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Algebra & Geometry Seminar

Monday, April 11th, 2022

Time: 4:30 p.m.  Place: Online via Zoom (contact Kaveh Mousavand for Zoom link)

Speaker: Emily Cliff (Universite de Sherbrooke)

Title: Moduli spaces of principal 2-group bundles and a categorification of the Freed-Quinn line bundle

Abstract: A 2-group is a higher categorical analogue of a group, while a smooth 2-group is a higher categorical analogue of a Lie group. An important example is the string 2-group in the sense of Schommer-Pries. We study the notion of principal bundles for smooth 2-groups, and investigate the moduli "space" of such objects. In particular in the case of flat principal bundles for a finite 2-group over a Riemann surface, we prove that the moduli space gives a categorification of the Freed--Quinn line bundle. This line bundle has as its global sections the state space of Chern--Simons theory for the underlying finite group. We can also use our results to better understand the notion of geometric string structures (as previously studied by Waldorf and Stolz--Teichner). This is based on joint work with Dan Berwick-Evans, Laura Murray, Apurva Nakade, and Emma Phillips.

Website details here:

Math & Stats Department Colloquium

Friday, April 8th, 2022

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

Speaker: Runmin Wang (Southern Methodist University)

Title: Statistical Inference for Change Points in High-Dimensional Data

Abstract: Estimation and testing of change points in high-dimensional data have wide applications in many disciplines, such as biological science, economics and finance. In this talk, we introduce a new U-statistic based approach to both problems and show its advantage over several existing methods via theory and simulations. The talk consists of two parts. In the first part, we will introduce a new test based on U-statistics for testing a mean shift in high-dimensional data. The test aims to detect dense alternatives and is tuning parameter free. At the core of our theory, we show weak convergence of a sequential U-statistic based process, and derive the limiting distribution under both the null and alternatives. In the second part, we will discuss a change point location estimator which maximizes a new U-statistic based objective function. Under mild and easily interpretable assumptions, we derive its convergence rate and asymptotic distribution after suitable centering and normalization. A comparison with the popular least squares based approach illustrates the theoretical advantage of ours. A bootstrap-based approach is also proposed to construct a confidence interval with accurate coverage, which is corroborated by simulation results. We shall illustrate our method using a real data example at the end of the talk.

Runmin Wang is an Assistant Professor in the Department of Statistical Science at Southern Methodist University. He got his Ph.D. in Statistics from the University of Illinois at Urbana-Champaign in 2020. His research interests include change point inference, high-dimensional data and time series analysis.

Algebra & Geometry Seminar

Monday, April 4th, 2022

Time: 4:30 p.m.  Place: Online via Zoom (contact Kaveh Mousavand for Zoom link)

Speaker: Matthew Mastroeni (Iowa State University)

Title: Chow rings of matroids are Koszul

Abstract: The Chow ring of an algebraic variety is an algebro-geometric analog of the cohomology ring of a smooth manifold that encodes important information about the intersections between its subvarieties. Feichtner and Yuzvinsky computed a presentation for the Chow ring of a smooth toric variety associated to a matroid (and some other data) which is now called the Chow ring of the matroid. These rings have garnered significant attention in recent years thanks to their role in establishing long-standing conjectures on the combinatorics of matroids, including the resolution of the Heron-Rota-Welsh Conjecture by Adiprasito, Huh, and Katz and the resolution of the Top-Heavy Conjecture by Braden, Huh, Matherne, Proudfoot, and Wang.

From a commutative algebra standpoint, Chow rings of matroids are very nice graded Artinian Gorenstein rings defined by quadratic relations, and so, a natural conjecture posed by Dotsenko is that the Chow ring of a matroid is always Koszul. In this talk, we will discuss how the combinatorics of a matroid influences algebraic properties of its Chow ring, culminating in recent joint work with Jason McCullough giving an affirmative answer to Dotsenko’s conjecture.

Website details here:

Statistics Seminar

Wednesday, March 30th, 2022

Time: 10:00 a.m.  Place: Online via Zoom (contact Brian Ling for Zoom link)

Speaker: Chuan-Fa Tang (Department of Mathematical Sciences at University of Texas at Dallas)

Title: Taylor's law for semivariance and higher moments of heavy-tailed distributions

Abstract: The power law relates the population mean and variance is known as Taylor's law proposed by Taylor in 1961. We generalize Taylor's law from the light-tailed distributions to heavy-tailed distribution with infinite mean. Instead of population moments, we consider the power-law between the sample mean and many other sample statistics, such as the sample upper and lower semivariance, the skewness, the kurtosis, and higher moments of a random sample. We show that, as the sample size increases, the preceding sample statistics increase asymptotically in direct proportion to the power of the sample mean. These power laws characterize the asymptotic behavior of commonly used measures of the risk-adjusted performance of investments, such as the Sortino ratio, the Sharpe ratio, the potential upside ratio, and the Farinelli-Tibiletti ratio, when returns follow a heavy-tailed nonnegative distribution. In addition, we find the asymptotic distribution and moments of the number of observations exceeding the sample mean. We propose estimators of tail-index based on these scaling laws and the number of observations exceeding the sample mean and compare these estimators with some prior estimators.

Curves Seminar

Monday, March 28th, 2022

Time: 11:00 a.m.  Place: Jeffery Hall Room 222, or Online via Zoom (contact Deepanshu Prasad for Zoom link)

Speaker: Deepanshu Prasad

Title: Broken Circuit Complexes for a Matroids

Abstract: I will construct a standard $\mathcal{K}$-basis for the Orlik-Solomon algebra $A(\mathcal{A})$ by using broken circuit module and hence, prove that the Orlik-Solomon algebra is a free $\mathcal{K}$-module.

