joint work with Berger and Li.

- December 1, 2:00 pm

Markus Hunziker, University of Georgia

"Hilbert Series of Determinantal Varieties"Abstract: The Hilbert series of the graded ring associated to a projective variety contains a lot of information about the variety, e.g., dimension, degree, arithmetic genus etc. If the variety is nice then the Hilbert series can be uniquely written in the form h(t)=f(t)/(1-t)^n, where f(t) is a polynomial with non-negative integer coefficients and f(1) is not equal to zero. In this talk I will consider the Hilbert series of determinantal varieties (including symmetric and skew-symmetric determinantal varieties). These varieties arise naturally in many branches of mathematics, e.g., in classical invariant theory. I will give a formula for the coefficients of the\ numerator f(t) of the Hilbert series as dimensions of representations of cert\ain compact Lie groups. The work presented is joint work with Thomas Enright.

- January 27, 2:00 pm

Pavel Etingof, M.I.T.

"Finite dimensional representations of the rational Cherednik algebra"Abstract: The Cherednik algebra was introduced by Cherednik 10 years ago to prove the famous Macdonald conjectures about multivariable orthogonal polynomials. Since then it was realized that it is itself a very important algebraic object, which is intimately related to many parts of representation theory. I will define the (rational) Cherednik algebra, and describe its finite dimensional representations.

- February 3, 2:00 pm

Gordana Todorov, Northeastern University

"Representation dimension and the finitistic dimension conjecture"Abstract: Recent work on representation dimension has given new impetus to an old unsolved problem known as the Finitistic Dimension Conjecture. We know that the Finitistic Dimension Conjecture is true for all algebras of representation dimension at most three (Igusa-Todorov). Representation dimension, introduced by M. Auslander, is always finite (Iyama) and for many classes of algebras is now known to be at most three. No algebra is known to have representation dimension larger than three.

- February 10, 2:00 pm

Jason Starr, M.I.T.

"Rational curves on Fano hypersurfaces"Abstract: I will discuss some results due to Joe Harris, Mike Roth and myself about the birational geometry of spaces of rational curves on Fano hypersurfaces: reducibility/irreducibility, dimensions, types of singularities and Kodaira dimensions. I will explain the relation of these results to the open question of which Fano hypersurfaces are unirational, and, time permitting, to a conjecture of de Jong and myself generalizing Lang's theorem that a function field of transcendence degree 2 is a "C_2 field".

- February 17, 2:00 pm

Vladimir Retakh, Rutgers University

"Universal algebra of pseudo-roots of noncommutative polynomials"Abstract: The quadratic algebra Q_n of pseudo-roots of noncommutative polynomials introduced by I. Gelfand, R. Wilson and the speaker has a rich and interesting combinatorial structure. In particular, it gives a new characterization of the algebra of noncommutative symmetric functions. In the talk I am going to discuss some solved and unsolved problems of the algebra Q_n.

- February 24, 2:00 pm

Laura F. Matusevich, Harvard University

"Hypergeometric systems and local cohomology"Abstract: Hypergeometric systems are a very interesting class of systems of PDEs that can be studied using combinatorial (commutative) algebra. There are many open problems in this area, and the one I will discuss concerns the dimension of the solution space (the "holonomic rank") of such a system. All the available evidence points at a very strong connection between rank fluctuations and the local cohomology of the underlying toric ring. The precise relationship is a joint conjecture with Ezra Miller. The talk will be self contained: I will assume no hypergeometric or toric background, and among other things, I will show how to combinatorially compute the local cohomology of a toric (or semigroup) ring.

- March 10, 2:00 pm

Peter Trapa, University of Utah and Harvard

"Spherical unitary representations of split groups"Abstract: The purpose of this talk is to give an elementary overview of Barbasch's classification of the spherical unitary dual for split classical groups (over a real or p-adic field). The answer can be organized in terms of a remarkably simple kind of functoriality and, at least formally, suggests nonobvious relationships between the spherical unitary duals of different Lie groups. I'll state at least one such relationship carefully. (It can be interpreted as a generalized representation-theoretic Shimura correspondence.) Virtually all of the results discussed here are due to Barbasch; the new ones are joint with him, as well as with Adams and Vogan.

- March 17, 2:00 pm

Andrei Zelevinsky, Northeastern University

"Cluster algebras of finite type"Abstract: Cluster algebras were introduced a few years ago in a joint work with Sergey Fomin. They are a new class of commutative rings designed to provide an algebraic framework for the study of canonical bases and total positivity in semisimple Lie groups. Every cluster algebra has a distinguished family of generators called cluster variables. The main structural result of the theory obtained so far is a complete classification of cluster algebras of finite type, i.e., those having finitely many cluster variables. We find it quite striking that this classification turns out to be one more instance of the famous Cartan-Killing classification. I will discuss this classification result and main ideas of its proof.

