Mathematics and Statistics

Understanding the role of creative thinking, problem solving, and collaboration in mathematics and statistics; exploration of research and careers in the mathematical and statistical sciences; building community with fellow mathematics and statistics majors.

Discrete math for Liberal Arts students: graphs (circuits, trees, directed graphs), discrete growth, symmetry, and select topics from planning and scheduling, linear programming, tiling, voting methods, and unit conversions. Not open to students who have completed MAT 1505.

Examining fairness in our personal lives and in society: Voting systems and power indices, strategic political positioning spatial models, fair division, congressional district apportionment, game theory, the GINI index of economic inequality, gerrymandering.

Course in an area of pure or applied mathematics or statistics. May be repeated for credit if areas of topical focus are different. Designed specifically to satisfy the core requirement in mathematics and statistics, for students in the humanities and social sciences.

Discrete and continuous dynamics of biological systems: discrete dynamical systems, sequences, functions, discrete and continuous limits, the derivative, the integral, methods and applications of differentiation and integration, Taylor polynomials, modeling with differential equations, Euler's method, applications to Biology.

Mathematical and statistical modeling in the Life Sciences. Topics selected from: dynamical systems, diffusion, Markov, Bayesian, connectionist, and information theory models, applied to epidemiology, ecology, neuroscience and neuron signaling, cell and molecular biology, genetics, physiology, psychology, and other areas. Pre-requisites MAT 1312 or Equivalent

Calculus for Liberal Arts students: polynomial, rational and transcendental functions, the derivative, numerical and graphical introduction to integration.

Functions, limits, and basic definitions of differential and integral calculus. Techniques of differentiation and integration. The Fundamental Theorem of Calculus. Applications in various areas of business and economics.

Limits, transcendental functions (logarithms, exponential functions, inverse trigonometric functions), differentiation (definition, tangent lines, rates of change, techniques, implicit differentiation, related rates), applications of differentiation (graphing, optimization), indeterminate forms and L'Hopital's Rule. Use of a computer algebra system, eg. MAPLE.

Integration (indefinite, definite), applications of integration (area, volume, applications to physics and economics, etc.), methods of integration, approximate integration (trapezoidal and Simpson's rules), improper integrals, differential equations, infinite sequences and series. Continued use of a computer algebra system.

Internship Elective

Internship Elective

Vectors, matrices, and matrix algebra; systems of linear equations; matrix inverses; least squares problems; eigenvalues and eigenvectors; using Python for computational linear algebra; applications from areas such as data science, computer graphics, graph algorithms, and web search.

Parametric equations; polar, cylindrical, and spherical coordinates; vectors and the geometry of space; vector functions (derivatives, integrals, curvature, etc.); partial derivatives; optimization; multiple integration and its applications; vector calculus (line integrals, vector analysis). Continued use of a computer algebra system.

Mathematical proofs: direct, indirect, induction. Logic, set theory, relations, functions. Optional topics from algebra, number theory, combinatorics, and analysis.

First order and linear second order differential equations, matrices and linear equation systems, eigenvalues and eigenvectors, and linear systems of differential equations.

Internship

Internship

Lecture course in an area of mathematics or statistics. May be repeated for credit if topics are different.

Explore techniques for solving mathematical problems, including problems typical of the Putman Mathematical Competition. Students solve and present solutions to problems posed.

Vectors, matrices, transpose and inverse of a matrix, systems of linear equations, the four fundamental subspaces, eigenvalues and eigenvectors, symmetric matrices, matrix factorizations, applications such as information retrieval, ranking web pages, graphs and networks, least squares, and data compression.

Real numbers, sequences, convergence, supremum and infimum, completeness of the reals, continuous functions, Intermediate Value Theorem, differentiable functions, Mean Value Theorem, Riemann integral, Fundamental Theorem of Calculus, Taylor's Theorem.

Theory of vector spaces, linear transformations, basis, and dimension. Selected topics from orthogonal transformations, least squares, eigenvalues and eigenvectors, similarity, diagonalization, matrix decompositions, infinite dimensional transformations.

Topics selected from groups and subgroups, cyclic groups, permutation groups, isomorphisms, direct products, cosets and Lagrange's Theorem, normal subgroups and factor groups, group homomorphisms, the Fundamental Theorem of Finite Abelian Groups, rings, fields.

Development of mathematics from ancient times to the birth of calculus in the seventeenth century.

Induction, permutations and combinations, general counting methods, generating functions, recurrence relations, principle of inclusion-exclusion, graph theory, trees, planarity, crossing numbers, Hamiltonian cycles, Eulerian tours.

Numerical and computational aspects of root-finding methods, interpolation and polynomial approximation, numerical differentiation and integration, approximation theory.

Model formulation, the modeling process, graphs as models, examples from the physical and social sciences, parameter estimation, experimental modeling, dimensional analysis, modeling discrete and continuous dynamic behavior, implementation.

