Mathematics and Statistics Undergraduate Course Descriptions
MATH-1001: Pre-Calculus
Prereq: None
Description: This course is intended as a preparation for MATH-1350: Calculus I. Topics include: algebraic operations, factoring, exponents and logarithms, polynomials, rational functions, trigonometric functions, inverse trigonometric functions, and the logarithmic and exponential functions.
MATH-1004: Mathematics in Society
Prereq: None
Description: The course provides an overview of mathematical concepts as they relate to various disciplines. Topics will be chosen from set theory, logic, combinatorics, probability, statistics, and voting theory. It is a course for students who require a general overview of mathematics, especially those majoring in liberal arts, the social sciences, business, nursing and allied health fields. This course is not designed to prepare students for any specific future mathematics course. This course meets the Quantitative Reasoning and Data Literacy (QRDL) core requirement.
MATH-1040: Probability and Statistics
Prereq: None
Exclusions: Cannot be taken if the student has already taken ECON 2110, GOVT 2201, IPOL 3270, MATH 2140 or OPAN 2101.
Description: This course will introduce students to the basic concepts, logic, and issues involved in statistical reasoning, as well as basic statistical methods used to analyze data and evaluate studies. The major topics to be covered include methods for exploratory data analysis, an introduction to sampling and experimental design, elementary probability theory and random variables, and methods for statistical inference including simple linear regression. The objectives of this course are to help students develop a critical approach to the evaluation of study designs, data and results, and to develop skills in the application of basic statistical methods in empirical research. An important feature of the course will be the use of statistical software to facilitate the understanding of important statistical ideas and for the implementation of data analysis. College Economics and Political Economy majors should enroll in ECON 2110 or MATH 2140. This course does NOT count towards a mathematics credit. Mathematics majors/minors and Statistics minors should enroll in MATH 2140.
MATH-1350: Calculus I
Prereq: None
Description: This course is an introduction to single variable calculus. It covers calculus of single variable functions, limits, continuity, derivatives, Mean Value Theorem, applications of the derivative, L’Hôpital’s Rule, antiderivatives, Riemann sums, the indefinite and definite integral, basic techniques of integration, and the Fundamental Theorem of Calculus. In this course, students review and extend their knowledge of the exponential, logarithmic, trigonometric and inverse trigonometric functions.
MATH-1360: Calculus II
Prereq: MATH 1350, AP Calc AB score of 5, or AP Calc BC score of 4.
Description: This is the second course in the Calculus sequence and is a continuation of MATH-1350. Topics include techniques of integration, applications of the definite integral, improper integrals, sequences and series including Taylor’s theorem and power series, and polar and parametric curves.
MATH-1505: Data Visualizations & Graphics
Prereq: Any introductory statistics course, MATH 1040, ECON 2110, GOVT 2201, IPOL 3270, MATH 2140, OPAN 2101, or AP Statistics score of 5.
Description: Effective graphical displays of data allow the viewer to understand the structure of a complex dataset at a glance. Graphical displays have many purposes from an initial exploration of a dataset to presentation of final results. This course provides an introduction to using the statistical software package R to create appropriate and illuminating visual displays of data. Properties that make a visual display of data clear, accurate, and powerful will be discussed and tools to create effective visualizations will be taught. In addition, students will learn some of the most common abuses of data visualization and how to recognize (and correct) misleading displays of data. The course has a final project component where students will need to present and defend their results. Classroom material will be accompanied by hands-on experience using statistical software.
MATH-2010: Graph Theory
Prereq: None
Description: This course covers types of graphs such as trees, networks, Eulerian and Hamiltonian graphs, tournaments and DAGS. Properties such as planarity, bipartiteness, and connectivity are related. We consider vertex and edge colorings of graphs and graph drawings with applications to VLSI chips, RNA folding, and traffic signal design.
