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College of Arts and Sciences

2014-2015 Seminars and Colloquia

department of mathematics

Boston College Distinguished Lecturer in Mathematics Series

Jordan Ellenberg, Vilas Distinguished Achievement Professor in Mathematics, University of Wisconsin will be the Distinguished Lecturer in our annual lecture series. Prof. Ellenberg will give three lectures on February 16-18, 2015

Monday, February 16th at 4:30 pm in McGuinn 121 "How to get rich playing the Massachusetts Lottery"

Abstract: From 2005 to 2012, a group of friends who met as MIT undergraduates won over 3 million dollars playing a poorly designed game in the Massachusetts lottery. How did they do it, and how did they get away with it? Their strategy, it turns out, involved the theory of combinatorial designs. I’ll explain what combinatorial designs are, what they have to do with lotteries, their relation with geometry over finite fields, and the 2014 breakthrough of Peter Keevash that solved one of the major open problems in the subject.

Tuesday, February 17th at 5:00 pm in Gasson 305 "Arithmetic statistics over function fields, or: the topology of numbers"

Abstract: The talk will introduce the fruitful analogy between number fields like Q and function fields over finite fields, like F_q(t).  We will concentrate especially on the asymptotics of enumerative questions coming from number theory,  a subject often known as “arithmetic statistics.”   It turns out that when you ask arithmetic statistics questions over F_q(t), an unexpected relation to algebraic topology emerges.  We will explain, for instance, why the questions “how many squarefree integers are there between N and 2N” and “how many primes are there between N and 2N” transform into a question about the cohomology of Artin’s braid group, and why “1-1/q” is the algebraic geometer’s version of “6/pi^2.”  I’ll also talk about a deeper example, connecting the  Cohen-Lenstra conjectures about average behavior of class groups of number fields with the cohomology of moduli spaces of curves called Hurwitz spaces; in particular, we will explain how topological theorems about stable cohomology of mapping spaces provide the only known proofs of statements of Cohen-Lenstra type over function fields.  (Joint work with Akshay Venkatesh and Craig Westerland, and with Tom Church and Benson Farb.)

Wednesday, February 18th at 3:00 pm in Gasson 305 "Arithmetic statistics over function fields, II: recent developments"

Abstract: In this talk, I’ll talk about recent developments stemming from the theme of the first talk, arguing that almost every interesting question in arithmetic statistics is connected to an interesting question in geometry when transposed to the function field setting.  In particular, the topological viewpoint provides a kind of machine for generating geometrically motivated (and hopefully correct!) conjectures, giving a coherent common grounding for many popular conjectures in arithmetic statistics, while suggesting many others that haven’t yet been seriously investigated.  In the rare contexts when the geometrically motivated conjecture conflicts with an existing conjecture, the geometrically motivated conjecture seems to win.


BC-MIT Number Theory Seminar

Organizers: Ben Howard and David Geraghty at BC, and Bjorn Poonen at MIT.

March 10, 2015 at BC Cushing 209

3:00 Ben Brubaker (U. Minnesota)

Title: "Matrix coefficients for p-adic groups and Hecke algebra modules"

Abstract: I'll report on continuing work to categorize important matrix coefficients for representations of p-adic groups (includingspherical, Whittaker, and Bessel functionals among others) in terms of representations of Iwahori-Hecke algebras. We'll showhow these characterizations lead to simple computations of the functionals on distinguished test vectors (e.g. the spherical vector), which has applications to global constructions in automorphic forms. This is based on joint work with Bump, Friedberg, and Licata.

4:30 David Harbater (U. Penn)

Title: Local-global principles for torsors

Abstract: Many algebraic structures can be classified by torsors, i.e. by principal homogenous spaces for algebraic groups. This holds, for example, for quadratic forms and central simple algebras. Local-global principles for such algebraic structures can then be obtained from local-global principles for torsors. This has been studied classically in the case of torsors over global fields. This talk, on recent and continuing work with Julia Hartmann and Daniel Krashen, will focus on the case of semi-global fields (one-variable function fields over complete discretely valued fields).

