Coursework

Machine Learning and Statistics

EN.553.741: Machine Learning II - Johns Hopkins University (Spring 2022)
Statistical learning theory, including some concentration inequality, generalization bounds and VC dimension. Problems and algorithms for unsupervised data analysis, including dimension reduction, manifold learning and clustering.

EN.553.739: Statistical Pattern Recognition Theory and Methods - Johns Hopkins University (Spring 2022)
Topics in the theory, methods, and applications of machine learning from an explicitly statistical perspective.

EN.553.740: Machine Learning I - Johns Hopkins University (Fall 2021)
Fundamental concepts of machine learning (such as generalization error, Bayes estimators and the bias vs. variance dilemma). A collection of learning algorithms for classification and regression. Linear and kernel regression, support vector machines, lasso, logistic regression, decision trees and neural networks. Includes background material on inner-product spaces, reproducing kernels and on optimization.

EN.553.730: Statistical Theory - Johns Hopkins University (Fall 2021)
Distribution theory for statistics of normal samples, exponential statistical models, the sufficiency principle, least squares estimation, maximum likelihood estimation, uniform minimum variance unbiased estimation, hypothesis testing, the Neyman-Pearson lemma, likelihood ratio procedures, the general linear model, the Gauss-Markov theorem, simultaneous inference, decision theory, Bayes and minimax procedures, chi-square methods, goodness-of-fit tests, and nonparametric and robust methods.

ISE 365: Applied Data Mining - Lehigh University (Spring 2021)
Introduction to the data mining process including business problem understanding, data understanding and preparation, modeling and evaluation, and model deployment. Emphasis on hands-on data preparation and modeling using techniques from statistics, artificial intelligence, such as regression, decision trees, neural networks, and clustering. A number of application areas are explored.

Applications to Medicine and Biology

EN.553.784: Mathematical Foundations of Computational Anatomy - Johns Hopkins University (Fall 2023)
Fundamental concepts and methods that pertain the analysis of the variation of anatomical shapes extracted from medical images. Basic properties of the most important shape representations (landmark, curves, surfaces, images…), distances and discrepancy measures that allow for their comparison, and nonlinear optimal control methods that underlie the Large Deformation Diffeomorphic Metric Mapping (LDDMM) family of registration algorithms. Shape averaging methods and template-centered representations for the analysis of shape datasets.

EN.553.650: Computational Molecular Medicine - Johns Hopkins University (Spring 2022)
Computational systems biology has emerged as the dominant framework for analyzing high-dimensional “omics” data in order to uncover the relationships among molecules, networks and disease. In particular, many of the core methodologies are based on statistical modeling, including machine learning, stochastic processes and statistical inference. We will cover the key aspects of this methodology, including measuring associations, testing multiple hypotheses, and learning predictors, Markov chains and graphical models. In addition, by studying recent important articles in cancer systems biology, we will illustrate how this approach enhances our ability to annotate genomes, discover molecular disease networks, detect disease, predict clinical outcomes, and characterize disease progression.

EN.580.631: Introduction to Computational Medicine - Johns Hopkins University (Fall 2021)
Computational medicine is an emerging discipline in which computer models of disease are developed, constrained using data measured from individual patients, and then applied to deliver precision health care. This course will cover computational anatomy. Students will learn how to: model anatomies using magnetic resonance imaging data; compare anatomies via mappings onto anatomical atlases; discover anatomical biomarkers of disease; analyze changes in the connectivity of anatomies in disease.

HMS 316: Social Epidemiology - Lehigh University (Fall 2020)
Social epidemiology is the study of the distribution and social determinants of health and disease in human populations. This course introduces the basic principles of epidemiological study design, analysis and interpretation, covering topics such as how a disease spreads across populations and how public health interventions can help control or reduce the spread of disease. This course also reviews epidemiology as a social science by reviewing the social causes and consequences of health.

MATH 371: Readings in Mathematical Biology - Lehigh University (Spring 2020)
Phase space analysis of higher-order systems; equilibrium points, limit cycles, linearization, stability analysis; Predator-prey models, Poincare-Bendixson theorem, Bendixson criterion; perturbation analysis; Poincare maps; Bifurcations and chaotic dynamics; Routh Hurwitz Criterion; Survival functions; applications to epidemiology.