MATH-2020: Combinatorics
Prereq: None
Description: Main topics in the course include permutations and combinations, recurrence relations, and the Principle of Inclusion/Exclusion with applications to counting derangements and surjections. We also cover the 12-fold way, the Robinson-Schensted correspondence, Latin Squares and the Euler conjecture, design theory, and projective planes.
MATH-2140: Intro Math Statistics
Prereq: MATH 1360
Exclusion: ECON 2110
Description: This course provides an introduction to probability theory and statistical inference. The first half of the course introduces fundamentals in probability. Topics to be covered include basic probability principles, enumeration methods, properties of random variables, common discrete and continuous distribution functions, and expected values. The second half of the course focuses on the core of statistical inference and deals with the central limit theorem, maximum likelihood estimation, confidence intervals, hypothesis testing, and the linear regression model. Statistical software will be used to illustrate concepts and to perform data analysis.
MATH-2240: Math for Machine Learning
Prereq: MATH-1360
Description: This course is designed to lay a strong mathematical foundation for students who wish to pursue AI or Data Science related topics. The course will cover matrix algebra, differentiation of functions of many variables, integration on high dimensional domains, basic problems of machine learning, such as linear regression, and the use of continuous optimization to solve them. Students will gain insight into why modern tools like Large Language Models (LLM) work or fail in different circumstances. This course is suitable for math majors, math/statistics minors, as well as students from other departments.
MATH-2250: Linear Algebra
Prereq: MATH 1360
Description: This course presents the basic theory and methods of finite dimensional vector spaces and linear transformations on them. Topics include: matrices and systems of linear equations; vector spaces, bases, and dimension; linear transformations, kernel, image, matrix representation, basis change, and rank; scalar products and orthogonality; determinants, inverse matrices; eigenvalues, eigenvectors, diagonalization of symmetric matrices, positive definite matrices, spectral theorem for Hermitian matrices; linear discrete dynamical systems via matrix iteration.
MATH-2370: Multivariable Calculus
Prereq: MATH 1360
Description: This is a first course in differential and integral calculus of functions of several variables. After the introduction of vectors and the 2 and 3-dimensional Euclidean space, functions of several variables are discussed. Functions of two variables will be visualized by surfaces in the three-dimensional space. Partial derivatives and the total derivative of real-valued and vector-valued functions, the chain rule, directional derivatives, extrema of real-valued functions, constrained extrema and Lagrange multipliers, double and triple integrals will be covered. Time permitting, the course will conclude with fundamental theorems of vector calculus, including Green’s, Gauss’s and Stokes’s theorems.
MATH-2410: Ordinary Differential Equation
Prereq: MATH 2370
Description: This course provides an introduction to the theory, techniques, and applications of ordinary differential equations. Topics include first order equations, second order linear equations, series solutions, the method of Laplace transforms, systems of equations, Euler’s Method, some bifurcation theory, an introduction to nonlinear equations and stability theory.
MATH-2420: Discrete Dynamical Systems
Prereq: MATH 2250 & MATH 2370
Description: This course is an introduction to the mathematical theory of discrete dynamical systems, in which the state of a changing system at a given time is a fixed function of the previous state, and the long term behavior of the system is determined by iterating (i.e. self-composing) the function over and over. This theory is used to analyze ecological models, financial processes, differential equations, and more, and also illustrates how a system with seemingly simple rules can lead to surprisingly complicated behavior. In this course we’ll cover the following topics in one-dimensional dynamics: fixed point analysis, bifurcation theory, the quadratic family, chaotic systems, symbolic dynamics, Sharkovsky’s theorem, and an introduction to fractals and complex dynamics. The focus of the course will be on theory supplemented by computational experimentation.
MATH-2430: Convex Geometry, Func. & App.
