Department of Mathematics and Statistics

Friday, November 17th, 2017

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Mokshay Madiman

Title: The convexifying effect of Minkowski summation

Abstract: For a compact subset $A$ of $R^d$, let $A(k)$ be the Minkowski sum of k copies of A, scaled by $1/k$. By a 1969 theorem of Emerson, Folkmann, Greenleaf, Shapley and Starr, $A(k)$ approaches the convex hull of A in Hausdorff distance as k goes to infinity; this fact has important applications in a number of areas including mathematical economics. A few years ago, the speaker conjectured that the volume of A(k) is non-decreasing in k, or in other words, that when the volume deficit between the convex hull of A and A(k) goes to 0, it actually does so monotonically. While this conjecture holds true in dimension 1 (as independently observed by F. Barthe), we show that it fails in dimension 12 or greater. Then we consider whether one can have monotonicity of convergence of when non-convexity is measured in alternate ways. Our main positive result is that Schneider’s index of non-convexity of A(k) converges monotonically to 0 as k increases; even the convergence does not seem to have been known before. As a by-product, we also obtain optimal rates of convergence. We also obtain analogous results for the Hausdorff distance to the convex hull, as well as for the inner radius, and demonstrate applications to discrepancy theory. Joint work with Matthieu Fradelizi (Marne-la-Vallée), Arnaud Marsiglietti (CalTech), and Artem Zvavitch (Kent State).

Mokshay Madiman (University of Delaware): Mokshay Madiman has been an Associate Professor in the Department of Mathematical Sciences at the University of Delaware since January 2013. Dr. Madiman received his Ph.D. degree in applied mathematics from Brown University in 2005. From 2005 to 2012, he worked at the Department of Statistics at Yale University, New Haven, CT, rst as a Gibbs Assistant Professor, then as an Assistant Professor, and nally as an Associate Professor of Statistics and Applied Mathematics. From 2014 to 2017, he was also an Adjunct Professor of Mathematics at the Tata Institute of Fundamental Research, Mumbai. He has spent a semester each in visiting positions at the Tata Institute; the Indian Institute of Science, Bangalore; Princeton University; and the Institute for Mathematics and its Applications, Minneapolis. Dr. Madiman's research is primarily in probability and information theory, but also interacts with combinatorics, functional analysis, and statistics.

Friday, November 10th, 2017

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Hector Pasten

Title: In Quest of Arithmetic Derivatives

Abstract: There are well-known arithmetic analogies between integers and polynomials (or more generally, holomorphic functions), specially with respect to solutions of Diophantine equations. A crucial aspect that is missing in this analogy is that for polynomials and holomorphic functions one can use derivatives, while for integers there is no direct substitute of this operation. After some motivation, I will discuss a couple of possible candidates for arithmetic derivatives.

Hector Pasten (Harvard University): Hector Pasten (Chile, 1988) is since 2014 a Benjamin Peirce Fellow at Harvard University. He was also a member of the Institute for Advanced Study at Princeton (2015-2016). Pasten received a Ph.D. in 2010 from Universidad de Concepcion in the subject of mathematical logic and non-archimedean analysis. Then he received a Ph.D. in 2014 from Queen's University in the subject of number theory. Among other distinctions, Pasten was awarded the Governor-General of Canada Academic Gold Medal (2014), the Doctoral Prize of the Canadian Mathematical Society (2015), and the Mathematical Council of the Americas Prize (2017). His current research is in number theory and related areas, addressing topics such as decidability of arithmetic structures, Diophantine approximation, and analogies between number theory and value distribution.

Friday, November 3rd, 2017

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Nasser Sadeghkhani

Title: Bayesian Predictive Density Estimation with Additional Information

Abstract: In the context of Bayesian theory and decision theory, the estimation of a predictive density of a random variable represents an important and challenging problem. Often the times there is some additional information at our disposal which is unduly being ignored. In this talk, we deal with strategies to take into account this kind of information, in order to obtain e ective and sometimes better performing predictive densities than others in the literature.

Nasser Sadeghkhani (Queen's University): Nasser Sadeghkhani earned his Ph.D in Mathematics (Statistics) from the Universite de Sherbrooke in 2017 under the supervision of Eric Marchand. He recently joined the Department of Mathematics and Statistics at Queen's University as a Coleman Postdoctoral Fellow. Dr Sadeghkhani's research interests include Bayesian Statistics, Multivariate Statistics, Survival Analysis, Decision Theory, and Predictive Inference.

Friday, October 27th, 2017

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Daniel Wise

Title: The Cubical Route to Understanding Groups

Abstract: Cube complexes have come to play an increasingly central role within geometric group theory, as their connection to right-angled Artin groups provides a powerful combinatorial bridge between geometry and algebra. This talk will introduce nonpositively curved cube complexes, and then describe the developments that recently culminated in the resolution of the virtual Haken conjecture for 3-manifolds, and simultaneously dramatically extended our understanding of many in nite groups.

Daniel Wise (McGill University): Daniel Wise earned his Ph.D. in Mathematics from Princeton University in 1996, with a thesis on "Non Positively Curved Squared Complexes, Aperiodic Tilings and Non-Residually Finite Groups". After a NSF Postdoctoral Fellowship at UC Berkeley (1996-1997), he became H.C. Wang Assistant Professor at Cornell (1997-2000), Visiting Assistant Professor at Brandeis (2000-2001), and Assistant Professor at McGill University (2001-2004). Wise was promoted to Associate Professor in 2004, Full Professor in 2009, and appointed James McGill Professor in 2013. He has also served as Chair of the Institut Henri Poincare (2015-2016). Prof Wise received the Oswald Weblen Prize in Geometry in 2013, and the Je ery-Williams the CRM-Fields-PIMS Prizes in 2016. In 2014, he was ICM speaker and became Fellow of the Royal Society of Canada and, in 2016, he became Guggenheim Fellow. Wise's research is dedicated to the theory of in nite groups - with applications to Geometry and Topology. Specically, he studies geometric group theory, metric spaces of nonpositive curvature, residually finite groups, subgroup separability, 3-dimensional manifolds, coherence.

Wednesday, October 25th, 2017

Time: 1:30 p.m.  Place: Jeffery Hall 319

Speaker: Richard Leyland

Title: Galois representations of CM Elliptic Curves

Abstract: We introduce the notion of the mod-N Galois representation $\rho_{E/F,N}$ attached to an elliptic curve $E/F$. Motivated by conjectures of Frey and Mazur, we aim to determine all elliptic curves $E’/F$ such that $\rho_{E’/F,N}\cong \rho_{E/F,N}$ in the case where both $E$ and $E’$ have complex multiplication. In this talk, we introduce the problems, terminology currently being worked on and present a small result in the case where $E$ and $E’$ have complex multiplication by different imaginary quadratic fields.