2019-20 Colloquia talks

Reduce the Computation in Jackknife Empirical Likelihood for Comparing Two Correlated Gini Indices

In this talk, I introduce a new version of the jackknife empirical likelihood method in constructing a confidence interval for the difference of two Gini indices from paired samples. This approach is motivated by the need to reduce the computation cost associated with the profile jackknife empirical likelihood method proposed by Wang and Zhao (2016). This is done by using a plug-in approach and avoiding the maximization over a nuisance parameter used by the previous method. I establish the Wilks' theorem. I also investigate two calibration methods, the adjusted jackknife empirical likelihood and the bootstrapped calibrated jackknife empirical likelihood, to improve coverage accuracy for small samples. I show through simulations that the proposed methods perform better than Wang and Zhao's methods in terms of coverage accuracy and require less computation. A real data application proves that the proposed methods work perfectly in practice.

Studying Knots and Three-Manifolds through Maps on Surfaces

Knots and three-manifolds are fundamental mathematical objects studied in topology and geometry. They can each be associated to maps on surfaces - knots can be represented as closed braids and three-manifolds can be represented as open book decompositions. In this talk I will introduce this perspective, discuss why it is useful, and explain some of my past work that explores when properties of these maps dictate topological information about the corresponding knot or manifold.

Fusion Systems

What happens when you take a finite group and try to forget all information about it, except that information in a "neighborhood" of a prime p? You get a "p-local finite group", of course, also called a "p-fusion system". I'll explain what a fusion system is, give a couple of small examples, and interpret some old results on finite groups in this new setting. At the end, I plan to describe briefly what a fusion system can tell you about structures associated to the group in topology and in representation theory.

Blow Up Chemical Reaction Networks

In this talk, I would explain what is a toric dynamical system associated to a chemical reaction network. In the study of chemical reaction networks, the toric methods introduced by Gatermann were formalized by Craciun, Dickenstein, Shiu and Sturmfels in the paper "Toric Dynamical Systems" in 2009. We then define the blow up algebras associated to ideals. Finally, we explore the relation between blow up algebras and chemical reaction networks. This is joint work with David Cox and Gabriel Sosa.

The Stability Problem for KdV Solitons

In this talk we will discuss various approaches to the stability problem for soliton solutions of the KdV equation. The talk will include discussions of orbital stability and asymptotic stability in Sobolev spaces.

Fast Conversion in Redundant Signed Radix 2 Arithmetic

Numbers in the radix n signed digit number system can be added using fully parallel units, i.e., without carry propagation. However, because the signed digit system is redundant, other operations require a unique canonical form. The non-adjacent form, or minimal representation, available in radix 2, provides such a form to each integer value. While the conversion to this canonical form cannot be parallelized, we show that it can be parallelized up to a carry-propagation signal that, up to a constant factor, has the same or shorter critical path (circuit-level complexity) as binary addition using standard adder constructions. This work is relevant to applications in fast arithmetic and cryptography, where signed digit systems and nonadjacent forms are used to evaluate kP (k an integer, P a point on an elliptic curve) in work of Koblitz and others on elliptic curve cryptosystems.

Statistical Methodologies for Neuroconnectivity Analysis Using Autistic fMRI Data

The human brain is an amazingly complex network. Aberrant activities in this network can lead to various neurological disorders such as multiple sclerosis, Parkinson’s disease, Alzheimer’s disease and autism. fMRI has emerged as an important tool to delineate the neural networks affected by such diseases, particularly autism. In this talk, we will present a special type of mixed-effects model together with an appropriate procedure for controlling false discoveries to detect disrupted connectivities for developing a neural network in whole brain studies. Results are illustrated with a large dataset known as Autism Brain Imaging Data Exchange (ABIDE) which includes 361 subjects from 8 medical centers.

Toeplitz Operators on Constrained Subalgebras

Toeplitz operators are a type of linear transformation defined on a vector space of functions. Specifically, they are transformations whose matrix representations in the basis {zⁿ, n ≥ 0} have constant diagonals. Toeplitz operators are known to model various quantum processes. The space of functions, A, that form the domain of our Toeplitz operators are functions defined on the complex unit disc (complex values z such that |z| ≤ 1) which are differentiable inside the disc and continuous on the unit circle. A lot of work has gone into understanding when the associated quantum processes can be reversed – that is, when these Toeplitz operators are invertible. This further raises questions about their eigenvalues. In this talk we will discuss modern work in this direction. Afterward, we will ask what happens when the underlying algebra A is replaced with a constrained subalgebra of A. In particular, the subalgebra of functions in A that agree at two prescribed points in the unit disc. This talk will include discussions of recent joint work between myself and Michael Jury at the University of Florida and between myself and Ben Russo at Farmingdale State College.

I Never Met a Datum I Didn’t Like

Data are quite important. And with big data, there are more and more data elements to contend with. The 3 V’s of big data: velocity, volume, and variety attest to this. But are all data created equal? NO. So the statistician has an ongoing and increasingly important role to assure relevant, representative data are being analyzed. This talk will discuss where data analytics meets statistics and some of the great potential and, yes, the pitfalls, of our deriving useful information from all that data. It also includes examples from the author’s real life and the Supreme Court!

This talk is aimed at a general audience!

Sixth Satya Mishra Memorial Lecture

It’s Not What We Said, It’s Not What They Heard, It’s What They Say They Heard

Statisticians have long known that success in our profession frequently depends on our ability to succinctly explain our results so decision makers may correctly integrate our efforts into their actions. However, this is no longer enough. While we still must make sure that we carefully present results and conclusions, the real difficulty is what the recipient thinks we just said. This presentation will discuss what to do, and what not to do. Examples, including those used in court cases, executive documents, and material presented for the President of the United States, will illustrate the principles.

Galois Orders

Galois rings and Galois orders form a class of algebras that contain many important examples such as U(gl_n), the universal enveloping algebra of the general linear Lie algebra. We will look at a simple example, how to realize U(gl_n) as a Galois order, and an extension of U(gl_n).

Geometric Constructibility Through Origami

Historically, a compass and a straightedge were used to construct specific geometric objects. A famous historical problem was how to use these tools to construct a line segment of length the cube root of two. In modern times, using abstract algebra, this is known to be impossible. A benefit of living in modern times, however, is that we have other ways of building geometric objects. In particular, we can use origami. In this talk we will discuss the formal axioms of origami constructions, show how origami can be used to construct a line segment of length the cube root of two, and explore the limits of origami constructions. This will be a hands on talk: be ready to fold paper.

Graphs and Links in Surfaces

Knots and links are usually represented by diagrams in the plane or the 2-sphere. More generally, they can be represented in surfaces S of higher genus, and we envision them in the thickened surface S x [0,1]. In this joint work with Susan Williams we explain how graphs and a discrete version of the Laplacian operator from physics can be used to help us understand knots and links in thickened surfaces.