Department of Mathematics and Statistics

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

Department Colloquium - Michael Perlman (Queen's University)

Michael Perlman (Queen's University)

Friday, April 16th, 2021

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Michael Perlman (Queen's University)

Title: Measuring hypersurface singularities via differential operators and Hodge theory.

Abstract: Given a polynomial with complex coefficients, its set of zeros is a geometric object known as an algebraic hypersurface. We will discuss two invariants defined via differential operators that can detect and measure singularities of these hypersurfaces: the Bernstein-Sato polynomial and the Hodge ideals. Via the example of the hypersurface defined by the n x n determinant, we will illustrate that these invariants are two sides of the same coin: the mixed Hodge structure.

Michael Perlman is Coleman Postdoctoral Fellow in the Department of Mathematics and Statistics at Queen's University. He obtained his Ph.D. in Mathematics in May 2020 from the University of Notre Dame. His research is in Algebraic Geometry, Commutative Algebra, and their interactions with Representation Theory.

Department Colloquium - Qiyang Han (Rutgers University)

Qiyang Han (Rutgers University)

Friday, March 26th, 2021

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Qiyang Han (Rutgers University)

Title: Multiple isotonic regression: limit distribution theory and confidence intervals.

Abstract: In the first part of the talk, we study limit distributions for the tuning-free max-min block estimators in multiple isotonic regression under both fixed lattice design and random design settings. We show that at a fixed interior point in the design space, the estimation error of the max-min block estimator converges in distribution to a non-Gaussian limit at certain rate depending on the number of vanishing derivatives and certain effective dimension and sample size that drive the asymptotic theory. The limiting distribution can be viewed as a generalization of the well-known Chernoff distribution in univariate problems. The convergence rate is optimal in a local asymptotic minimax sense. In the second part of the talk, we demonstrate how to use this limiting distribution to construct tuning-free pointwise nonparametric confidence intervals in this model, despite the existence of an infinite-dimensional nuisance parameter in the limit distribution that involves multiple unknown partial derivatives of the true regression function. We show that this difficult nuisance parameter can be effectively eliminated by taking advantage of information beyond point estimates in the block max-min and min-max estimators through random weighting. Notably, the construction of the confidence intervals, even new in the univariate setting, requires no more efforts than performing an isotonic regression for once using the block max-min and min-max estimators, and can be easily adapted to other common monotone models. This talk is based on joint work with Hang Deng and Cun-Hui Zhang.

Qiyang Han is an Assistant Professor of Statistics at Rutgers University. He received his Ph.D. in Statistics from University of Washington in 2018. He is broadly interested in mathematical statistics and high dimensional probability. His current research is concentrated on abstract empirical process theory and its applications to nonparametric function estimation, Bayes nonparametrics, and high dimensional statistics.

Department Colloquium - Mike Hill (UCLA)

Mike Hill (University of California, Los Angeles)

Friday, March 19th, 2021

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Mike Hill (UCLA)

Title: Counting exotic spheres.

Abstract: The circle, surfaces, and three manifolds have essentially one smooth structure on them: there is a unique way to "do calculus" on these. For dimensions at least 5, ordinary Euclidean space does too. In 1956, Milnor shocked the mathematical community by showing that this is not the case for spheres: the 7-sphere has "exotic" smooth structures! In this talk, I will discuss the question "how many distinct smooth structures are there on a given sphere?'' In particular, I will describe some work addressing when the only smooth structure on a sphere is the usual one.

Mike Hill is a Professor at the University of California, Los Angeles. His research focus is in algebraic topology. Prof. Hill completed his Ph.D. at the Massachusetts Institute of Technology in 2006. Prior to joining UCLA in 2015, he was a faculty member at the University of Virginia. He is an editor for Mathematische Zeitschrift, Documenta Mathematica and the Transactions of the American Mathematical Society.

Department Colloquium - Boris Hasselblatt (Tufts University)

Boris Hasselblatt (Tufts University)

Friday, March 12th, 2021

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Boris Hasselblatt (Tufts University)

Title: Symmetries of dynamical systems.

Abstract: Examples of dynamical systems with symmetries are not particularly rare, but it is often clear that perturbations can break such symmetry. Thus, it is natural to expect that the presence of symmetries is exceptional. We describe some results about continuous, discrete, and trivial symmetry groups of flows (continuous-time dynamical systems) in the categories of smooth flows and of continuous flows.