Math & Stats Department Colloquium

Friday, April 1st, 2022

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

Speaker: Timothy Chan (University of Toronto)

Title: An Inverse Optimization Approach to Measuring Clinical Pathway Concordance

Abstract: Clinical pathways outline standardized processes in the delivery of care for a specific disease. Patient journeys through the healthcare system, however, can deviate substantially from these pathways. Given the positive benefits of clinical pathways, it is important to measure the concordance of patient pathways so that variations in health system performance or bottlenecks in the delivery of care can be detected, monitored, and acted upon. This paper proposes the first data-driven inverse optimization approach to measuring pathway concordance in any problem context. Our specific application considers clinical pathway concordance for stage III colon cancer. We develop a novel concordance metric and demonstrate using real patient data from Ontario, Canada that it has a statistically significant association with survival. Our methodological approach considers a patient’s journey as a walk in a directed graph, where the costs on the arcs are derived by solving an inverse shortest path problem. The inverse optimization model uses two sources of information to find the arc costs: reference pathways developed by a provincial cancer agency (primary) and data from real-world patient-related activity from patients with both positive and negative clinical outcomes (secondary). Thus, our inverse optimization framework extends existing models by including data points of both varying “primacy” and “alignment”. Data primacy is addressed through a two-stage approach to imputing the cost vector, whereas data alignment is addressed by a hybrid objective function that aims to minimize and maximize suboptimality error for different subsets of input data.

Timothy Chan is the Canada Research Chair in Novel Optimization and Analytics in Health, a Professor in the department of Mechanical and Industrial Engineering, the Director of the Centre for Analytics and AI Engineering, the Associate Director, Research and Thematic Programming of the Data Sciences Institute, and a Senior Fellow of Massey College at the University of Toronto. His primary research interests are in operations research, optimization, and applied machine learning, with applications in healthcare, medicine, sustainability, and sports.

Statistics Seminar

Wednesday, March 23rd, 2022

Time: 10:00 a.m.  Place: Online via Zoom (contact Brian Ling for Zoom link)

Speaker: Na Li (Queen's University)

Title: Bootstrap adjustment for predictive classification

Abstract: In clinical practice, it is important to identify a subgroup of patients who may benefit more in terms of a clinical outcome from a given treatment. The subgroup is usually induced by a continuous predictive biomarker and an associated unknown cutpoint, and this predictive classification problem is formulated as testing the significance of the interaction between the treatment and the subgroup indicator. Two commonly adopted procedures, minimum p-value and profile tests, are not reliable due to the inflated Type I error and/or identifiability issues. We propose bootstrap-based adjustments for various types of outcomes and establish their asymptotic validity. The proposed methods are applied to clinical trial data.

Curves Seminar

Monday, March 21st, 2022

Time: 11:00 a.m.  Place: Jeffery Hall Room 222, or Online via Zoom (contact Deepanshu Prasad for Zoom link)

Speaker: Alexandre (Sasha) Zotine

Title: Broken Circuit Complexes for a Matroids

Abstract: In this talk, we'll introduce broken circuits, broken circuit complexes, and relate the coefficients of the characteristic polynomial of a matroid to its broken circuit complexes.

Math & Stats Department Colloquium

Friday, March 25th, 2022

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

Speaker: Peter Olver (University of Minnesota)

Title: Fractalization and Quantization in Dispersive Systems

Abstract: The evolution, through spatially periodic linear dispersion, of rough initial data produces fractal, non-differentiable profiles at irrational times and, for asymptotically polynomial dispersion relations, quantized structures at rational times. Such phenomena have been observed in dispersive wave models, optics, and quantum mechanics, and lead to intriguing connections with exponential sums arising in number theory. Ramifications and recent progress on the analysis, numerics, and extensions to nonlinear wave models, both integrable and non-integrable, will be presented. Time permitting, recent related results for the Fermi-Pasta-Ulam problem will also be discussed.

Peter Olver is a Full Professor in the School of Mathematics at the University of Minnesota since 1985. He served as the Head of the Department from 2008 to 2020. He is a Fellow of the American Mathematical Society, the Society for Industrial and Applied Mathematics (SIAM), and the Institute of Physics, UK. His research interests revolve around the applications of symmetry and Lie groups to differential equations. Over the years, he has contributed to a wide range of fields, including mathematical physics, fluid mechanics, elasticity, quantum mechanics, Hamiltonian mechanics, the calculus of variations, differential geometry, classical invariant theory, computer vision, geometric numerical methods.

Statistics Seminar

Wednesday, March 16th, 2022

Time: 10:00 a.m.  Place: Online via Zoom (contact Brian Ling for Zoom link)

Speaker: Guanhua Fang (Baidu USA)

Title: Advances in Machine Learning with Heavy-tailed Distributions

Abstract: In many fields such as telecommunications, survival analysis, quality control, online recommendation, and reinforcement learning, one often encounters situations where the data sources behave normally most of the time, but sometimes could become hetero genous and unstructured. Learning in these applications should consider reward distributions with tails heavier than the normal distribution.

In the literature, a remarkable M-estimator proposed by Catoni (2012) has been shown to be rate-optimal in mean estimation problems with finite variance condition. During this talk, I will discuss more advanced statistical learning results based on Catoni-type estimators especially in the situations with infinite variance or presence of contaminated observations. Several interesting applications are given to show how new theory can be adapted into machine learning tasks to achieve better performance.