Courses
Below are extended descriptions of certain courses to help you plan your academic pathway in mathematics.
1000
Extended Description: Math 1005 serves as a bridge between algebra and calculus, equipping students with the foundational skills necessary for success in higher mathematics. The course covers a wide range of topics, including functions and their properties, such as polynomial, rational, exponential, logarithmic, and trigonometric functions. Students learn to analyze and graph these functions, solve equations and inequalities, and apply mathematical concepts to real-world problems. Additional focus is often placed on advanced topics such as sequences, series, and mathematical modeling. By emphasizing critical thinking and problem-solving skills, Math 1005 prepares students for the rigorous demands of calculus and other advanced mathematics courses.
Extended Description: Math 1100 is a calculus class designed to develop a solid understanding of essential calculus topics, including limits, derivatives, and integrals. The course engages students in a thorough exploration of various types of functions, such as polynomial, rational, exponential, logarithmic, and trigonometric functions, with a strong emphasis on their properties, transformations, and graphical interpretations. Students learn to apply analytical techniques to solve complex equations and inequalities, fostering a deep comprehension of function behavior, including continuity and asymptotic analysis. Additionally, Math 1100 introduces preliminary concepts related to derivatives and integrals, paving the way for more advanced studies in calculus. By emphasizing problem-solving, critical thinking, and real-world applications of mathematical theories, the course prepares students for the rigorous challenges of further calculus topics and fosters a strong mathematical intuition necessary for their academic and professional pursuits.
Extended Description: Math 1200 is the first semester of a two-semester calculus sequence that emphasizes differential calculus, particularly in the context of optimization and its applications in the social sciences. Students begin with a review of essential functions, including exponential, logarithmic, and trigonometric functions, followed by topics such as limits, differentiation of various functions, related rates, and curve sketching. The course is structured to reinforce students’ understanding of calculus concepts, ensuring that those without prior calculus experience can successfully navigate its challenges.
Extended Description: Math 1201 is the second semester of the calculus sequence, focusing on integration techniques and their application to modeling problems commonly encountered in the social sciences, particularly through the use of differential equations. Key topics include antiderivatives, the Fundamental Theorem of Calculus, techniques of integration, and methods for solving first-order differential equations. The course also introduces linear approximation and basic probability, enhancing students' mathematical modeling skills in their respective fields of study.
Extended Description: Math 1300 is an intensive introduction to Calculus designed to provide students with a mathematically rigorous foundation. The course covers key concepts including limits and continuity, derivatives, optimization, and basic integration, with an emphasis on formal mathematical definitions and logical reasoning. Students will solve complex problems while focusing on enhancing their understanding of the underlying mathematical principles. The objectives are to facilitate comprehension of mathematical arguments, develop computational skills using various methods, explore relationships among concepts, and prepare for more advanced topics in Math 1301 and other higher-level mathematics courses. Ultimately, Math 1300 aims to deepen students' mathematical thinking and communication skills while fostering a substantial understanding of truth in mathematics.
Extended Description: Math 1301 is an advanced, rigorous extension of the Math 1300 sequence, focusing on deepening students' understanding of Calculus through formal definitions and logical reasoning. The course addresses key topics such as transcendental functions, integration techniques, differential equations, sequences and series, and calculus in various coordinate systems. Its primary objectives include mastering mathematical definitions, enhancing computational skills, investigating interconnections among concepts, and laying a robust foundation for further studies in advanced mathematics. Emphasizing comprehension and the reasoning behind mathematical truths, Math 1301 prepares students for higher-level courses while reinforcing sound mathematical thinking.
2000
Extended Description: Multivariable Calculus, as a central topic in the curriculum, extends the principles of single-variable calculus to functions of multiple variables. The course introduces fundamental concepts such as vectors, curves, and surfaces in space, providing a framework for analyzing objects in three or more dimensions. Key topics include limits and continuity, partial differentiation, multiple integration (double and triple integrals), and vector calculus. The curriculum also delves into essential theorems like Green's, Stokes', and the Divergence Theorem. Its primary objectives are to build a strong foundation for advanced mathematics, science, and engineering by developing critical skills in generalizing calculus operations and applying them to higher-dimensional spaces.