Boris Hasselblatt is a professor of mathematics as well as former department chair and Associate Provost at Tufts University. He obtained his PhD from Caltech, before moving to Tufts in 1989. His research centers on smooth, geometrically motivated, and topological dynamical systems with hyperbolic behavior. He has coauthored the book Introduction to the Modern Theory of Dynamical Systems, which is the reference in dynamics, and one of the top 100 most cited books in mathematics. He is a Fellow of the American Mathematical Society, held the Chaire Jean Morlet at CIRM in Marseille, as well as many visiting positions, including at the ETH Z\"{u}rich, the University of Tokyo and the IHES. He cofounded the Journal of Modern Dynamics, Electronic Research Announcements - Mathematical Sciences, and Mathematics Research Reports, and he currently serves as the eleventh Secretary of the American Mathematical Society.

Department Colloquium - Caroline Colijn (Simon Fraser)

Caroline Colijn (Simon Fraser University)

Friday, March 5th, 2021

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Caroline Colijn (Simon Fraser University)

Title: Mathematical models and the case for vaccinating essential workers sooner

Abstract: We develop the first mathematical model incorporating long COVID and COVID complications. It also includes an age- and essential-work contact structure. We compare the model to BC data on incident COVID-19 cases, hospitalizations and deaths. With good agreement we proceed to compare the benefits of different vaccine rollout strategies. We find that where the pandemic is controlled with widespread distancing, limiting the transmission rate, strategies that prioritize high-contact workers early in the program have considerable benefits compared to oldest-first orders. Benefits hold for infections, hospitalizations and deaths, as well as for long COVID and chronic impacts. We estimate large net monetary benefits saved by such a program. Finally, we illustrate the intuition behind this result using a simple model accounting for a higher risk of a severe outcome in some groups (eg the elderly) vs a higher risk of exposure and transmission in others (eg high-contact workers).

Caroline Colijn joined Simon Fraser University's Department of Mathematics in 2018 as a Canada 150 Research Chair in Mathematics for Evolution, Infection and Public Health. Her work is at the interface of mathematics and the epidemiology and evolution of pathogens. Prof. Colijn was recently awarded the Radio-Canada Scientist of the Year for 2020.

Department Colloquium - Idris Assani (University of North Carolina)

Idris Assani (University of North Carolina)

Friday, February 26th, 2021

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Idris Assani (University of North Carolina)

Title: Recurrence and Return Times

Abstract: We present some pointwise results on the topic of the pointwise convergence of multiple ergodic averages. These include our work on the averages along the cubes, on the Furstenberg averages and on the return times. We also raise some open questions.

Idris Assani is a Professor in the Department of Mathematics of at the University of North Carolina at Chapel Hill. His research area is in ergodic theory. He is the author of the research monograph Wiener Wintner Ergodic Theorems about mathematics related to the Wiener–Wintner theorem, and is also the editor of many volumes of collected papers in ergodic theory. Prof. Assani was named as one of the inaugural fellows of the American Mathematical Society in 2012 and one of the 66 most influential mathematicians throughout its history by The Royal Society on World Maths Day in 2017.

Department Colloquium - Yihong Wu (Yale University)

Yihong Wu (Yale University)

Friday, February 12th, 2021

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Yihong Wu (Yale University)

Title: Self-regularizing Property of Nonparametric Maximum Likelihood Estimator in Mixture Models

Abstract: Introduced by Kiefer and Wolfowitz 1956, the nonparametric maximum likelihood estimator (NPMLE) is a widely used methodology for learning mixture models and empirical Bayes estimation. Sidestepping the non-convexity in mixture likelihood, the NPMLE estimates the mixing distribution by maximizing the total likelihood over the space of probability measures, which can be viewed as an extreme form of overparameterization. In this work we discover a surprising property of the NPMLE solution. Consider, for example, a Gaussian mixture model on the real line with a subgaussian mixing distribution. Leveraging complex-analytic techniques, we show that with high probability the NPMLE based on a sample of size n has $O(\log n)$ atoms (mass points), significantly improving the deterministic upper bound of n due to Lindsay 1983. Notably, any such Gaussian mixture is statistically indistinguishable from a finite one with $O(\log n)$ components (and this is tight for certain mixtures). Thus, absent any explicit form of model selection, NPMLE automatically chooses the right model complexity, a property we term self-regularization. Statistical applications and extensions to other exponential families will be given. Time permitting, we will discuss some recent results on optimal regret in empirical Bayes and the role of NPMLE. This is based on joint work with Yury Polyanskiy (MIT).

