2015-16 Colloquia talks

A Homological Relation in Negative Degrees

Abstract: We visit one of the problems arising in homological algebra, dealing with a theory called "Tate cohomology," which carries some algebraic information of the object it is defined on. Let that object be any finite dimensional Hopf algebra A over a field k. The aim of this talk is to extend a known relation of the ordinary cohomology and Hochschild cohomology of A to negative degrees. In particular, we prove that the Tate cohomology of A is an algebra direct summand of its Tate-Hochschild cohomology. Some terminologies will come from a semester of abstract algebra, but all necessary definitions and background will be given and no prior knowledge is expected.

Graded Algebras

Abstract: A graded algebra A over a field F is an algebra written as the direct sum of subspaces called graded components such that for any two components U and V there is a component W such that UV is a subset of W. If graded components can be labeled by elements of a group G, U by g, V by h, and W by gh, then we say that A is a G-graded algebra. For example, the algebra A of polynomials in several variable is graded by the group Z of integers, the graded components being spans of monomials of a fixed degree. If the number of variables is m then there is a natural grading by the m-th power of Z, the components being spans of monomials of a fixed degree in each of the m variables. The full classification of all possible group gradings on polynomial algebras is unknown because this problem is closely connected to the description of all automorphisms on polynomial algebras. At the same time, the classification of gradings on simple associative, Lie or Jordan algebras is now complete, in the case when the ground field of coefficients is algebraically closed of characteristic greater than 3. We also have a classification of gradings on large classes of simple Lie algebras of Cartan type. In this lecture I will try to outline the classification and explain the methods used for this work.

The General Automorphism Group of Affine Space

Abstract: The group of automorphisms of affine space, called the general automorphism group, is a fundamental object of study in algebraic geometry. There are several ways of studying this group. First, one can consider some natural subgroups, and try to describe the subgroup lattice. Another approach is to try and describe the structure of this group as an infinite dimensional algebraic variety. We will describe some recent work on both of these approaches to understanding the group, including a potential application to the Jacobian conjecture.

Differentiating Representations of Algebraic Groups in Characteristic p

Abstract: Let G be a semisimple algebraic group over an algebraically closed field k (for example, SL_n(k)). Representations for G "differentiate" to representations for its Lie algebra, and if k has characteristic 0, then this gives an equivalence between the finite-dimensional representations of each object. In characteristic p, there is a much looser correspondence, leading to various interesting problems which will be introduced in this talk.

Convergence of Ergodic Averages and Model Theory

Abstract: The classical Mean Ergodic Theorem of von Neumann proves the existence of pointwise limits of averages for any cyclic group K={Uⁿ:n∈Z} acting on some Hilbert space H via powers of a unitary transformation U. In this talk we prove convergence of averages for the simplest nontrivial case beyond von Neumann, namely for averages of the sequence {zⁿ²} for a fixed unit complex number z. The proof uses the "generalized first-order" theory of models of Banach structures, whose basic principles we explain. We also discuss some connections with number theory as well as ongoing work on a general model-theoretic framework that allows reinterpreting and extending recent results akin to Walsh's 2014 ergodic theorem for multiple ergodic averages.

Metrizability in Generalized Inverse Limits

Abstract: For the metric arc I=[0,1] and continuum-valued bonding relation f closed in I², the inverse limit lim{I,f,ω} is the subspace of the countable power I^ω containing sequences x satisfying x(n)∈f(x(n+1)). A recent trend in continuum theory is to generalize this notion to lim{I,f,L}, where L is an arbitrary linear order. When L=ω, the inverse limit is a subspace of the metrizable space I^ω; however, we will show that when L is uncountable, the inverse limit cannot be metrizable unless f is trivial. Furthermore, when L is an uncountable well order, it will be shown that the inverse limit is not even Corson compact.

Arithmetic Properties of Combinatorial Sequences

Abstract: Given a sequence of integers that counts a family of combinatorial objects, it is natural to ask about its long-term behavior. Traditionally researchers have been interested in asymptotic growth rates. However, one can also ask about number theoretic properties, such as the density of attained residues modulo a prime power and p-adic asymptotics. Such questions can be answered by partial interpolations of a sequence to the p-adic integers.

