2016-17 Colloquia talks

Regularity of Powers of Edge Ideals

Abstract: Let R=k[x₁, ..., x_n] be a polynomial ring and I be a homogeneous ideal of R. It is a celebrated result that regularity of powers of I is asymptotically a linear function.

However, the form of the linear function and at which power of I regularity of powers attains that linear function are not generally known. Let G be a graph and I(G) be its edge ideal. In this talk, we focus on powers of I(G) and answer this question when G is a forest, cycle, and a unicyclic graph by explicitly computing the regularity of I(G)^s.

Empirical Likelihood for Change Point Detection in Time Series Models

Abstract: The Empirical Likelihood (EL) method introduced by Owen (1988) is a widely used non-parametric tool for constructing confidence regions due to its appealing asymptotic distribution of the likelihood-ratio-type statistic which is same as the one under the parametric settings. However, the EL method was introduced to be used with independent data, hence it becomes difficult to apply it to dependent data such as time series models. An EL-based detecting procedure for structural changes in time series data is developed. The asymptotic results such as the asymptotic null distribution of the test statistic, the consistency of the estimator of the change location and the test are established. Further, simulations are conducted to investigate the performance of the proposed test under various scenarios. Additionally, real data are used to illustrate the detection procedure.

Mixture Designs: Models and Methods

Abstract: Mixture designs are widely in use in industrial and pharmaceutical experiments. We will review the models and present their salient features. Then we will discuss methods of estimation of model parameters and related data analysis. Optimality aspects will also be highlighted.

Mantel’s Theorem, Its Proof, and Turan’s Theorem

Abstract: Mantel’s Theorem (1907) states that if G is a triangle-free graph of order n, then the maximum number of edges in G is at most n²/4. We will prove this theorem. Furthermore, we demonstrate that this is a sharp inequality by exhibiting some triangle-free graphs on n vertices with exactly n²/4 edges. Then we consider a generalization of Mantel’s Theorem, namely, Turan’s Theorem, which gives an upper bound for the number of edges in a graph of order n that contains no (r+1)-clique (Mantel’s Theorem is Turan’s Theorem with r = 2). Finally, we will give the extremal graphs for Turan’s Theorem.

Berezinians and Darboux Transformations on the Superline

Abstract:

Coherent and Incoherent Numbers of a Knot

Abstract: A complementary region of an oriented knot diagram on the two-sphere is said to be coherent (resp. incoherent) if the orientation of its boundary is coherent (resp. incoherent). In this talk, we investigate the numbers of all coherent regions and all incoherent regions of an oriented knot diagram, and define two numerical invariants, called the coherent number and the incoherent number, of a(n oriented) knot. Then we investigate the relation between our two invariants and other numerical knot invariants such as the canonical genus, the braid index and the crosscap number of a knot. We also characterize the knots with coherent number up to four. This is joint work with Reiko Shinjo (Kokushikan University).

Purely Sequential Estimation of a Negative Binomial Mean with Applications in Ecology

Abstract: We discuss a set of purely sequential strategies to estimate an unknown negative binomial mean under different forms of loss functions. We develop point estimation techniques where the thatch parameter may be known or unknown. Both asymptotic first-order efficiency and risk efficiency properties will be elaborated. The results will be supported by an extensive set of data analysis carried out via computer simulations for a wide variety of sample sizes. We observe that all of our purely sequential estimation strategies perform remarkably well under different situations. We also illustrate the implementation of these methodologies using real datasets from ecology, namely, weed count data and data on migrating woodlarks.

Adaptive Group Sequential Designs with Population Enrichment and Endpoint Assessment in Phase 3 Randomized Controlled Trials

Abstract: Phase 3 randomized controlled trials remain the gold standard to establish efficacy and safety of an experimental intervention. However, phase 3 designs which do not consider the heterogeneity of patient subgroups yield an intervention benefit (if any) which is true "on average" for the overall population of interest. On the other hand, a confirmatory trial may produce a negative result despite the experimental intervention benefiting some subgroups, due to a diluted overall effect estimate driven by nonresponsive subgroups. The US Food and Drug Administration (FDA) recognized the need for new confirmatory clinical trial designs and methods which account for patient subgroup heterogeneity. Use of co-primary outcomes have been recommended by regulatory agencies for the evaluation of an experimental intervention benefit for several disease processes. In this talk, I will introduce two new confirmatory clinical trial design methodologies which prospectively incorporate subgroup heterogeneity through population enrichment and assess co-primary endpoints simultaneously.