Prereq: MATH 2250 & MATH 2370
Description: Convexity is an important concept in advanced mathematics, with important applications to optimization, economics, materials science, and other fields. In this elementary introduction, we will study convex sets, hyperplanes, linear functionals, separating hyperplanes, supporting hyperplanes, Helly’s Theorem, Kirchberger’s Theorem, the isoperimetric problem, polytopes, Euler’s formula, convex duality, linear programming, convex functions, distance functions, continuity and differentiability properties of convex functions, and optimization and convexity. We will consider examples taken from optimization, economics and materials science.
MATH-2460: Intro to Mathematical Biology
Prereq: MATH 2250 & MATH 2370
Description: Mathematical and statistical techniques are central to biology today. This course examines applications of mathematics in biological contexts including genetics, ecology, physiology, neuroscience and epidemiology. It draws on diverse mathematical tools and introduces students to principles that underlie mathematical modeling in diverse scientific and technical domains.
MATH-2540: Regression Analysis
Prereq: MATH 2140 or ECON 2110
Exclusion: PSYC 5004
Description: This course provides an in-depth coverage of regression analysis. After reviewing matrix algebra and simple linear regression using matrix notation, the course will focus on inference and model building in multiple linear regression. Regression inference, handling of categorical regressors, variable selection, interaction effects, multicollinearity, model diagnostics will be thoroughly covered. The course concludes with one-way and multi-way analysis of variance (ANOVA) models. Statistical concepts will be accompanied by hands-on data analysis using the R statistical software.
MATH-2620: Statistical Learning and Data Science
Prereq: MATH 2540 or ECON 2120 or GOVT 2201. A background in linear algebra is recommended.
Description: This course will introduce students to tools for handling large, heterogeneous datasets. Topics to be discussed include data visualization, data analysis using statistical and machine learning methods, and interpretation and communication of results. We will cover dimension reduction techniques (SVD, PCA), unsupervised clustering methods (k-means, hierarchical), supervised learning methods (linear regression, logistic regression, k-nearest neighbor, classification and regression trees, random forests, support vector machines), performance evaluation (bias-variance trade-off, cross-validation), variable selection methods (shrinkage methods, variable importance). The R statistical software will be used.
MATH-2625: Biostatistical Methods
Prereq: MATH 2140 or ECON 2110
Description: This course introduces students to some of the basic statistical techniques used in the analysis of data resulting from biomedical and biological research. Data from these sources can often have small samples, be categorical, or include time-to-event outcomes. Consequently, we will cover the derivation and application of introductory methods for nonparametric statistics, contingency tables, and survival analysis. Topics include signed rank tests, rank sum tests, the delta method, exact tests, Pearson’s chi-squared test, risk ratios, various odds ratio estimators, survivor and hazard functions, Kaplan-Meier and Nelson-Aalen estimators, log-rank tests, and multiple testing adjustments. This course emphasizes the writing up and presentation of statistical results for consumption by a broader audience. Statistical computing is performed using the statistical freeware tool, R.
MATH-2640: Advanced Regression Methods
Prereq: MATH 2540, PSYC 5004, or ECON 2120
Description: This course expands on the concepts learned in Math-2540 and introduces students to advanced regression methods. We will start with a review of linear regression models, including model building and model diagnostics. The course will focus on generalized linear models (GLM) for non-Gaussian response data, including models for categorical outcomes (logistic, multinomial logit, proportional odds models) and models for count data (Poisson, negative binomial, zero-inflated models). If time permits, other advanced topics chosen by the instructor will be discussed. Statistical concepts will be accompanied by hands-on data analysis using the R statistical software.
MATH-2645: Applied Time Series Analysis
Prereq: MATH 2140 or ECON 2110
Description: This course introduces students to the theory and application of time series methods for data that are collected over time. Topics include exploratory data analysis tools, methods for detrending, and seasonal adjustment of data, regression methods including regression with autoregressive errors, smoothing techniques including exponential smoothing, modeling and forecasting based on the ARIMA class of models, and ARCH/GARCH models. Time permitting, advanced topics including neural network methods for time series, and methods for analyzing multivariate time series. Students gain hands-on experience of applied time series methods for real data sets using the statistical software, R. Examples will be drawn from a variety of disciplines including business, finance, economics, health, environmental studies, and ecology. Students are required to complete a final data analysis project in which they apply the knowledge they gained in the course to real data.
