Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Department Colloquium - Bob Ross Barmish (U. of Wisconsin-Madison)

Bob Ross Barmish

Friday, December 1st, 2017

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

Speaker: Bob Ross Barmish, University of Wisconsin-Madison

Title: When the expected value is not expected

Abstract: Our motivation for the research to be described is derived from the following fact: The expected value of a random variable~$X$, denoted~$\mathbb{E}(X)$, is often inconsistent with what human beings may actually expect based on psychological considerations. This is particularly important when predictions involving life-threatening situations arise. To bring this issue into sharp focus, this seminar begins with a set of questions related to the recent rampage of Hurricane Irma in the State of Florida. When the use of empirical data leads to an expected value forecast of storm surge wave height which is unduly pessimistic, will the "boy who cried wolf" effect be in play the next time a hurricane approaches the mainland? On the other hand, if the formally calculated expected wave height is too optimistic, might it be the case that many will take inadequate protective measures? Based on such considerations, we define a new alternative to~$\mathbb{E}(X)$ which we believe is quite useful for "mission critical" situations with downside risk being of paramount concern. We call this new metric the {\it Conservative Expected Value} and denote it by~$\mathbb{CEV}(X)$. In this talk, we provide the technical definition of the $\mathbb{CEV}$, compare it with the classical expected value and describe some aspects of the rich mathematical theory which accompanies it. We also include a description of some of the studies we have conducted using the~$\mathbb{CEV}$ to evaluate historical data.

Bob Ross Barmish (University of Wisconsin): B. Ross Barmish received his Ph.D. degree in electrical engineering from Cornell University in 1975. From 1975 to 1978, he was an Assistant Professor of Engineering and Applied Science at Yale University. From 1978 to 1984, he was as an Associate Professor of Electrical Engineering at the University of Rochester. He joined the University of Wisconsin{Madison in 1984, where he is currently a Professor ofelectrical and computer engineering. His research interests include robustness of dynamical systems, building a bridge between feedback control theory, and trading in complex financial markets. Prof. Barmish was a recipient of the Best Paper Award (1986-1989 and 1990-1992) from the International Federation of Automatic Control, and the 2013 Bode Prize by the IEEE Control Systems Society.

Free Probability Seminar - Pei-Lun Tseng (Queen's University)

Thursday, November 30th, 2017

Time: 4:00-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Pei-Lun Tseng

Title: Infinitesimal Laws of non-commutative random variables, II

Abstract:  In this talk, we will focus on a single type B variable, and introduce the corresponding infinitesimal law. In addition, we will also define the free additive convolution for infinitesimal laws, and to see the relation between the type B laws and the infinitesimal laws.

Free Probability and Random Matrices Seminar Webpage:

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.

Number Theory - Multiple Speakers

Wednesday, November 29th, 2017

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

1st Speaker:  M. Ram Murty

Title:  Hilbert’s tenth problem over number fields

Abstract:   Hilbert's tenth problem for rings of integers of number fields remains open in general,

although a conditional negative solution was obtained by Mazur and Rubin assuming some unproved conjectures about the Shafarevich-Tate groups of elliptic curves. In this talk, we highlight how the non-vanishing of certain L-functions is related to this problem. In particular, we show that Hilbert's tenth problem for rings of integers of number fields is unsolvable assuming the automorphy of L-functions attached to elliptic curves and the rank part of the Birch and Swinnerton-Dyer conjecture. This is joint work with Hector Pasten.

2nd Speaker:  François Séguin

Title:  A lower bound for the two-variable Artin conjecture

Abstract:  In 1927, Artin conjectured that any integer other than -1 or a perfect square generates the multiplicative group Z/pZ× for infinitely many p. In a 2000 article, Moree and Stevenhagen considered a two-variable version of this problem, and proved a positive density result conditionally to the generalized Riemann Hypothesis by adapting a 1967 proof by Hooley for the original conjecture. During this talk, we will prove an unconditional lower bound for this two-variable problem. This is joint work with Ram Murty and Cameron Stewart.

3rd Speaker:  Arpita Kar

Title:  On a Conjecture of Bateman about $r_5(n)$

Abstract:  Let $r_5(n)$ be the number of ways of writing $n$ as a sum of five integer squares. In his study of this function, Bateman was led to formulate a conjecture regarding the sum $$\sum_{|j| \leq \sqrt{n}}\sigma(n-j^2)$$ where $\sigma(n)$ is the sum of positive divisors of $n$. We give a proof of Bateman's conjecture in the case $n$ is square-free and congruent to $1$ (mod $4$). This is joint work with Prof. Ram Murty.

4th Speaker:  Siddhi Pathak

Title:  Derivatives of L-series and generalized Stieltjes constants

Abstract:  Generalized Stieltjes constants occur as coefficients of (s-1)^k in the Laurent series expansion of certain Dirichlet series around s=1. The connection between these generalized Stieltjes constants and derivatives of L(s,f) for periodic arithmetical functions f, at s=1 is known. We utilize this link to throw light on the arithmetic nature of L'(1,f) and certain Stieltjes constants. In particular, if p is an odd prime greater than 7, then we deduce the transcendence of at least (p-7)/2 of the generalized Stieltjes constants, {gamma_1(a,p) : 1 \leq a < p }, conditional on a conjecture of S. Gun, M. Ram Murty and P. Rath.

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.

Number Theory - Jung-Jo Lee (Queen's University)

Wednesday, November 22nd, 2017

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

Speaker: Jung-Jo Lee

Title: Cyclotomic units, values of the Riemann zeta function and p-adic measures

Abstract:  I would explain the connection between the cyclotomic units and the values of complex Riemann zeta function via higher logarithmic derivative maps. We can use Mahler transform to construct a p-adic zeta function.

Note: I consider a series of talks, each "hopefully" self-contained.

Talk 2 : Euler system of cyclotomic units and Iwasawa's main conjecture

Talk 3 : Euler system of Heegner points and Birch and Swinnerton-Dyer conjecture

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.

Number Theory - Peng-Jie Wong

Wednesday, November 15th, 2017

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

Speaker: Peng-Jie Wong

Title: Base change and the Langlands reciprocity conjecture

Abstract: In light of Artin reciprocity, Langlands enunciated a reciprocity conjecture asserting that any complex Galois representation is automorphic. Around 1980, Langlands established certain cyclic base change and proved his reciprocity conjecture for 2-dimensional Galois representations with projective image isomorphic to $A_4$. In this talk, we will try to give a motivation of the Langlands reciprocity conjecture and discuss its relation to base change. If time permits, we will explain how the conjectural base change leads to Langlands reciprocity for all 3-dimensional Galois representations with solvable image.

This is a joint work with M. Ram Murty and V. Kumar Murty