- March 20, 2:00 pm

Greg Kuperberg, University of California, Davis

"The quantum central limit theorem (and its relation to random matrices and random words)" - March 24, 2:00 pm

Alexander Postnikov, M.I.T.

"Gromov-Witten invariants and toric tableaux"Abstract: We present a formula for products of Schubert classes in the quantum cohomology ring of the Grassmannian. It is given in terms of a generalization of Schur symmetric polynomials for shapes that are naturally embedded in a torus. The construction implies symmetries of the Gromov-Witten invariants (the structure constants of the quantum cohomology) with respect to the action of S_3, (Z/nZ)^2, and S_2. The last symmetry is a certain "strange duality" of the GW-invariants that inverts the quantum parameter q. We solve a problem posed by Fulton and Woodward about characterization of the powers of q that occur with nonzero coefficients in the quantum product of two Schubert classes.

- March 31, 2:00 pm

Lassina Dembélé, Brandeis University

"Explicit computation of Hilbert modular forms on Q(\root 5)"Abstract: This talk will be dedicated to some computational aspects of Hilbert modular forms. I will present an algorithm which can be used to compute such forms on Q(\root 5). This algorithm is a nice application of the Jacquet-Langlands correspondence. I will then present some of the geometric objects that correspond to some of the forms we compute.

- April 14, 2:00 pm

Anna Lachowska, M.I.T.

"Modular group action in the center of the small quantum group"Abstract: The small quantum group Uf was introduced by Lusztig as a certain finite dimensional Hopf algebra associated to a semisimple complex Lie algebra Lg and a primitive l-th root of unity q in C. According to Lyubashenko and Majid, in many cases Uf admits a bijective action of two operators obeying the modular identities. This action stabilizes the center Z of Uf, and can be used to study its structure. We will start from the obvious subalgebra in Z, which arises from the Grothendieck ring of the category of finite dimensional Uf-modules, and construct the smallest modular-invariant subspace in Z that contains it. This gives many more central elements than was previously known, in particular we get the whole ceegy to calculate it for a large class of PEL moduli problems.

- September 23, 2:00 pm

Alexander Polishchuk, Boston University

"Holomorphic vector bundles on noncommutative tori" - September 30, 2:00 pm

Mark Reeder, Boston College

"Tame Langlands correspondence for p-adic groups"Abstract: This is an attempt to parametrize representations of a reductive p-adic group in terms of geometric and arithmetic data attached to conjugacy classes in a dual group. I will discuss a very simple case, where the representations are easy to construct, and only the parametrization is interesting. Even here, we find it essential to consider representations of several related groups (inner forms) at once, but the representations are tied together by certain numerical invariants (formal degrees).

- October 7, 2:00 pm

Richard Hain, Duke University and Harvard

"The algebraic cycle C-C^- and Moriwaki's inequality"Abstract: Moriwaki's line bundle is a line bundle on the Deligne-Mumford moduli space of stable genus g curves, which has non-negative degree on almost all complete curves. In this talk, I will explain how Moriwaki's result follows naturally when one studies the metrized line bundle over this moduli space one constructs to compute the (archimedean) height of the algebraic cycle C-C^- in the jacobian of the curve C. I will also explain the analogue of this inequality for the moduli space of pointed stable curves, which arises when one studies the height of the degree 0 cycle (2g-2)x - K_C in the curve C. The coefficients that appear in both inequalities are, in some sense, "structure constants" of mapping class groups.

- October 14, 2:00 pm

Lawrence Ein, University of Illinois at Chicago and Harvard

"Jet schemes and singularities of pairs"Abstract: We will discuss joint work with R. Lazarsfeld, M. Mustata and T. Yasuda on jet schemes and singularities of pairs. We describe how to compute some of the invariants of singularities using jet schemes. As an application, we are able to give a proof for the inversion of adjunction conjecture, when the ambient space is smooth.

- October 21, 2:00 pm

Jerzy Weyman, Northeastern University

"Combinatorics of semi-invariants of quivers" - October 28, 2:00 pm

Alex Martsinkovsky, Norto the algebraic description of Cend_n and classification of its representations. I will also discuss several conjectures about associative and Lie subalgebras of Cend_n. - November 11, 2:00 pm

Frank Calegari, Harvard University

"p-adic modular forms"Abstract: Early this century, Ramanujan observed that coefficients of the Delta function Delta(tau) = q \prod_{n=1}^{infty} (1-q^n)^{24} satisfy certain congruences. Ramanujan also found congruences satisfied by coefficients of the modular function j. Although Ramanujan's congruences for Delta were "explained" by Serre and Swinnerton-Dyer using Deligne's theory, it took Katz's construction of p-adic modular forms to provide a good theoretical basis for congruences for j. In this talk, I will explain some of the history of p-adic modular forms, how explicit computations can really be carried out, and some recent results and conjectures that have implications for classical modular forms.

