College of Arts and Sciences

# 2013-2014 Seminars and Colloquia

## department of mathematics

#### BC Distinguished Lecturer in Mathematics Series

Richard Schwartz, Chancellor's Professor of Mathematics   Brown University           March 24-26, 2014

March 24th 4:00 PM  Gasson 305

Title: "Polygonal Outer Billiards"

Abstract: polygonal outer billiards is a dynamical system,like ordinary billiards, except that the ball moves outside the table (according to different rules) rather than inside the table. When the shape is a convex polygon, the game produces beautiful and mysterious tilings of the plane. I'll survey what is known about this game, and then concentrate on my solution of the Moser-Neumann problem, which asks whether there there is a shape relative to which some point moves arbitrarily far away from the origin during the game.

March 25th 4:00 PM  Gasson 301

Title: "Pentagrams and Rational Maps"

Abstract: I'll discuss three polygon iterations: the midpoint map, the pentagram map, and a third one, which I call "projective heat flow". All three maps follow the same general scheme. You start with a polygon and then the map produces a new, modified,polygon. The midpoint map is a classical and easily-understood map, closely related to heat flow. The pentagram map turns out to be discrete integrable system, and is now well-studied. The third one is sort of a marriage of the midpoint map and pentagram map, and probably this is the first time it has been studied. I'll explain what the projective heat flow map does to pentagons.

March 26th 4:00 PM  Stokes S113

Title: "Thomson's 5 Electron Problem"

Abstract: Thomson's problem asks how N electrons arrange themselves on the sphere so as to minimize their total electrostatic potential. The case N=5 had been open for about a century, even though everyone long believed that the triangular bi-pyramid was the energy-minimizing configuration. I'll explain my rigorous computer-assisted proof that the T.B.P. really is the solution to Thomson's 5 electron problem.

### BC-MIT Number Theory Seminar

Organizers: Sol Friedberg and Ben Howard at BC, and Sug Woo Shin and Bjorn Poonen at MIT.