Mathematical Induction

Abstract: Augustus De Morgan coined the term "mathematical induction" in 1838. It was an accident. "An intuitive truth" is how Felix Klein described this important method of proof, one that "carries us over the boundary where sense perception fails." We survey the history of mathematical induction from Euclid to Bertrand Russell. Along the way we will prove that all ponies are red.

Introduction to Lie Algebras with Coefficients in a Ring R

Abstract: Examples of Lie algebras with coefficients in polynomial rings, and quotient rings will be presented. In particular, Lie algebras formed as tensor products of simple Lie algebras and rings, where the rings are polynomial rings, or quotients of polynomial rings. Several examples will be described, such as affine Lie algebras, elliptic algebras, and n-point algebras. The three-point algebra is perhaps the simplest nontrivial example of a Krichever–Novikov algebra beyond an affine Kac–Moody algebra.

Knots, Graphs, Geometry and Densities

Abstract: Knot theory has a fascinating history of using techniques from diverse fields of mathematics. In this talk we will explore the interactions between knot theory, graph theory and hyperbolic geometry. After giving some background in knots and geometry, we will focus on two natural knot invariants, a geometric quantity called the volume density, and a diagrammatic quantity called the determinant density. We will talk about recently discovered interesting relationships between the spectra of volume and determinant densities, and explore natural questions and conjectures motivated by this study. This is joint work with Ilya Kofman and Jessica Purcell.

75 Years of Apology: G.H. Hardy's "A Mathematician’s Apology"

Abstract: The year 2015 was the 75th anniversary of the appearance of G.H. Hardy’s "A Mathematician's Apology." When its second edition appeared in 1967, one well-known critic added to his praise a warning: "it won’t make a nickel for anyone." In this talk, intended for a general audience, we explore the history of the book as well as the controversy that continues to surround it.

Shrinkage Methods Utilizing Auxiliary Information from External Data Sources to Improve Prediction Models with Many Covariates

Abstract: We consider predicting an outcome Y using a large number of covariates X. However, most of the data we have to fit the model contains only Y and W, which is a noisy surrogate for X, and only on a small number of observations do we observe Y, X, and W. We develop Ridge-type shrinkage methods that trade-off between bias and variance in a data-adaptive way to yield smaller prediction error using information from both datasets. We also demonstrate how the problem can be treated in a full Bayesian context with different forms of adaptive shrinkage. Finally, we introduce the notion of a hyper-penalty for guiding choices of the tuning parameter to perform adaptive shrinkage.

Our work is motivated by the rapid development of genomic assay technologies. In our application, mRNA expression of a selected number of genes is measured by both quantitative real-time polymerase chain reaction (qRT-PCR, X) and microarray technology (W) on a small number of lung cancer patients. In addition, only microarray measurements (W) are available on a larger number of patients. For future patients, the goal is to predict survival time (Y) using qRT-PCR (X). The question of interest is whether the large dataset containing only W aid with prediction of Y using X.

The high-dimensionality of the problem, the large fraction of missing covariate information, and the fact that we are interested in a prediction model for Y|X (rather than Y|W) make this a non-standard statistical problem. The general idea of integrating/leveraging information from existing diverse data sources to boost prediction has broader application in contemporary scientific studies. This is joint work with Philip S. Boonstra and Jeremy MG Taylor from the Department of Biostatistics, University of Michigan.

The Average Degree of Irreducible Representations of a Finite Group

Abstract: A representation of degree n of a group G over a field F is a way to represent elements in G by n x n invertible matrices with entries in F in such a way that the rule of group operation corresponds to matrix multiplication. Perhaps the best way to describe representations is through characters. The character afforded by a representation is a function on the group which associates to each group element the trace of the corresponding matrix. In the representation theory and character theory of finite groups, one of the main problem is to study the influence of the degrees of irreducible representations/characters on the structure of groups.

We will present some recent results on the average degree of irreducible representations of finite groups. In particular, we show that there is a tight connection between the average degree and important global characteristics of finite groups such as solvability, nilpotency, and commutativity, as well as p-local characteristics such as p-solvability, p-nilpotency, and the normality of Sylow p-subgroups. These are joint works with Lewis, Maroti, Moreto, and Tiep.

