Colloquia

Machine Learning methods for Cyber Security

Given the continuing advancement of networking applications and our increasing dependence upon software-based systems, there is a constant need to develop new and advanced security techniques for defending modern information technology (IT) systems from malicious cyber-attacks. This presentation introduces novel methodologies to build relevant strategies for system administrators (defenders). A security model is developed to analyze the system's overall performance and predict the key decision makers' likely behavior influencing the protection structure. The initial objective here is to create a reliable intrusion detection mechanism to help identify malicious attacks at a very early stage, i.e., to minimize potentially critical consequences and damage to system privacy and stability. Furthermore, another key objective is also to develop effective intrusion prevention (response) mechanisms. Along these lines, a machine learning-based solution framework is developed consisting of two modules. Precisely, the first module prepares the system for analysis and detects whether or not there is a cyber-attack. The second module analyzes the type of breach and formulates an adequate response. A decision agent is built in the second module to investigate the environment and make appropriate decisions in the case of uncertainty. This agent starts by conducting its analysis in a completely unknown milieu but continually learns to adjust its decision-making based upon the provided feedback. The overall system is designed to operate in an automated manner without any intervention from administrators or other cybersecurity personnel. Human input is essentially only required to modify some critical model (system) parameters and settings.

Optimization of Dynamic Objective Functions Using Path Integrals

Path integrals are used to find an optimal strategy for a firm under a Walrasian system. We define dynamic optimal strategies and develop an integration method to capture all non-additive non-convex strategies. We also show that the method can solve the non-linear case, for example Merton-Garman Hamiltonian system, which the traditional Pontryagin maximum principle cannot solve in closed form. Furthermore, we assume that the strategy space and time are inseparable with respect to a contract. Under this assumption we show that the strategy spacetime is a dynamic curved Liouville-like 2-brane quantum gravity surface under asymmetric information and that traditional Euclidean geometry fails to give a proper feedback Nash equilibrium. Cooperation occurs when two firms’ strategies fall into each other’s influence curvature in this strategy spacetime. Small firms in an economy dominated by large firms are subject to the influence of large firms. We determine an optimal feedback semi-cooperation of the small firm in this case using a Liouville-Feynman path integral method. In later parts we use path integrals in different scenarios such as choosing a player in a cricket team.

Linear Mixed Model Selection via Minimum Approximated Information Criterion

The analyses of correlated, repeated measures, or multilevel data with a Gaussian response are often based on models known as the linear mixed models (LMMs). LMMs are modeled using both fixed and random effects. With a primary focus on the simultaneous selection and estimation of fixed effects only in LMMs, I designed a penalized log-likelihood procedure referred to as the minimum approximated information criterion for LMMs (lmmMAIC). This method is used to find a parsimonious model that better generalizes data with a group structure.

Combinatorial and Algebraic Aspects of Semigroups

A numerical semigroup is the set of all integers which can be written as a non-negative integer combination of a fixed set of positive integers which are relatively prime. The non-negative integers which are not in the numerical semigroup is called the set of holes of the semigroup. In 1978, Herbert Wilf posed a question about the density of this set of holes which is still widely open and has become known as Wilf’s conjecture. In the first portion of this talk we will discuss Wilf’s conjecture and a recent extension of it to higher dimensions which is joint work with C. Cisto, G. Failla, Z. Flores, C. Peterson, and R. Utano. In the remainder of the talk we will focus on the representation of a semigroup as a quotient of a polynomial ring by a so-called toric ideal. A Gröbner basis for the toric ideal possesses a wealth of information about the semigroup, and it is of particular interest both in pure and applied algebraic geometry to find a Gröbner basis of low degree. Time permitting we will touch on two projects where we find a Gröbner basis of quadrics for combinatorially meaningful classes of toric ideals. These two projects are joint with C. Francisco, J. Schweig, J. Mermin, and G. Sosa, and, respectively, B. Jabbar Nezhad.

Knot Invariants, Categorification, and Representation Theory

In my talk, I will provide a survey highlighting connections between representation theory, low-dimensional topology, and algebraic geometry central to my current research. I will recall basic facts about the representation theory of the Lie algebra sl₂ and discuss how these relate to the construction of knot invariants such as the well-known Jones polynomial. I will then introduce certain algebraic varieties called Springer fibers and explain how they can be used to geometrically construct and classify irreducible representations of the symmetric group. These two topics turn out to be intimately related. More precisely, I will demonstrate how one can study the topology of certain Springer fibers using the combinatorics underlying the representation theory of sl₂. On the other hand, I will show how Springer fibers can be used to categorify certain representations of sl₂. As an application, one can upgrade the Jones polynomial to a homological invariant which distinguishes more knots than the polynomial invariant. Based on my exposition, I will discuss future research directions and explore how this picture might generalize to other Lie types beyond sl₂.

Combinatorial Questions Arising from Biomolecular Processes

We will look at combinatorial problems that describe biomolecular processes, namely DNA self-assembly and DNA:RNA interactions. In the first part, we show how DNA self-assembly processes give rise to new graph invariants and how these invariants are related to known chromatic parameters. We also discuss how these invariants help in building molecules that can optimally assemble into a target structure. In the second part of the talk, we use formal grammars to model DNA:RNA interactions and the formation of R-loops. By identifying patterns specific to R-loops using experimental data, we can expand the current grammar into a probabilistic model to predict locations favorable for R-loops in a given DNA sequence.