Gelfand-Zeitlin Theory; where Linear Algebra, Geometry, and Representation Theory Meet

Abstract:

Nonparametric Cross-Dimensional Inference for High-Dimensional Dependent Data

Abstract: Modern data in genetics, genomics, image processing, neuroscience and text mining often contain measurements of a very large number of features that are subject to systematic internal interdependencies or external perturbations. As such, they are high-dimensional and highly dependent. This poses great challenges for scalable, accurate and reproducible inference for such data. Existing methods may be unscalable, incapable to deal with complicated dependence or suboptimal. To mitigate these issues, we propose a "Nonparametric Cross-Dimensional Inference" methodology that leverages the blessing of dimensionality and latent variable structures for simultaneous inference for high-dimensional dependent data. Our method is accurate, fast, flexible and scalable. It complements the expectation-maximization algorithm and variational Bayes methods, and can be optimal. Its excellent performances are supported by theory, simulation studies, and applications to human population genetics, RNA-seq gene expression study in yeast, and recovery profile for post-trauma patients.

Central Values of L-functions

Abstract: Analyzing the special values of L-functions has been a significant target of research ever since the Riemann zeta function was introduced in the eighteenth century, as they play an important role in many fields including number theory. For example, it is known that the zeros of the Riemann zeta function are deeply connected to the distribution of prime numbers. In this talk, I will survey various L-functions and results particularly concerning the vanishing or nonvanishing of their central values.

Dynamical Modular Curves and Applications

Abstract: The field of arithmetic dynamics is motivated largely by various analogies between number theory/arithmetic geometry and the theory of discrete dynamical systems. On the number theory side, I will start with a brief overview of the arithmetic of elliptic curves. Many of the results in this area rely on the geometry and arithmetic of modular curves, which encapsulate certain information about elliptic curves and their torsion points. I will then explain some of the analogies between elliptic curves and dynamical systems, including a dynamical version of a modular curve. Finally, I will describe various properties and applications of dynamical modular curves appearing in the literature and in past and ongoing projects of mine.

A Contest With Complementarities

Abstract: In many situations, we observe complementarities, where the value of one item is dependent on the ownership of related items. In this talk, we will extend this idea of complementarity to game theory problems, and derive optimal strategies for a game containing complementarities. We will Also illustrate some practical limits on the size of such games if we wish to maintain tractability, and present some experimental results on the ability of people to actually find the optimal solutions.

On the Hazard Functions of Kumaraswamy Distributions and Associated Inference

Abstract: Kumaraswamy distributions are a family of distributions with two shape parameters and with bounded support. This family of distributions can be applied to model many distributions very similar to Beta distributions. The tractability of Kumaraswamy distributions makes it more applicable than Beta distributions. Specifically we show that the hazard function of a Kumaraswamy distribution is either bathtub or increasing. In reliability and survival analysis, as it is often of interest to determine the point at which the hazard function reaches its minimum, we propose different estimators of that point.

Partitions with a Fixed Number of Odd Parts

Abstract: In this talk we will give a brief introduction to the theory of partitions, and some classical partition identities. We will later move on to new combinatorial results that came out of the recent study of partitions with a fixed number of odd-indexed and even-indexed odd parts. In particular, among other things, we will see a generalization of Savage and Sills' "new little Gollnitz theorem," talk about the generating functions for the number of partitions with a fixed BG-rank, and a result concerning alternating sums of parts of partitions.

Segregation and Robin Hood's Surrogates

Abstract: The concept of segregation is perhaps most easily introduced in the context of the distribution of students of differing ethnicity in the various schools in a particular urban school district. At issue is whether the ethnic distribution of students is different, at times markedly different, from school to school. Mathematically, this corresponds to a question about dependence in a given two-way contingency table. In this talk we describe a suitable segregation ordering and discuss its relationship with a spectrum of segregation measures which have been proposed in the literature.