- November 18, 2:00 pm

Nicholas Shepherd-Barron (University of Cambridge)

"Smoothness of deformation spaces in mixed characteristics via tangent lifting"Abstract: I shall describe the notion of a slightly ramified extension of local rings, and use this to prove a criterion for the smoothness of deformation spaces (or local moduli spaces) in mixed characteristic in terms of lifting the tangent space of the problem. From one point of view, this generalizes Cartier's proof that group schemes are smooth in characteristic zero. In consequence, the deformation spaces of certain Calabi-Yau varieties are smooth in mixed characteristic.

- November 25, 2:00 pm

James Propp, Brandeis University

"Laurent polynomials, dynamics, and combinatorics"Abstract: The nth iterate of the map (x,y) -> (y,(y^2+1)/x) is of the form (x,y) -> (f(x,y), g(x,y)), where the rational functions f and g turn out to belong to the ring Z[x,1/x,y,1/y] of Laurent polynomials. This is just one example of a very widespread phenomenon that governs iteration of many multivariate rational maps. I'll talk about how combinatorial models can help one understand specific instances of this "Laurent phenomenon"; e.g., for the specific mapping described above, the relevant combinatorial model is the dimer model on a 2-by-n grid. Such models can yield positivity theorems that seem hard to prove by purely algebraic means.

- December 2, 2:00 pm

Maxim Vybornov, M.I.T.

"Quiver varieties and Beilinson-Drinfeld Grassmannians of type A"Abstract: We construct Nakajima's quiver varieties of type A in terms of Beilinson-Drinfeld Grassmannians of type A. This gives a compactification of quiver varieties and a decomposition of affine Grassmannians into a disjoint union of quiver varieties. The construction provides a geometric framework for skew (GL(m),GL(n)) duality and identifies the natural basis of weight spaces in Nakajima's construction with the natural basis of multiplicity spaces in tensor products which arises from affine Grassmannians. This is joint work with I. Mirkovic.

- January 28, 2:00 pm

Catherine O'Neil, M.I.T.

"Explicit descent over X(3) and X(5)"Abstract: We motivate the theory of descent on elliptic curves: descent was originally created to measure the solutions to diophantine equations (specifically those that cut out elliptic curves) but has evolved more recently into a way of measuring the failure of a local-to-global principle for elliptic curves known as the Tate-Shavarevich group. We write down explicit models for genus one curves that are required to perform descent in the special case that the elliptic curve has full level structure.

- February 4, 2:00 pm

Emrah Paksoy, Brandeis University

"A method to calculate the cohomology ring of stable weighted point configurations in CP^1"Abstract: The orbit space of stable weighted configurations of n points in CP^1 forms a non-singular algebraic variety C_n(m). In a similar way we define D_n(m), the categorical quotient of the space of semi-stable configurations with respect to the action of PSL_2(C). If m=(m_1,...,m_n) is an n-tuple of weights (positive real numbers) associated to the n points, it is known that D_n(m) is birationally equivalent to M_n, the variety of space polygons in Euclidean space R^3. I will talk about a method to calculate the cohomology ring of M_n (or D_n(m)) and an ampleness criterion to determine which of these spaces are Fano.

- February 11, 2:00 pm

Matt Baker, Harvard University

"Lower bounds for canonical heights over abelian extensions" - February 25, 2:00 pm

Jonathan Weitsman, University of California, Santa Cruz

"The topology of Hamiltonian loop group spaces" - March 4, 2:00 pm

Fernando Rodriguez-Villegas, Harvard University

"Periods L-functions and hyperbolic geometry"Abstract: Inspired by Deninger's work, David Boyd found many experimental relations between the Mahler measure of certain two variable polynomials and the zeta function of an associated number field at s=2. I will describe how these identities can be proved and the role played by the underlying hyperbolic geometry.

- March 18, 2:00 pm

Klaus Kuennemann, University of Regensburg, Germany

"Heights of cycles on Jacobians"Abstract: Let C be a curve over a number field with Jacobian J=Jac(C). We recall the definition of the Neron-Tate height of algebraic cycles on J and compute this height for special cycles on J in the case of good reduction. We explain the obstructions in the case of bad reduction and give some partial results.

