Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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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.

Dynamics, Geometry, & Groups - Alena Erchenko (Stony Brook)

Thursday, April 1st, 2021

Time: 1:30 p.m Place:

Speaker: Alena Erchenko (Stony Brook University)

Title: Riemannian Anosov extension.

Abstract: Consider a smooth Riemannian manifold Σ with strictly convex spherical boundary, hyperbolic trapped set (possibly empty) and no conjugate points. We show that Σ can be isometrically embedded into a closed Riemannian manifold with Anosov geodesic flow. We will explain one of the main ingredients which is the analysis of the behavior of Jacobi fields. We also discuss some applications of the main result. This is a joint work with Dong Chen and Andrey Gogolev.

Statistics & Biostatistics - Bang Liu (University of Montreal)

Thursday, April 1st, 2021

Time: 4:00pm Place:

Speaker: Bang Liu (University of Montreal)

Title: Data, Knowledge, and Logic: Modeling and Reasoning for Natural Language Understanding.

Abstract: Existing deep learning-based techniques for different NLU tasks are mostly data-intensive and domain-sensitive. However, creating large-amount and high-quality training datasets for NLU tasks, e.g., question answering, are both expensive and time-consuming. In this talk, we will introduce our research on data generation, knowledge expansion, and reasoning over graphs. Specifically, for data augmentation, we generate large-scale and high-quality question-answer pairs from unlabeled text to augment the training data for question answering. For knowledge expansion, we create and expand an ontology with newly discovered concepts or entities to capture the emerging knowledge in the world and keep the ontology dynamically updated. For reasoning over graphs, we propose a reinforcement learning-based relational reasoning framework, R5, that reasons over relational data and learns the underlying compositional logical rules. Our long-term vision is to design low-resource, knowledge-empowered, and transferable NLU systems and apply them to different domains.

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.

Dynamics, Geometry, & Groups - Maxime Fortier-Bourque (Glasgow)

Thursday, March 25th, 2021

Time: 1:30 p.m Place:

Speaker: Maxime Fortier-Bourque (University of Glasgow)

Title: Geometric inequalities via trace formulas.

Abstract: The systole of a metric space is the length of its shortest closed geodesic and its kissing number is the number of distinct homotopy classes of closed geodesics realizing the systole. Given a moduli space of metric spaces, like the space of flat tori of a given dimension or the space of closed hyperbolic surfaces of a given genus, the goal is to find how large these quantities can get. After stating what is known so far, I will describe ongoing joint work with Bram Petri in which we obtain new upper bounds on the systole and kissing number of hyperbolic surfaces. Our approach is based on the Selberg trace formula and is inspired by work of Cohn and Elkies on the maximal density of sphere packings in Euclidean spaces.

Statistics & Biostatistics - Wen Teng (Queen's University)

Thursday, March 25th, 2021

Time: 4:00pm Place:

Speaker: Wen Teng (Queen's University)

Title: A non-parametric simultaneous confidence band for biomarker effect on the restricted mean survival time.

Abstract: Study of prognostic and predictive biomarkers play an important role in the design and analysis of clinical trials. The Cox proportional hazards model is often used to study the biomarker main effect and the treatment-biomarker interaction effect for survival data. The estimated effects can be biased if the proportional hazards assumption is violated. The restricted mean survival time is becoming popular in clinical studies for having a clear intuitive interpretation. In this paper, we first propose non-parametric methods to make statistical inference for the one-sample problem of the biomarker effect on the restricted mean survival time; we then extend the methods to the two-sample problem for studying the difference in the biomarker effects between samples, or treatment groups in clinical trials. For a given biomarker, the restricted mean survival time is estimated by kernel smoothing methods adjusted by the inverse probability of censoring weight. We prove the consistency for the estimates and develop simultaneous confidence bands for the biomarker effects on the restricted mean survival time. The simultaneous confidence bands were evaluated in extensive simulation studies and were found to have good finite sample performance. We then apply the proposed methods to a breast cancer study con-ducted by the Breast International Group (BIG) to illustrate how the Ki67 biomarker affects the survival time of patients, compared between the treatment groups.

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.

Dynamics, Geometry, & Groups - Ioannis Iakovoglou (IMB)

Thursday, March 18th, 2021

Time: 1:30 p.m Place:

Speaker: Ioannis Iakovoglou (Institut de Mathématiques de Bourgogne)

Title: Anosov flows in dimension 3: a dynamical game describing the actions of surgeries on the foliations.

Abstract: From every Anosov flow in dimension 3 it is possible to construct infinitely many others via Dehn-Goodman-Fried surgery. Similarly, to the theorem of Lickorish-Wallace stating that any (closed orientable and connected) 3-manifold is obtained by Dehn surgeries on the 3-sphere, conjecturally the same thing happens for (transitive with orientable foliations) Anosov flows in dimension 3. Motivated by this question, in a recent work with C.Bonatti we propose a dynamical game on the plane as a means to understand the foliations of an Anosov flow after surgery. In this talk, I will introduce this dynamical game on the plane together with some interesting questions around it and their relation with Anosov flows in dimension 3.

Statistics & Biostatistics - Wei Tu (Queen's University)

Thursday, March 18th, 2021

Time: 4:00pm Place:

Speaker: Wei Tu (Queen's University)

Title: Differential privacy in health data

Abstract: The protection of individual patient privacy is essential in health care research. Privacy-protecting data analysis has a long history under the name of “statistical disclosure control” in statistics. Differential privacy, emerging from the theoretical computer science literature, has become popular over the last decade due to its intuitive formulation and formal privacy guarantee, and is at its early stages of implementation in industry, government and academia. In this talk, I will introduce the framework of differential privacy and present a few applications in health research. Specifically, a differentially private Kaplan-Meier estimate using the recently proposed Gaussian differential privacy framework will be presented, as well as differential private learning in training clinical prediction task using EHR and medical imaging data.