 September 17, 2013 at MIT, roiom 4-163 Melanie Matchett Wood (Wisconsin) "Cohen-Lenstra moments and local conditions" 3:00-4:00 p.m. In 1984, Cohen and Lenstra gave conjectures predicting the distribution of the odd parts of class groups of imaginary quadratic fields, and predicting a different distribution for the odd parts of class groups of real quadratic fields. We will describe these conjectures, and their analogs in the function field case, as well as a refinement which naturally arises for function fields. We will prove many instances of this refinement in the function field case, which also include function field instances of a conjecture of Bhargava on how local conditions on the quadratic field do not affect the class group distribution. These results suggest a further conjecture on the interaction of local conditions and class groups, which we can prove in some number field and function field cases. Bianca Viray (Brown) "Computing obstructions to the local-to-global principle on Enriques surfaces" 4:30-5:30 p.m. In 1970, Manin showed that the Brauer group can obstruct the existence of global rational points, even when there exist points everywhere locally. Conjecturally, this Brauer-Manin obstruction explains all failures of the local-to-global principle in the case of (geometrically) rational surfaces. However, beyond rational surfaces, for example in the case of Enriques surfaces and K3 surfaces, there is little evidence in support (or against) such a conjecture. This lack of evidence stems from a difficulty in computing the Brauer-Manin obstruction for such surfaces, specifically in computing the so-called transcendental part of the Brauer group. In this talk, we explain how to compute the transcendental Brauer element on any Enriques surface, and, more generally, how to compute the 2-torsion Brauer classes on any double cover of a ruled surface. This is joint work with Brendan Creutz. October 22, 2013 at BC Fulton 230 Directions 3:00-4:00 pm Kirsten Eisentrager (Penn State) Title: "Hilbert's Tenth Problem for function fields of positive characteristic" Abstract: Hilbert's Tenth Problem in its original form was to find an algorithm to decide, given a multivariate polynomial equation with integer coefficients, whether it has a solution over the integers. In 1970 Matiyasevich, building on work by Davis, Putnam and Robinson, proved that no such algorithm exists, i.e. Hilbert's Tenth Problem is undecidable. Since then, analogues of this problem have been studied by asking the same question for polynomial equations with coefficients and solutions in other commutative rings. In this talk we will discuss some recent undecidability results for function fields of positive characteristic. 4:30-5:30pm Thomas Tucker (Rochester) Title: "Integral points in two-parameter orbits" Abstract: Let K be a number field, let f : P^1 --> P^1 be a nonconstant rational map of degree greater than 1 that is not conjugate to a powering map, let S be a finite set of places of K, and suppose that u,w ∈ P^1(K) are not preperiodic under f. We prove that the set of (m,n) ∈ N^2 such that f^m(u) is S-integral relative to f^n(w) is finite and effectively computable. This may be thought of as a two-parameter analog of a result of Silverman on integral points in orbits of rational maps.  We also discuss so-called Bang-Zsigmondy variants of this question.  This represents joint work Corvaja, Sookdeo, and Zannier. November 19, 2013 at MIT, room 4-163 3:00-4:00pm Roman Holowinsky (The Ohio State University) "Hybrid subconvexity bounds for Rankin-Selberg convolutions" We'll discuss several recent results concerning the subconvexity problem for L-functions.  In all of the cases we shall consider, one benefits analytically from the particular size or form of the L-function's conductor.  The results presented are joint with either Ritabrata Munshi, Nicolas Templier or Zhi Qi. ------------------------- 4:30-5:30pm Michael Larsen (Indiana University) "Type A images of Galois representations and maximality" I will talk about recent joint work with Chun Yin Hui about images of motivic Galois representations of "Type A", i.e., for which every simple composition factor of the Zariski closure of the image is of type A in the Cartan-Killing classification. February 11, 2014 at BC, Cushing 209 Directions 3:00-4:00 pm Tasho Kaletha (Princeton) Title: Rigid inner forms and endoscopy Abstract: The basic version of the local Langlands conjecture predicts a correspondence between Langlands parameters and packets of representations of a given reductive group G defined over a local field F. The more refined version enhances the space of parameters by including representations of certain finite groups and then predicts a correspondence between enhanced parameters and individual representations of G. This refinement is needed in many application, one example being the multiplicity formula for discrete automorphic representations. While the basic version is easy to state for any G, a precise statement of the refined version was not known for general reductive groups over p-adic fields (including classical groups like special linear, symplectic, and special orthogonal groups over division algebras). In this talk, we will propose a uniform and precise statement of the refined local Langlands conjecture for arbitrary connected reductive groups over local fields of characteristic zero. It is based on the construction of a canonical gerb over such a field, whose arithmetic properties lead to a normalization of the endoscopic objects involved in the local Langlands conjecture. Time permitting, we will discuss evidence for the validity of this statement, which includes the case of real groups, as well as certain classes of representations of p-adic groups. 4:30 - 5:30 pm Florian Herzig (Toronto) Title: On mod p local-global compatibility for GL3 in the ordinary case Abstract: Suppose that rhobar : G_{Qp} -> GL3(Fpbar) is a maximally nonsplit, ordinary, Fontaine-Laffaille Galois representation. Then its "extension class" is determined by an invariant in Fpbar. In a global situation, under suitable hypotheses, we show that this invariant can be calculated using GL3(Qp)-representation theory. This is joint work in progress with Stefano Morra. March 11, 2014 at MIT Junecue Suh (Harvard) Eric Urban (Columbia) April 1, 2014 3:00 PM at BC Cushing 209 Directions Nicolas Templier (Princeton) Title: Families of L-functions and their Symmetry Abstract: We consider families of automorphic representations on reductive groups. We establish a quantitative Plancherel equidistribution theorem for the Satake parameters of these families. We proceed via the trace formula by proving uniform bounds for orbital integrals and for trace characters of discrete series representations. The theorem is strong enough to apply to the Katz-Sarnak heuristics on zeros of L-functions making it possible to conjecture the universality class of the distribution of zeros in families. Joint works with S.-W.Shin, P.Sarnak and J.-L.Kim. Zhiwei Yun (Stanford) Title:  Epipelagic representations and rigid local systems  Abstract: Reeder and Yu have constructed in a uniform way certain supercuspidal representations of p-adic groups called "epipelagic representations", using invariant theory studied by Vinberg et al. In the function field case, we will realize these epipelagic representations as local components of automorphic representations, and construct the corresponding Langlands parameters, i.e., local systems over P^1 minus two points. These local systems can be computed explicitly for classical groups, and they give new families of local systems (with monodromy in all types of groups, classical or exceptional) that are expected to be rigid.

#### 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

##### September 26, 2013

Yulan Quin (Tufts)
Title: Boundaries of CAT(0) spaces with Right Angles

##### October 3, 2013

Ailsa Keating (MIT)
Title: Tori in four-dimensional Milnor fibres

##### October 10, 2013

Peter Lambert-Cole (LSU)

##### October 17, 2013

Corrin Clarkson (Columbia)
Title: 3-Manifold Mutations Detected by Geegaard Floer Homology

##### October 24, 2013

Lukas Lewark (Durham)

##### October 31, 2013

Jessica Banks (UQAM)

##### November 7, 2013

Gabor Lippner (Harvard)

##### November 21, 2013

Liam Watson (Glasgow)

#### BC Number Theory/Algebraic Geometry Seminar

To be determined.

#### 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.