Newton’s Method, Complex Dynamics, and Converging Polynomials

Newton’s method is a standard algorithm from calculus for finding roots of equations. We rephrase Newton’s method in terms of iterating a function and use the possible success or failure of the method at different starting points to motivate concepts from complex dynamics. Next, we examine the family of quadratic polynomials to understand the concept of parameter spaces in complex dynamics. Finally, we touch upon joint work with Devin Becker and Lorelei Koss. This work examines the complex dynamics of families of polynomials that converge to families of entire maps.

Estimation and Clustering in Popularity Adjusted Block Model

We consider the Popularity Adjusted Block model (PABM) introduced by Sengupta and Chen (2018). We argue that the main appeal of the PABM is the flexibility of the spectral properties of the graph which makes the PABM an attractive choice for modeling networks that appear in biological sciences. We expand the theory of PABM to the case of an arbitrary number of communities which possibly grows with a number of nodes in the network and is not assumed to be known. We produce estimators of the probability matrix and the community structure and provide non-asymptotic upper bounds for the estimation and the clustering errors. We use the Sparse Subspace Clustering (SSC) approach for partitioning the network into communities, the approach that, to the best of our knowledge, has not been used for clustering network data. The theory is supplemented by a simulation study. In addition, we show advantages of the PABM for modeling a butterfly similarity network and a human brain functional network.

Multi-Resolution Approximations of Gaussian Processes for Large Spatial Datasets

Gaussian processes are popular and flexible models for spatial, temporal, and functional data, but they are computationally infeasible for large datasets. We discuss Gaussian-process approximations that use basis functions at multiple resolutions to achieve fast inference and that can approximately represent any spatial covariance structure. Two special cases of this multi-resolution approximation (MRA) framework will be discussed, a taper version and a domain-partitioning (block) version. We describe theoretical properties and inference procedures, and study the computational complexity of the methods. We also present a block MRA for multivariate data, which can deal with massive datasets with multiple variables and provides great flexibility in capturing cross-dependence among processes.

Novel Statistical Methods with Application to Microbiome Data Analysis

Recent advances in low-cost metagenomic and amplicon sequencing techniques enable routine sampling of environmental and host-associated microbial communities across different habitats. The data produced by these large-scale surveys typically comprise the abundances (or compositions) of microbial taxa at different taxonomic levels. To investigate the dependency of additional covariate measurements such as metabolites or host phenotypes on the microbial abundance, we propose a general robust regression framework for compositional data and a negative binomial factor regression (NBFAR) model. Using the former approach, we infer parsimonious and robust statistical relationships between the microbial abundance data and the additional host-associated measurements. NBFAR provides an efficient framework to model the related over dispersed count microbial abundance data as outcomes and the high dimensional host-associated features as covariates, and then model the underlying dependency in terms of a low-rank and sparse coefficient matrix. We propose an efficient parameter estimation procedures for both the approaches and  make them freely available in the R package RobRegCC and NBFAR. We demonstrate the efficacy of the procedures in simulation studies and applications on the gut microbiome data. 

A Spatio-Temporal Model in a Longitudinal Diffusion Tensor Imaging Study

We focus on the theoretical and applied aspects of a spatio-temporal modeling for the reconstruction of in-vivo fiber tracts in white matter when a single brain is scanned with magnetic resonance imaging (MRI) on several occasions. The goal of this research is twofold: one is how to estimate the spatial trajectory of a nerve fiber bundle at a given time point in the presence of measurement noise and the other is how to incorporate a progressive deterioration of brain function into a hypothesis test. The type I error and the power of the test are evaluated via Monte Carlo simulations. Lastly, we apply the proposed method to a real longitudinal diffusion tensor imaging (DTI) study repeatedly measured for a single brain across time.

Monomial Ideals of Graphs

Given a homogeneous ideal I in a polynomial ring R=k[x₁,…,x_n], we can describe the structure of I by using its minimal free resolution. All the information related to the minimal free resolution of I is encoded in its Betti numbers. However, it is a difficult problem to express Betti numbers of any homogeneous ideal in a general way. Due to this difficulty, it is common to focus on coarser invariants of I or particular classes of ideals.

In this talk, we consider monomial ideals associated to graphs. We will discuss the Castelnuovo-Mumford regularity, projective dimension, and extremal Betti numbers of such ideals and provide formulas for these invariants in terms of the combinatorial data of their associated graphs. Results presented in this talk are from joint work with Biermann, O’Keefe, Lin, and Casiday.

A New Approach to Wallis’ Integral

The theory of differentiation is based on a small number of well-defined rules. Given a class of special functions, it is possible to use these rules to obtain all derivatives of the functions in the class. On the other hand, there is no universal algorithm for integration. It is a priori unclear why the integral of Exp(-x²) is difficult to obtain.

The talk will present a new algorithm developed in the context of integrals coming from Feynman diagrams. It consists of a small number of rules that, up to now, has produced a large number of evaluations. Most of these rules are heuristics. The analysis and proofs related to them are open questions. The method will be illustrated with Wallis’ integral, one of the first examples of a closed-evaluation.

The talk will also include interesting mathematical questions that have appeared in our study of definite integrals.

Technology and the Abstract Mathematician

While my research program is disjoint from the computational side of mathematics, that does not mean my work as both a scholar and instructor isn't benefited from clever use of technology. In this talk I'll demonstrate several of my favorite tools that make my work more efficient, buying me more time for doing math. Note that each tool I'll demonstrate has a free version hosted "in the cloud", that is, all that's required is an updated web browser and internet connection to use.

Stop Me If You’ve Heard This Theorem Before

Mathematics is the basis of comedy. In this talk we will back up this seemingly preposterous claim. The only prerequisite is a sense of humor. There will be one performance only.