MATH-2650: Computational Statistics
Prereq: MATH 2140 or ECON 2110
Description: Modern statistics increasingly relies on computationally intensive methods and techniques. This course introduces students to some of those commonly used methods. It assumes knowledge of concepts from introductory mathematical statistics including probability distributions, expectations, maximum likelihood estimation, and hypothesis testing. While some familiarity with the open source software R is preferred, no prior programming experience is expected. Topics will include random number generation, deterministic methods like Fisher scoring, rejection sampling, importance sampling, permutation tests, jackknifing, cross validation, bootstrap resampling, elementary Bayesian statistics, Gibbs sampling, as well as the expectation-maximization (EM) algorithm and variational inference. All computation will be performed in R.
MATH-2700: Mathematics of Deep Learning
Prerequisites: MATH-2250 & MATH-2370. Students must also have experience programming in Python to the extent of basic coding structures such as loops, functions, and conditionals.
Description: This course will explore different deep learning architectures and their associated mathematics. The architectures we will consider are fully connected neural nets, convolutional neural nets, recurrent neural nets and transformers. Associated with these architectures, we will explore basic ideas in optimization including gradient methods applied to convex and nonconvex optimization, the role of convolution in signal processing, basic models of time series, and probabilistic models in high dimensional spaces. There will likely be some variation from this plan depending on student interest and time constraints. Application of the architectures to datasets will be done through the python package Pytorch. By the end of the course, students should be able to write python code to implement the architectures. However, we will strike a balance between application and theory. By the end of the course, students should be familiar with associated ideas and results in mathematics, particularly relating to computation and modeling in high dimensions. Students who are mainly interested in implementation of the architectures are likely better off taking some of the other neural net classes offered at Georgetown.
MATH-2800: Intro to Proof/Prob-Solving
Prereq: MATH 1360 with a minimum grade of B
Description: This course is designed to help students transition from the computational concepts covered in 1000-level courses to the more theoretical concepts covered in 3000-level courses. Students learn to manipulate abstract definitions, determine whether statements are true or not, and prove or disprove statements. Special emphasis is placed on learning to read, write, and critique proofs. The different techniques covered are backward/forward proof, proof by contradiction, proof by contraposition, and proof by induction. Students will apply these methods to a variety of problems involving numbers, functions, sets, relations, and cardinality. This course is a prerequisite for many upper level mathematics courses.
MATH-2900: History of Mathematics
Prereq: MATH 2800 or permission of instructor
Description: This course explores the evolution of mathematical ideas from ancient civilizations—Babylon, Egypt, and Greece—through major developments in India, the Islamic world, and Europe, leading to modern mathematics. Topics include arithmetic, geometry, algebra, and calculus, with emphasis on how these concepts developed within their cultural, social, and technological contexts. Students will examine historical approaches to solving familiar problems in order to deepen their understanding of foundational mathematical ideas.
MATH-3200: Number Theory and Cryptography
Prereq: MATH 2250
Description: This course begins with a survey of cryptography from Roman times up to today’s high tech world. Students will learn how to encrypt messages and how to attempt to break codes. We will discuss the efficiency and security level of different encryption methods. To make the discussion rigorous, we will need tools/concepts/results from mathematics.
Mathematical content: mappings and inverse mappings, modular arithmetic, the additive group Z/n, the multiplicative group Z*/n, Euler’s phi function, Fermat’s Little theorem and Euler’s generalization, primitive roots, discrete logarithms, Diophantine equations, the Chinese Remainder Theorem, Wilson’s theorem, multiplicative functions, quadratic residues and the law of quadratic reciprocity.