This talk is aimed at a general audience!

Third Satya Mishra Memorial Lecture

What Population Does Your Sample Represent?

Abstract: Not infrequently data are collected to study a particular distribution or population but, because of the sampling mechanism used, the sample is not representative of the desired target distribution. For example, size biasing occurs when large items are more likely to be included in the sample than are small ones (a relatively frequent occurrence). Hidden truncation occurs when observations are only made subject to constraints on covariables (also more frequent than one might suspect). Some examples showing how one can draw inferences about the target population based on such biased samples will be discussed.

Schur-Weyl Duality and the Free Lie Algebra

Abstract: Schur-Weyl duality is a classical correspondence between subspaces of r-fold tensors of an n-dimensional vector space that are invariant under all linear operators and subspaces that are invariant under all permutations. This goes back to Schur's thesis in 1927. The space of r-fold tensors contains the space of homogeneous Lie polynomials of degree r. It turns out that Schur-Weyl duality induces a correspondence between subspaces of Lie polynomials that are invariant under all linear transformations and subspaces that are invariant under a certain subalgebra of the group algebra of the symmetric group of degree r. This subalgebra also arises in a surprisingly different context. Studying it's structure leads to intriguing combinatorial questions (and a sequence that does not appear in the Online Encyclopedia of Integer Sequences!).

This talk should be accessible to a general audience.

From Abstract Tensors to Category Theory and Beyond - Computer Assisted Knot Theory

Abstract: The abstract tensor formalism lies at the heart of so-call quantum topology. For example, the bracket expansion of the Jones polynomial can be expressed in terms of ∪ and ∩ abstract tensors. In this talk, I will present various algebraic axioms via the structure constants of the algebra and show how to go from these to diagrams and subsequently to categories. Then I'll give a brief description of the category of tangles and some analogues. Finally, I'll extend the ideas to those of 2-categories and show how the program globular can be used to implement computations in braided monoidal 2-categories.

An Analog of Euler's Theorem on Integer Partitions

Abstract: A partition of an integer is a way of writing it as a sum of positive integers (without regard to the order of the summands). It is a famous result of Euler that the number of ways to partition an integer into distinct parts is the same as the number of ways to partition it into odd parts. In this talk, we revisit that theorem (as well as some highlights of the theory of integer partitions) and exhibit a new analog for partitions of fixed perimeter.

This analog arose as a by-product of work on core partitions. A special case of an elegant result due to Anderson proves that the number of (s,s+1)-core partitions is finite and given by the Catalan numbers. Amdeberhan recently conjectured that (s,s+1)-core partitions into distinct parts are enumerated by Fibonacci numbers. We prove this conjecture by enumerating a more general two-parameter family of core partitions into distinct parts.

The talk is intended for a general mathematical audience.

A Stochastic Approximation Approach to Pairs Trading under a Regime-Switching Model

Abstract: This talk is concerned with a numerical method for pairs trading. In particular, we focus on a stochastic approximation algorithm. A pair of correlated stocks are selected and monitored. When the "spread" of the stock prices decreases to a certain level, the pairs trade is initiated by longing the stronger stock and shorting the weaker one, betting the eventual divergence of the "spread". The log difference of the pair is assumed to satisfy a regime-switching model, and the trade will be determined by two conditional probability threshold levels. The objective is to identify the optimal threshold levels so as to maximize an overall return. The effectiveness of the method is examined in a numerical example.

Lehmer’s Question and Graph Complexity

Abstract: D.H. Lehmer’s question about roots of polynomials with integer coefficients has remained an important open question for more than 80 years.

In this joint work with Susan Williams we show that Lehmer’s Question is equivalent to an elementary question about graph complexity and spanning trees.

The talk is intended for a general mathematical audience.

Joint Entropy of Type II Censored Survival Data

Abstract: Consider a life testing experiment with progressive type II censoring scheme. Shannon entropy, Awad sup entropy and Renyi entropy for the data collected are computed when the underlying distribution of the life time is a Rayleigh distribution with a single parameter. Then we examine the suitability of these three entropies as objective functions to choose an optimal progressive censoring scheme. Also under the same conditions the suitability of Fisher information and Kullback-Leibler divergence as objective functions are considered.