Yihong Wu is an Associate Professor in the Department of Statistics and Data Science at Yale University. He obtained his Ph.D.~in Electrical Engineering from Princeton University in 2011. He was a Postdoctoral Fellow at the University of Pennsylvania (2011 - 2012), and an Assistant Professor at the University of Illinois at Urbana-Champaign (2013 - 2016). He is broadly interested in theoretical and algorithmic aspects of high-dimensional statistics, information theory, and optimization. He has received many awards, including the Sloan Research Fellowship in Mathematics in 2018, the NSF CAREER award in 2017, the Simons-Berkeley Research Fellowship in 2015, and the Marconi Society Paul Baran Young Scholar Award in 2011.

Department Colloquium - David Banks (Duke University)

David Banks (Duke University)

Friday, February 5th, 2021

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: David Banks (Duke University)

Title: Statistical Challenges in Agent-Based Models.

Abstract: Agent-based models (ABMs) are computational models used to simulate the actions and interactions of agents within a system. Usually, each agent has a relatively simple set of rules for how it responds to its environment and to other agents. These models are used to gain insight into the emergent behavior of complex systems with many agents, in which the emergent behavior depends upon the micro-level behavior of the individuals. ABMs are widely used in many fields, and this talk reviews some of those applications. However, as relatively little work has been done on statistical theory for such models, this talk also points out some of those gaps and recent strategies to address them.

David Banks is a Professor of the Practice of Statistics at Duke University. His research areas include models for dynamic networks, dynamic text networks, adversarial risk analysis, human rights statistics, agent-based models, forensics, and certain topics in high-dimensional data analysis. He is past-president of the Classification Society, and has twice served on the Board of Directors of the American Statistical Association. He is currently the president of the International Society for Business and Industrial Statistics. He is a fellow of the American Statistical Association and of the Institute of Mathematical Statistics. He won the American Statistical Association's Founders Award in 2015.

Department Colloquium - Milen Yakimov (Northeastern University)

Milen Yakimov (Northeastern University)

Friday, January 29th, 2021

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Milen Yakimov (Northeastern University)

Title: Noncommutative Discriminants.

Abstract: The notion of discriminant plays an important role in various algebraic, geometric and combinatorial settings. The discriminant of a noncommutative algebra is modeled on Dedekind's definition for algebraic number fields. The discriminants in the former class have many applications but have only been computed in few situations. We will present an introduction to this subject and will then describe three general theorems for computing discriminants of noncommutative algebras based on Poisson geometry, Representation Theory and Cluster Algebras, respectively. The three theorems can be applied to compute the discriminants of many families of algebras of wide interest: quantum matrices at roots of unity, quantum Weyl algebras, quantum Schubert cell algebras, algebras in noncommutative projective algebraic geometry and others. The talk is based on joint work with Kenneth Brown (Glasgow University), Bach Nguyen (Xavier University) and Kurt Trampel (University of Notre Dame).

Milen Yakimov is a Professor in the Department of Mathematics at Northeastern University. His research interests include noncommutative algebra, quantum groups, Poisson geometry, cluster algebras, representation theory and integrable systems. Before joining Northeastern University, he was the Michael F. and Roberta Nesbit McDonald Professor in the Department of Mathematics at the Louisiana State University. He became a Fellow of the American Mathematical Society in 2018.

Department Colloquium - Mohammad Farazmand (NC State)

Mohammad Farazmand (NC State University)

Friday, January 22nd, 2021

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Mohammad Farazmand (NC State University)

Title: Extreme Events: Dynamics, Prediction and Mitigation.

Abstract: A wide range of natural and engineering systems exhibit extreme events, i.e., spontaneous intermittent behavior manifested through sporadic bursts in the time series of their observables. Examples include ocean rogue waves, intermittency in turbulence, extreme weather patterns and epileptic seizure. Because of their undesirable impact on the system or the surrounding environment, the real-time prediction and mitigation of extreme events is of great interest. In this talk, I will discuss three aspects of extreme events. First, I introduce a variational method that unveils the mechanisms underpinning the formation of extreme events. Next, I show how this framework enables the data-driven, real-time prediction of extreme events. I demonstrate the application of this method with several examples, including the prediction of ocean rogue waves and the intermittent energy dissipation bursts in turbulent fluid flows. Finally, I will discuss a closed-loop adaptive control and a delay feedback control for mitigating extreme events.

Mohammad Farazmand is an assistant professor within the Department of Mathematics at North Carolina State University. He works on the fields of data-driven modeling and analysis of complex systems. He was previously a postdoctoral associate at the Massachusetts Institute of Technology and a J. Ford Postdoctoral Fellow at the Georgia Institute of Technology. He obtained his Ph.D. in 2014 from ETH Zurich under the supervision of George Haller.