Extended Description: This course integrates differential equations and linear algebra. The curriculum covers scalar differential equations, Laplace transforms, and systems of differential equations. It also explores matrix operations, determinants, the algebra of matrices, vector spaces, linear operators, and the concepts of eigenvalues and eigenvectors. This blend of topics provides a solid foundation in both subjects, emphasizing their applications in various fields.
Extended Description: Linear algebra is an essential area of mathematics that serves as a powerful tool for solving systems of linear equations, such as determining values of variables in equations like (x + 2y = 4) and (2x - y = 5). The course will introduce matrices as a method for organizing and manipulating numerical data, initially viewing them as a form of spreadsheet. As students advance, they will explore geometric interpretations of linear equations and understand matrices as functions that can manipulate spaces in two or more dimensions. This understanding positions linear algebra as a versatile framework that simplifies complex problems through rephrasing and problem-solving techniques. To support learning and communication, the course will emphasize the practice of articulating mathematical arguments clearly. Topics covered will include vector and matrix operations, linear transformations, properties of vector spaces, eigenvalues, and eigenvectors. The instructor anticipates an engaging and informative experience and encourages open communication for any questions that may arise.
Extended Description: Differential equations are a crucial tool for scientists and engineers to model various phenomena, making it essential to learn how to solve these equations and interpret their solutions. This course will cover techniques for solving linear, separable, and exact first-order differential equations, as well as higher-order linear equations with constant or variable coefficients. Additionally, applications of these equations in areas such as population modeling, Newtonian mechanics, mixing solutions, and mass-spring systems will be explored. Students will also use techniques to visualize and estimate solutions to differential equations, while gaining an understanding of the Laplace transform and power series as effective tools for solving these equations.
Extended Description: Math 2500 is an integrative course that combines the principles of multivariable calculus and linear algebra, providing students with a comprehensive understanding of how these two mathematical disciplines interact and apply to real-world problems. Students will explore topics such as partial derivatives, multiple integrals, and the gradient, divergence, and curl of vector fields, while also delving into linear algebra concepts such as vector spaces, linear transformations, and matrix operations. The course emphasizes the geometric interpretation of multivariable functions and the applications of linear algebra in solving systems of equations and understanding transformations in higher dimensions. Through a blend of theoretical discussions and practical problem-solving, students will develop a robust framework for analyzing complex mathematical problems and enhancing their mathematical reasoning skills, preparing for advanced coursework in mathematics and its applications in various fields.
Extended Description: This course covers fundamental topics in vector algebra and geometry, linear transformations, and matrix algebra. The curriculum explores real and complex vector spaces, systems of linear equations, and inner product spaces. It also delves into functions of several variables and vector-valued functions, including limits, continuity, and derivatives, and examines extremum and nonlinear problems. The course covers multiple integrals, line and surface integrals, and differential forms, incorporating integration on manifolds and theorems from Green, Gauss, and Stokes. A strong emphasis is placed on eigenvectors and eigenvalues, as well as the use of rigorous proofs.
Extended Description: This course integrates differential equations and linear algebra. The curriculum covers scalar differential equations, Laplace transforms, and systems of differential equations. It also explores matrix operations, determinants, the algebra of matrices, vector spaces, linear operators, and the concepts of eigenvalues and eigenvectors. This blend of topics provides a solid foundation in both subjects, emphasizing their applications in various fields.
Extended Description: This course provides a detailed exploration of differential equations, which are foundational to many scientific and engineering disciplines. Students will gain a solid understanding of first-order ordinary differential equations, second-order ordinary differential equations, matrix methods for linear systems, and autonomous systems. A key focus of the course is on rigorous proofs, which helps students deepen their comprehension of the underlying theory. By engaging with these proofs, students will enhance their critical thinking and mathematical writing skills, preparing them for advanced studies and practical applications.
Extended Description: Probablility and Statistics is an essential course for scientists and engineers, providing the fundamental tools for modeling real-world phenomena. The curriculum covers key principles of probability, including set theory, counting methods, and Bayes' Theorem. Students will define and analyze various discrete and continuous random variables, such as binomial, Poisson, normal, and exponential distributions. A major focus is on applying the Central Limit Theorem for large sample size approximations. The course culminates in the practical application of these concepts through population parameter estimation, confidence intervals, and hypothesis testing. The course also introduces linear regression for two variables.