April 7, 2015 at MIT Room E25-111 (Rescheduled from Feb. 10)

3:00-4:00pm Francis Brown (CNRS-IHES and IAS)

"Irrationality proofs, moduli spaces and dinner parties"

After introducing an elementary criterion for a real number to be irrational, I will discuss Apery's famous result proving the irrationality of zeta(3).  Then I will give an overview of subsequent results in this field, and finally propose a simple geometric interpretation based on a classical dinner party game.

4:30-5:30pm Bruno Klingler (Universite' Paris 7 and IAS)

"The hyperbolic Ax-Lindemann-Weierstrass conjecture"

The hyperbolic Ax-Lindemann-Weierstrass conjecture is a functional algebraic independence statement for the uniformizing map of an arithmetic variety. In this talk I will describe the conjecture, its role and its proof (joint work with E. Ullmo and A. Yafaev).   

April 14, 2015 at MIT Room 4-237

3:00 Alexander Gamburd (The Graduate Center, CUNY)

"Expander Graphs, Markoff Numbers and Strong Approximation"

After defining the terms in the title and describing relations between them I will discuss recent joint work with Bourgain and Sarnak on Markoff graphs (obtained by reducing Markoff tree modulo q). Our work yields results on diophantine properties of Markoff numbers, including, in particular, a theorem asserting that almost all of them are composite.

4:30 Dinesh S. Thakur (University of Rochester)

"Special values scenario in function field arithmetic"

We will give a survey of results, conjectures and techniques dealing with the arithmetic of special values in function field arithmetic.

September 16, 2014 at BC  McGuinn 334

3:00-4:00 p.m.

Liang Xiao (University of Connecticut, Storrs)

Title: Slopes of modular forms

Abstract: An interesting computation of Buzzard and Kilford suggested that the slope distribution of the Up operators on the space of overconvergent modular forms tends to form an arithmetic progression, when the Nybentypus character is highly divisible by p. Unfortunately, this was only verified for very small prime p and small tame level. I will explain a joint work with Daqing Wan and Jun Zhang, in which we work with overconvergent automorphic forms for a definite quaternion algebra instead. We prove certain weak version of this expectation for general p and general tame level structure.

4:30-5:30 p.m.

Kiran Kedlaya (University of California, San Diego)

Title: Cohomology of local systems on rigid analytic spaces

Abstract: Let K be a finite extension of Q_p. The notion of the etale fundamental group of a rigid analytic space has been introduced by de Jong. It is highly noncompact; consider for example Tate's uniformization of elliptic curves.

Let X be a smooth proper connected rigid analytic space over K. One then has a Riemann-Hilbert-style identification of continuous representations of the etale fundamental group of X on finite-dimensional Q_p-vector spaces with locally constant sheaves of finite-dimensional Q_p-vector spaces with respect to Scholze's pro-etale topology; these are what we call "etale Q_p-local systems" on X. We prove that the cohomology groups of such a sheaf are finite-dimensional Q_p-vector spaces.

The proof uses an extension of p-adic Hodge theory, especially the theory of (phi, Gamma)-modules, to the setting of etale fundamental groups, in order to transform the problem into something resembling the finiteness of cohomology of coherent sheaves on X (proved by Kiehl).

Joint work with Ruochuan Liu (Beijing).


October 14, 2014 at MIT Room 4-163
3:00-4:00 p.m.

Wei Zhang (Columbia U.)

Title: ATC, special parahorics and exotic good reduction

Abstract: I will report a joint work with M. Rapoport and B. Smithling, on an
"arithmetic transfer conjecture" (ATC). Similar to the arithmetic fundamental lemma conjecture (for unramified unitary groups), the transfer conjecture connects the derivative of a relative orbital integral to an intersection number on a formal moduli space of $p$-divisible groups associated to a special parahoric of a ramified unitary group. Miraculously, this formal moduli space has good reduction.