Cryptographic content: classical ciphers and their decryption (shift, affine, and Vigenere ciphers), key exchange protocols (eg, Diffie-Hellman), public key ciphers (eg, RSA).
MATH-3210: Abstract Algebra I
Prereq: MATH-2250 and MATH-2800
Description: This is a rigorous introduction to algebraic structures, particularly groups, rings, and fields, and their homomorphisms, with emphasis on proofs. Topics from group theory will include modular arithmetic, symmetry groups, permutation groups, group actions, and Sylow theory. Topics from ring theory will include integral domains, unique factorization domains, and polynomial rings.
MATH-3300: Differential Geometry
Prereq: MATH-2250 and MATH-2370
Description: This course discusses the geometry of curves and surfaces. The length and the curvature of a smooth curve in two and three dimensions will be defined and computed. Parametrized surfaces in three-dimensional space are introduced and analyzed. Examples of quadric surfaces and ruled surfaces are presented. Important concepts are the Gauss curvature and Mean curvature of surfaces. Finally the inner geometry of surfaces is discussed and the concept of a geodesic, which is a curve on a surface that minimizes the distance between two points, is introduced. If time permits some theorems of vector calculus that you may have seen in Multivariable Calculus are reviewed. This course will reinforce tools studied in Multivariable Calculus, such as the chain rule, as well as elements of Linear Algebra, in particular the Spectral Theorem.
MATH-3310: Analysis I
Prereq: MATH 2800
Description: This is the first part of the two semester advanced calculus sequence (MATH 3310 & MATH 4310) which provides a rigorous treatment of topics in calculus with the emphasis on proofs of major theorems. Topics include the basic properties of the real numbers and n-dimensional Euclidean space, the basic topology of metric spaces including compactness and connectedness, the theory of numerical sequences and series, and the properties of continuous functions, differentiable functions and integrable functions on the real line. This course is expected to result in an appreciation of some of the central theorems of analysis on the real line, such as the Mean Value Theorem, Inverse Function Theorem and Taylor’s formula..
MATH-3320: Functions of Complex Variables
Prereq: MATH-3310
Description: Complex numbers. Analytic functions including exponential, logarithmic and trigonometric functions of a complex variable. Geometric and mapping properties of analytic functions. Contour integration, Cauchy’s theorem, the Cauchy integral formula. Power series representations. Residues and poles, with applications to the evaluation of integrals. Conformal mapping and applications as time permits.
OVER/UNDER COURSES
MATH-4110: Topology
Prereq: MATH 3210
Description:This course serves as an introduction to topology, both point-set topology and algebraic topology. The first half of the course is devoted to the basics of point-set topology. Topics include topological spaces, basis for a topology, metric spaces, subspaces, product spaces, continuous maps and homeomorphisms, connectedness and path-connectedness, compactness, the Cantor set, quotient spaces, and surfaces. The remainder of the course explores some aspects of algebraic topology: homotopy and homotopy type, cell complexes, operations on spaces, homotopy equivalence, the fundamental group and its applications, Van Kampen’s theorem, and covering spaces.
MATH-4210: Abstract Algebra II
Prereq: MATH 3210
Description: This course will cover rings and their homomorphisms, ideals, prime and maximal ideals, Euclidean domains, principal ideal domains, unique factorization domains, polynomial rings, irreducibility criteria for polynomials, and Galois theory as much as time permits: fields and their extensions, splitting fields, separable and normal extensions, automorphism groups of fields, Galois extensions and modules.
