Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Department Colloquium

Special Colloquium - Jason Klusowski (Yale University)

Jason Klusowski

Wednesday, November 29th, 2017

Time: 3:30 p.m.  Place: Jeffery Hall 234

Speaker: Jason Klusowski

Title: Counting motifs and connected components of large graphs via subgraph sampling

Abstract: Learning properties of large graphs from samples is an important problem in statistical network analysis. We revisit the problem of estimating the numbers of connected components in a graph of $N$ vertices based on the subgraph sampling model, where we observe the subgraph induced by $n$ vertices drawn uniformly at random. The key question is whether it is possible to achieve accurate estimation, i.e., vanishing normalized mean-square error, by sampling a vanishing fraction of the vertices. We show that it is possible by accessing only sublinear number of samples if the graph does not contains high-degree vertices or long induced cycles; otherwise it is impossible. Optimal sample complexity bounds are obtained for several classes of graphs including forests, cliques, and chordal graphs. The methodology relies on topological identities of graph homomorphism numbers, which, in turn, also play a key role proving minimax lower bounds based on constructing random instances of graphs with matching structures of small subgraphs. We will also discuss results for the neighborhood sampling model, where we observe the edges between the sampled vertices and their neighbors.

Jason Klusowski (Yale University): Jason M. Klusowski obtained his B.Sc. in Mathematics and Statistics in 2013 from the University of Manitoba, receiving the Robert Ross McLaughlin Scholarship in Mathematics, the Faculty of Science Medal in B.Sc., and the Governor General's Silver Medal. In 2013 he was received the NSERC Alexander Graham Bell Canada Graduate Scholarship and joined Yale University where he is currently a Ph.D. candidate in Statistics and Data Science, under the supervision of Prof. Andrew Barron. Jason Klusowski's research interests include the theoretical and computational aspects of neural networks, approximation algorithms for networks, high-dimensional function estimation, mixture models, and shape constrained estimation.

Department Colloquium - Jeffery Rosenthal (University of Toronto)

Jeffery Rosenthal

Friday, November 24th, 2017

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

Speaker: Jeffery Rosenthal, University of Toronto

Title: Adaptive MCMC for Everyone

Abstract: Markov chain Monte Carlo (MCMC) algorithms, such as the Metropolis Algorithm and the Gibbs Sampler, are an extremely useful and popular method of approximately sampling from complicated probability distributions. Adaptive MCMC attempts to automatically modify the algorithm while it runs, to improve its performance on the fly. However, such adaptation often destroys the ergodicity properties necessary for the algorithm to be valid. In this talk, we first illustrate MCMC algorithms using simple graphical Java applets. We then discuss adaptive MCMC, and present examples and theorems concerning its ergodicity and efficiency. We close with some recent ideas which make adaptive MCMC more widely applicable in broader contexts.

Special Colloquium - Jenny Wilson (Stanford University)

Jenny Wilson

Wednesday, November 22nd, 2017

Time: 3:30 p.m.  Place: Jeffery Hall 234

Speaker: Jenny Wilson

Title: Dynamics, geometry, and the moduli space of Riemann surfaces

Abstract: The ordered configuration space $F_k(M)$ of a manifold M is the space of ordered k-tuples of distinct points in M. For a fixed manifold M, as k increases, we might expect the topology of these configuration spaces to become increasingly complicated. Church and others showed, however, that when M is connected and open, there is a representation-theoretic sense in which these spaces stabilize. In this talk, I will explain these stability patterns, and describe higher-order stability phenomena established in recent work joint with Jeremy Miller. This project was inspired by work-in-progress of Galatius--Kupers--Randal-Williams.

Jenny Wilson (Stanford University): Jenny Wilson obtained her B.Sc. (with Honours) in Mathematics from Queen's University in 2009, and her Ph.D. in Mathematics from the University of Chicago in 2014. She then joined Stanford University, where she is Szego Assistant Professor.  The awards received by Dr. Wilson while at Queen's University include the Irene MacRae Prize in Mathematics and Statistics, the Medal in Mathematics and Statistics, the Governor General's Academic Silver Medal, and the NSERC Undergraduate Student Research Award, all in 2009. She also received two NSERC Postgraduate Fellowships (PGS M in 2009-2010 and PGS D in 2011-2014), the McCormick Fellowship (2009-2011), the Lawrence and Josephine Graves Teaching Prize at the University of Chigago (2013), and the AMS-Simons Travel Grant (2015-2018). Jenny Wilson's research involves applications of commutative algebra and representation theory to study algebraic structures in topology and geometric group theory. In recent work, she has investigated
con guration spaces of points in a manifold, Torelli groups, and certain congruence subgroups.

Special Colloquium - Alex Wright (Stanford University)

Alex Wright (Stanford University)

Monday, November 20th, 2017

Time: 4:30 p.m.  Place: Jeffery Hall 118

Speaker: Alex Wright

Title: Dynamics, geometry, and the moduli space of Riemann surfaces

Abstract: The moduli space of Riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in Teichmuller dynamics lead to connections with homogeneous spaces, algebraic geometry, dynamics, and other areas. The talk will mention joint works with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.

Alex Wright (Stanford University): Alex Wright obtained his Ph.d. in Mathematics from the University of Chicago in 2014. He was awarded the Clay Research Fellowship in 2014, and joined Stanford University as a Visiting Fellow. In 2015, he was also Postdoctoral Fellow at the Mathematical Sciences Research Institute in Berkeley and Member of the Institute for Advanced Study in Princeton. Since 2016, Dr. Wright is Acting Assistant Professor at Stanford University. Other awards received by Alex Wright include the NSERC Julie Payette Award (2008-2009) and the NSERC Postgraduate Scholarship (2009-2012). Dr. Wright's research lies at the intersection of dynamical systems and algebraic geometry, and was published in the most prestigious mathematical journals, including Annals of Mathematics and Inventiones Mathematicae.

Department Colloquium - Mokshay Madiman (University of Delaware)

Mokshay Madiman (University of Delaware)

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.

Department Colloquium - Hector Pasten (Harvard University)

Hector Pasten (Harvard University)

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.

Department Colloquium - Nasser Sadeghkhani (Queen's University)

Nasser Sadeghkhani (Queen's University)

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.

Department Colloquium - Daniel Wise (McGill University)

Daniel Wise (McGill University)

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.