4:30-5:30 p.m.

Martin Olsson (University of California, Berkeley)

Title: Motivic invariants of l-adic sheaves

Abstract: I will give an overview of a project aimed at understanding the motivic nature of l-adic sheaves. I will survey motivating questions about independence of l and past results of Lafforgue, Drinfeld, and others. I will then discuss how to incorporate correspondences into the theory, recent results, and open questions.


November 18, 2014 at BC McGuinn 521

3:00-4:00 p.m.

Laurent Fargues (Directeur de Recherche CNRS, Institut de Mathématiques de Jussieu)

Title: G-bundles on the curve

Abstract: In my joint work with Fontaine, we have defined and studied a "curve" linked to p-adic Hodge theory. We moreover classified vector bundles on this curve. In this talk I will recall the structure of this curve. Then, given a reductive group G over the p-adic numbers, I will explain how one can classify G-bundles on this curve and link this to Kottwitz set B(G) of sigma conjugacy classes in G.

4:30-5:30 p.m.

Joseph H. Silverman (Brown University)

Title: Canonical heights and nef divisors on abelian varieties, with an application to arithmetic dynamics

Abstract: Let A/K be an abelian variety defined over a number field, and let D be a divisor on A. The Neron-Tate height q_D(P) = lim h_D(nP)/n^2 is a quadratic form q_D : A(K) --> R, and if D is ample, then q_D is positive definite on A(K) modulo torsion. I will discuss an extension of this theorem to the case that D is only assumed to be a nef divisor and will give, as an application, a proof for abelian varieties of the following conjecture in arithmetic dynamics: Let f : X --> X be a dominant rational self-map of a smooth projective variety, all defined over a number field. Let P be an algebraic point of X whose forward orbit by iterates of f is well-defined and Zariski dense in X. Then the f-orbit of P has maximal arithmetic complexity. (Joint work with Shu Kawaguchi)

BC Math Society/Mathematics Department Undergraduate Lectures

Wednesday, March 19, 2014 - Stokes S209

Title: The Framingham Heart Study and the Development of Cardiovascular Disease Risk Prediction Functions

Abstract: The Framingham Heart Study (FHS) began in 1948, under the direction of National Heart Institute (now the National Heart Lung and Blood Institute), with the objective of assessing risk factors that contribute to cardiovascular disease (CVD) by examining and following a large cohort of participants from Framingham, MA (n=5,209) who did not have CVD or overt symptoms of CVD. The participants have been periodically examined and followed through death, with approximately 100 of these original cohort participants still alive today. Since 1971, the original cohort’s adult children and their spouses, and since 2002, the grandchildren of the original cohort, have been examined and followed. Also, since 1994, participants reflecting a more diverse Framingham community have been examined and followed. Overall, FHS involves approximately 15,000 participants. The abundance of risk factor and follow-up data collected over the years has allowed FHS to be among the leaders in CVD risk prediction, both clinically and methodologically. Here, we will provide a brief history of FHS, discuss clinical and methodological development of CVD risk prediction, provide examples of how FHS risk prediction models are used by physicians today, and briefly discuss current research.


BC Geometry/Topology Seminar

Meets Thursdays at 4:00 pm in Carney 309


BC Colloquium Series

To be determined.


Boston Area Links

The Mathematical Gazette is published weekly by the Worcester Polytechnic Institute Mathematical Sciences Department. It provides a list of mathematical seminars and colloquia in the Massachusetts area.

BC Number Theory/Algebraic Geometry Seminar

BC Number Theory and Algebraic-Geometry Seminar

Meets Thursdays at 3:00 p.m. in Carney 309

BC Math Society Undergraduate Lecture Series

September 24, 2014, 3:00 p.m., Prof. Bill Keane (BC)
Title: Eureka! The Life and (Some of the) Work of Archimedes.