MATH-4307: Mathematics of Climate
Prereq: MATH 2410 or MATH 5052
Description: In this course, mathematics and statistics will be used to answer current questions of interest in climate science and sustainability, and climate science will be used to motivate and explain techniques from applied mathematics and statistics. Mathematical and statistical topics will be selected from theories and methodologies for dynamical systems, bifurcation theory, ordinary and partial differential equations, signal processing, regression analysis, extreme value theory, and data assimilation. The course will emphasize conceptual models that capture important aspects of the Earth’s climate system: Energy balance and temperature distribution, ocean circulation patterns such as the Gulf Stream and El Niño – Southern Oscillation, glaciation cycles, extreme weather events, and data assimilation for weather and climate. Some knowledge of R is required.
MATH-4310: Analysis II
Prereq: MATH 3310
Description: This course is about analysis in n-dimensional Euclidean spaces. Topics include the Euclidean spaces, metric spaces, functions of several variables, differentiation and integration of functions of several variables, and the theorems of vector calculus. Most of the topics will be familiar from Multivariable Calculus (MATH-2370). Many proofs will be presented and discussed. Students will be made familiar with some concepts of topology, the basic theory of metric spaces as well as the importance of compact sets and connected sets in analysis. This course is expected to result in an appreciation for some of the central theorems of analysis such as the Inverse and Implicit Function Theorems, the Change of Variable Formula in multiple integrals, and the Integral Theorems by Green, Gauss, and Stokes.
MATH-4311: Intro Partial Differential Equations
Prereq: MATH 2370 and MATH 2250; MATH 2410 recommended
Description: Partial differential equations are essential tools in applied mathematics. This course synthesizes and builds upon techniques and concepts from multivariable calculus, linear algebra and ordinary differential equations to solve and analyze key linear partial differential equations such as the Laplace equation, the heat equation, and the wave equation. Topics include initial and boundary value problems, maximum principles, Fourier series, Fundamental solutions and Green functions.
MATH-4314: Optimization
Prereq: MATH 2410 or MATH 5052
Description: Optimization problems arise in a variety of applied sciences. Examples are the most economical transport of goods from service points to destinations, the design of components in manufacturing, or the desire to minimize noises and vibrations in airplanes or cars. The mathematical formulation of optimization problems results in the task of minimizing or maximizing functions which are subject to additional conditions called constraints.
Some optimization problems, such as finding the extreme values (maxima and minima) of continuous functions of several variables, are already discussed in calculus. Of particular interest in this course are optimization problems with constraints, possibly given by additional equations, inequalities, or differential equations. According to the type and the nature of the constraint, different techniques of solving the optimization problem are to be developed. These include linear programming, nonlinear programming, variational problems, and optimal control. Students will see a good number of examples of optimization problems, study various analytical techniques of solving them, reinforce their knowledge of differential calculus of functions of several variables, and should gain an appreciation of two fundamental mathematical concepts: Linearity and Convexity.
MATH-4320: Introduction to Stochastic Differential Equations
Prereq: MATH-2410 and (MATH-2140 or ECON-2110) and (MATH-2240 or MATH-2250)
Description: Many real-world systems, from finance to physics to neuroscience, exhibit random behavior that can be modeled using stochastic differential equations (SDEs). This course provides a first introduction to SDEs. After reviewing some essential probability theory, we construct and analyze Brownian motion. We then develop Itô’s integral, including the chain and product rules. Finally, we study Itô SDEs, examining existence and uniqueness of solutions and exploring applications to partial differential equations and option pricing.
MATH-4460: Machine Learning: Theory & Application
Prereq: MATH-2250 & MATH-2370 & MATH-2140
Description: Machine Learning describes a collection of computational techniques for finding patterns in data. This course focuses on supervised machine learning, which deals with using historical labeled data to construct a predictor to accurately label future data. We will cover how to formulate the supervised learning problem in mathematical terms, describe a measure of performance, restrict a search space for constructing models for prediction, optimize performance over the search space, and check for generalization. We will also discuss statistical learning guarantees and of limits in such results. The course, while theory-centric, will provide substantive exposure to implementation through programming exercises, and will establish foundations for constructing architectural ingredients of neural networks, which are ubiquitous in modern machine learning practice.