Analysis and Probability

EN.553.797: Introduction to Control Theory and Optimal Control - Johns Hopkins University (Spring 2022)
A control system is a dynamical system on which one can act through a parameter that can be chosen freely at any point in time. In this class, we will be interested in two main problems. The first one is controllability, which studies conditions for the existence of controls allowing an initial point to be driven to any other point. The second one is optimal control, in which we will study methods to minimize a certain cost over all possible controls, possibly with endpoint constraints.

EN.553.701: Real Analysis - Johns Hopkins University (Fall 2021)
Fundamental topics in real analysis, such as set theory, the topology of Euclidean space, continuous mappings, uniform convergence, differentiable mappings, inverse & implicit function theorems, integration theory, Fourier series, and basics of differential equations.

MATH 463: Advanced Probability - Lehigh University (Spring 2021)
Measure theoretic foundations; random variables, integration in a measure space, expectations; convergence of random variables and probability measures; conditional expectations; characteristic functions; sums of random variables, limit theorems.

ISE 417: Nonlinear Optimization - Lehigh University (Spring 2021)
Advanced topics in mathematical optimization with emphasis on modeling and analysis of nonlinear problems. Convex analysis, unconstrained and constrained optimization, duality theory, Lagrangian relaxation, and methods for solving nonlinear optimization problems, including descent methods, Newton methods, conjugate gradient methods, and penalty and barrier methods.

MATH 322: Methods of Applied Analysis I - Lehigh University (Fall 2020)
Fourier series, eigenfunction expansions, Sturm–Liouville problems, Fourier integrals and their application to partial differential equations; special functions. Emphasis is on a wide variety of formal applications rather than logical development.

MATH 310: Random Processes and Applications - Lehigh University (Fall 2020)
Theory and applications of stochastic processes. Limit theorems, introduction to random walks, Markov chains, Poisson processes, birth and death processes, and Brownian motion. Applications to financial mathematics, biology, business and engineering.

MATH 402: Real Analysis II - Lehigh University (Spring 2020)
Metric spaces; introduction to Banach and Hilbert space theory; Fourier series and Fejer operators; general measure and integration theory, Radon–Nikodym and Riesz representation and theorems; Lebesgue–Stieltjes integral.

MATH 350: Introduction to Differential Geometry - Lehigh University (Spring 2020)
Local and global differential geometry of curves and surfaces in Euclidean 3-space; Frenet formulas for curves, regular surfaces, first fundamental form, Gauss map, second fundamental form, curvatures for curves and surfaces and their relations, minimal surfaces, the Gauss-Bonnet theorem, selected special topics and applications, such as hypersurfaces in Euclidean (n+1)-space, Levi-Civita connection, geodesics, parallel transport, Riemannian curvature tensor, etc.

MATH 316: Complex Analysis - Lehigh University (Spring 2020)
Concept of analytic function from the points of view of the Cauchy–Riemann equations, power series, complex integration, and conformal mapping.

MATH 401: Real Analysis I - Lehigh University (Fall 2019)
Set theory, real numbers; introduction to measures, Lebesgue measure; integration, general convergence theorems; differentiation, functions of bounded variation, absolute continuity; Lp spaces.

MATH 309: Probability with Applications and Simulations - Lehigh University (Fall 2019)
Foundations of probability; random variables; probability models; expectations and moment generating functions; joint and conditional distributions; functions of random variables. Introduction to fundamental ideas and techniques of stochastic modeling, with an emphasis on the applications. Monte Carlo simulation.

MATH 230: Numerical Methods - Lehigh University (Fall 2019)
Representation of numbers and rounding error; polynomial and spline interpolation; numerical differentiation and integration; numerical solution of nonlinear systems; numerical solution of initial and boundary value problems; Monte Carlo methods.

MATH 319: Introduction to Differential Equations - Lehigh University (Spring 2019)
Homogeneous and non-homogeneous linear differential equations, existence and uniqueness theorems, Gronwall’s inequality; systems of first order linear differential equations; autonomous first-order systems: critical points, stability, bifurcation; series and periodic solutions, Fourier series and their convergence; introduction to numerical simulation methods.

Benjamin Brindle

Coursework

Machine Learning and Statistics

Applications to Medicine and Biology

Analysis and Probability

Other Topics in Mathematics