From Integrals to Multi-Sum Identities

Abstract: The method of brackets is a collection of a few heuristic rules for symbolic evaluation of definite integrals. The method is useful for a large class of single and multiple integrals, including many involving special functions. One of the computational challenges of this method is the simplification of the resulting output, especially when consisting of multi-sum series. This talk will show how to tackle some of these double- and triple-sums using a recurrence-finding approach. Adding more power to the method of brackets, this approach produces multi-sum identities for special function expressions.

Supervised and Unsupervised Learning for Multi-Modal Data in Lung Cancer Detection

Abstract: Clinical research studies commonly acquire complementary multi-modal and multi-source data for each patient in order to obtain a more accurate and rigorous assessment of the disease status and likelihood of progression. Multi-modal feature learning and prediction are challenging when integrating these kind of large-scale biomedical data. Motivated from a study of lung cancer detection, we present a novel integrative learning framework for the joint analysis of multi-modal data. The method is a statistical ensemble built on several modern statistical learning techniques, including feature extraction on functional data, random forests, and supervised multidimensional scaling. We also discuss a fast unsupervised clustering method for big data using an adaptive density peak detection procedure. The proposed framework is evaluated by application to high-dimensional chemical sensor array data from the Cleveland Clinic lung cancer early detection project.

Computing with Matrix Groups

Abstract: Given a number of invertible matrices over a finite field, an obvious question is to find out more about the group they generate. Answering this question is the main object of Matrix Group Recognition and recently started to make steps from purely theoretical analysis to concrete calculations.

I will describe how a divide-and-conquer approach is used to reduce the problem to that of simple groups, how identity relations can be used to verify calculations based on random elements, and how the resulting information can be used for practical calculation on the computer, e.g. in the system GAP.

χ-Square Test Based on Random Cells with Recurrent Event Data

Abstract: Recurrent event data are often observed in a wide variety of disciplines including the biomedical, public health, engineering, economic, actuarial science, and social science settings. Consider n independent units that are monitored for the occurrence of a recurrent event. Suppose the interfailure times are independent and identically distributed with common distribution function F. Of interest in this talk is the problem of testing the null hypothesis that F belongs to some parametric family of distributions where the model parameter is unknown. I will present results pertaining to a general χ-square goodness of fit test where cell boundaries are data dependent, that is, cell boundaries that are cut free without being predetermined in advance. The test is based on an estimator of the model parameter known as the minimum χ-square estimator and a nonparametric maximum likelihood estimator of F. Large sample properties of the proposed test statistic under the null hypothesis and a Pitman-like shrinking alternative will be presented. The test is shown to outperform the fixed-cells based test for small samples, but both are asymptotically equivalent. A simulation study is conducted to assess the performance of the test under the null hypothesis and model parameter misspecification. Finally, the procedures are illustrated with a fleet of Boeing 720 jet planes' air conditioning system failures.

Flexible Cure Rate Models and Associated Inference

Abstract: In this talk, I shall first give a historical account of cure rate models and present an introduction to the problems and the models used. I shall then present a flexible family of cure rate models and discuss likelihood inference for this family of models. I shall elaborate on the use of EM-algorithm for this purpose, and also present some results on model fitting and model discrimination between some of the well-known cure rate models. Finally, I will use the developed methods on a cutaneous melanoma data and illustrate all the results.

Statistics: A to Z

Abstract: I will first give a brief account of how the field of Statistics developed over history, and also mention in the process some major advancements that took place in the growth of the field. Next, I will give a concise review of various aspects/topics of the field. Finally, I will then highlight some key applications of statistical methods to a wide range of problems arising from Science, Engineering, Medicine and Business.

Internships and More!

Abstract: In this talk, I will discuss my experiences as a summer 2015 intern at the U.S. Department of Energy, as well as the projects I worked on over the summer. I will also discuss the things I did to obtain the internship, give advice on how to get accepted, and explore other internship, summer research, and job opportunities offered for undergraduate and graduate students in the STEM (Science, Technology, Engineering, and Math) disciplines.

Graphs, Laplacians and Link Shadows

Abstract: Checkerboard doodling leads to a connection between knot theory and graph theory. The Laplacian matrix of a graph can be used to compute the number of components of the graph's medial link.