Study of popular games, toys, and puzzles using recursions, counting techniques, graph theory, group theory, probability, Markov chains, and other mathematical tools.

Basic tools of financial markets; options; asset price random walks; estimation of parameters; arbitrage put-call parity; Black-Scholes Model; implied volatility; portfolio-optimization; hedging.

Deterministic methods: mathematical optimization, linear programming, formulation and solution techniques, duality, integer linear programming, transportation problem, assignment problem, network flows, dynamic programming.

Probabilistic methods: brief review of probability, transient and asymptotic properties of Markov chains, queuing models and their applications, Markov decision processes, forecasting, game theory, simulation.

Topics selected from affine, hyperbolic, spherical, elliptic, Euclidean or projective geometry.

Congruences, quadratic reciprocity, Diophantine equations; applications.

Algebra of complex numbers, analytic functions, Cauchy- Riemann equation, Laplace equations, conformal mapping, integrals of complex functions, Cauchy's theorem, power series, Taylor's theorem, Laurent's theorem, residues, entire functions.

Topological equivalence, connectedness, compactness, topology of subsets of Rn, manifolds, topological embeddings, topological spaces.

Geometry of curves and surfaces, curvature, first and second fundamental forms, minimal surfaces, use of MAPLE.

Supervised study of selected topics or problems in mathematics, student presentations. May be repeated for credit if content is different.

Lecture course in an area of applied mathematics. May be repeated for credit if topics are different.

Reading in a selected branch of mathematics under the direction of a member of the staff. May be repeated for credit.

Reading in a selected branch of mathematics under the direction of a member of the staff. May be repeated for credit.

Reading in a selected branch of mathematics under the direction of a member of the staff. May be repeated for credit.

Displaying and summarizing data, basic probability concepts, normal distributions, sampling distributions, estimation for a single population parameter, regression and correlation. Not open to students who have completed MAT 1505.

Probability concepts, hypothesis testing, inferences about means, variances and proportions, contingency tables, analysis of variance. Not open to students who have completed MAT 1505.

Descriptive and inferential statistics: graphical displays, estimation, & hypothesis testing. Restricted to nursing students; others by special permission only.

Introduction to statistics including topics such as study design, graphical and numerical descriptive statistics, bivariate data analysis, probability, sampling distributions, confidence intervals, hypothesis testing, goodness of fit tests, analysis of variance; resampling and simulation using statistical software; interpreting output from and understanding selected algorithms used in statistical packages.

Statistical concepts and methods with applications in biological and life sciences; data visualization, descriptive statistics, probability distributions, interval estimation and hypothesis testing for one and two variables, statistical software.

Statistical concepts and methods useful in analyzing problems in all areas of business. Descriptive statistics, probability, sampling distributions, confidence intervals, hypothesis testing, regression analysis, and time series. Applications in various areas of business and economics.

Lecture course on a topic from Statistics. May be repeated for credit if topics are different.

Application of Bayesian statistical procedures. Implementation using the programming language R. Bayes's Theorem. Bayesian statistical inference. Various types of prior distributions. Computer-intensive methods. Assessing the prior. Robustness analysis. Writing Bayesian statistical reports.

Data displays and summarization, probability distributions, point and interval estimation, hypothesis testing, categorical data analysis, regression and correlation.

Simple and multiple linear regression, including prediction, correlation, model building, multicollinearity, influential observations, and model fit; ANOVA for designed experiments, including completely randomized, randomized block and factorial designs; Time Series including linear time series models, moving averages, autoregressive and ARIMA models, estimation and forecasting.

Combining and summarizing real-world data to inform decision-making and predictions; data wrangling, visualization, text mining, ethics; uses R programming language. Some programming experience recommended.

Analysis of categorical response data, including contingency tables, logistic regression, multinomial logistic regression, and generalized linear models.

Completely randomized, randomized block, and Latin square designs; full and fractional factorial designs; nested and split plot designs; response surface methology.

Time series modeling, forecasting, and diagnostics, with an emphasis on applications from business and the sciences.

Regression methods for analyzing medical and biological data, including multiple regression, logistic regression, survival modeling, and longitudinal mixed modeling.

One-sample, two-sample, and multi-sample rank tests; nonparametric confidence intervals; permutation tests; nonparametric regression and correlation; the bootstrap.

Supervised and unsupervised data mining techniques, including clustering, classification, and association rule learning.

Probability, random variables, joint distributions, expected values, limit theorems, distributions derived from the normal distribution.

Transformation of random variables, Distributions related to the normal, Central Limit Theorem, Law of Large Numbers, Point estimation, Maximum likelihood estimation, Bias, Consistency, Sufficiency, Confidence intervals, Hypothesis testing, Likelihood ratio tests.

Supervised study of selected topics or problems in statistics, with student presentations and papers. May be repeated for credit if content is different.

Lecture course in an area of statistics. May be repeated for credit if topics are different. Prerequisites: Dependent on Topic.