- April 22, 2:00 pm

Mihnea Popa, Harvard University

"Regularity on abelian varieties"Abstract: I will introduce the notion of Mukai regularity for coherent sheaves on abelian varieties, defined via conditions on the supports of the cohomologies of Fourier-Mukai transform complexes. In a very special form, depending on the choice of a polarization, this resembles and in fact strenghtens the classical notion of Castelnuovo-Mumford regularity. The techniques are in part a generalization of an approach of Mumford-Kempf-Lazarsfeld made possible by the use of Mukai's theory. I will then explain how this can be applied to a variety of questions related to abelian varieties and more general irregular varieties, ranging from the appearance of new interesting invariants for line bundles, to effective results on linear series and defining equations, and to a study of special classes of vector bundles.

- April 29, 2:00 pm

Stephen Debacker, Harvard University

"Harmonic analysis on reductive p-adic groups" - May 6, 2:00 pm

Avner Ash, Boston College

"p-adic deformation of arithmetic cohomology"Abstract: There are theories, due independently to myself and Stevens on one hand, and Hida on the other, of nontrivial p-adic families of Hecke eigenclasses in the cohomology of a congruence subgroup of GL(n,Z) with twisted coefficients. When n = 2, every ordinary cuspidal eigenclass (equivalently: every ordinary modular newform) can be fitted into such a family. For larger n, this is not necessarily the case. I will discuss especially what happens for n = 3.

- September 24, 2:00 pm

Harry Tamvakis, Brandeis University

"Quantum cohomology of Lagrangian and orthogonal Grassmannians"Abstract: We describe the multiplicative structure of the (small) quantum cohomology ring of the Grassmannians parametrizing maximal isotropic subspaces of a vector space equipped with an orthogonal or symplectic form. No prior knowledge of quantum cohomology or Gromov-Witten invariants will be assumed. This is joint work with Andrew Kresch.

- October 1, 2:00 pm

Laurent Berger, Brandeis University

"An introduction to p-adic Hodge theory"Abstract: I will introduce the main objects that play a part in p-adic Hodge theory, and apply that to an example (Tate's elliptic curve): I will construct the "p-adic periods" of such a curve. Then, I will discuss the so-called "Fontaine monodromy conjecture" (which is now a theorem); this is related to Kiran Kedlaya's talk of Oct. 15th.

- October 15, 2:00 pm

Kiran Kedlaya, University of California, Berkeley

"A p-adic local monodromy theorem" - October 22, 2:00 pm

Michael Kleber, Brandeis University

"Schur, Weyl, and Young: a bit of combinatorial representation theory"Abstract: Classical and modern representation theory makes use of a host of elegant combinatorial tools. One of the oldest and easiest to appreciate is a construction of the representations of SL(n) based on Young diagrams. In its full generality, "Schur-Weyl duality" is a powerful technique, but this original case is concrete and readily accessible. The same approach applied to the group of symplectic or orthogonal matrices is no harder to describe, but it holds a few surprises even for experts, including one recent and unexplained link to quantum groups.

- October 29, 2:00 pm

Payman Kassaei, Brandeis University

"Congruences between modular forms on Shimura curves" - November 5, 2:00 pm

Anders Buch, M.I.T.

"Quantum cohomology of Grassmannians"Abstract: The (small) quantum cohomology ring of a Grassmann variety encodes the enumerative geometry of rational curves in this variety. By using degeneracy loci formulas on quot schemes, Bertram has proved quantum Pieri and Giambelli formulas which give a complete description of the quantum cohomology ring. In this talk I will present elementary new proofs of these results which rely only on the definition of Gromov-Witten invariants and standard facts about cohomology of Grassmannians.

- November 12, 2:00 pm

Romyar Sharifi, Harvard University

"The Galois Lie algebra associated to the projective line minus three points"Abstract: There is a canonical representation of the absolute Galois group of the rational numbers in the outer automorphism group of the fundamental group of the projective line minus three points. Ihara has associated to this a certain graded Lie algebra over the p-adic integers. Deligne has conjectured that, when tensored with the p-adic rationals, this Galois Lie algebra becomes free on one generator in each odd degree starting with 3. I will describe what is known or expected regarding the structure of the Galois Lie algebra itself.

- November 19, 2:00 pm

Takashi Kimura, Boston University

"Higher spin structures and integrable hierarchies" - November 26, 2:00 pm

Hans Wenzl, UCSD and Harvard University

"Centralizer algebras for minuscule representations"Abstract: Using a path description, we can determine a minimal set of generators for the centralizer of the action of a compact Lie group on tensor powers of a minuscule representation. For spinor representations and vector representations of classical type (except D_N), they already appear in the second tensor power. For the vector representation of D_N resp minuscule representations of E_N, N=6,7, an additional generator in the N-th (resp (N-1)-st) tensor power is needed.

- December 10, 2:00 pm

Tom Graber, Harvard University

"Rationally connected varieties"