Extended Description: MATH 2820 is an introductory course in Probability and Mathematical Statistics that builds on a foundation of MATH 2300. The first half of the curriculum focuses on classical probability theory and combinatorics. The second half introduces mathematical statistics, covering topics such as random variables, estimation, and hypothesis testing. Unlike a data-focused course, the main objective of MATH 2820 is to develop the essential mathematical and probabilistic foundations of statistics for students pursuing careers in science.
Extended Description: This course focuses on the practical application of statistical procedures to real-world data, building upon the theory from MATH 2820. The primary objectives include learning the basics of R programming, preprocessing and visualizing data, and estimating parameters. Students will also learn to check model assumptions and effectively report their findings.
Extended Description: MATH 2821, a follow-up to MATH 2820, shifts its focus from foundational theory to applied, data-driven statistical analysis. This course emphasizes experimental design and statistical inference procedures. Key topics include one-sample, two-sample, and paired t-tests; chi-square tests; and the analysis of variance for various designs, such as k-sample and factorial designs. A significant portion of the course is also dedicated to inference procedures related to regression data. Essentially, MATH 2821 provides a comprehensive overview of Applied Statistics for students with a prior background in Mathematical Statistics.
3000
Extended Description: This course provides a comprehensive exploration of the major developments in mathematics, tracing its evolution from ancient times to the early twentieth century. It is designed to offer a balanced perspective, delving into both the rich historical context and the fundamental mathematical content that shaped the field. The curriculum emphasizes a rigorous approach, with assignments focused on working through numerous exercises and theorems to solidify understanding. The course is structured to be particularly beneficial for teacher candidates, offering them a deep and well-rounded perspective on the subject they will teach.
Extended Description: MATH 3010 is a proof-based course designed to elevate students' mathematical skills beyond standard textbook exercises. The main objectives are to teach formal proof writing and to expand students' problem-solving experience with more challenging problems. The curriculum exposes students to various areas of mathematics, including topics from calculus, linear algebra, abstract algebra, and number theory, building upon prior knowledge without requiring advanced prerequisites. This course is particularly beneficial for students interested in preparing for math competitions, such as the Virginia Tech Regional Math Contest or the Putnam Problem Solving Competition.
Extended Description: Introduction to Analysis serves as a crucial bridge between introductory calculus and advanced analysis courses. This course re-examines the fundamental concepts of calculus, including the real number system, sequences and limits, continuity, derivatives, and the Riemann integral. While the topics may seem familiar, the primary focus is on understanding and constructing rigorous mathematical proofs. This makes the course excellent preparation for students pursuing graduate studies in mathematics, as well as for secondary mathematics teachers and actuaries who require a deep, foundational understanding of calculus principles.
Extended Description: This course explores the beautiful properties of the complex number system and analytic functions. Students will discover how trigonometric, hyperbolic, and exponential functions are interconnected in the complex plane. The curriculum also covers practical applications in engineering and science, such as the Fourier transform and boundary value problems in electrostatics and heat flow. The study's central tools will be Cauchy's integral theorem and the theory of residues. This course is designed for upper-level undergraduate and graduate students in mathematics, engineering, and science.
Extended Description: This course's principal objective is to solve boundary value problems involving partial differential equations, with a strong background in calculus and differential equations being essential. It explores the heat, wave, and potential equations, deriving mathematical models from physical intuition. The primary solution method taught is separation of variables, due to its wide use and versatility. Other techniques covered include D'Alembert's solution, series solutions, numerical methods, and Laplace transforms. The course highlights how these concepts are applied in fields such as classical and quantum mechanics, elasticity, and fluid mechanics.
Extended Description: This course is a deep dive into Fourier Analysis. The curriculum covers a wide range of topics, including Fourier Series, convolutions, and various types of convergence. It also explores the Fourier Integral in one and multiple dimensions, including the Fourier inversion formula and Plancherel Theorem. The course delves into advanced concepts such as the Radon and X-Ray transforms, Time-Frequency Analysis, Wavelet Transform, and Sampling Theory. The course also touches on Finite Fourier analysis and Fourier analysis on Abelian groups.
Extended Description: This is a rigorous, proof-based mathematics course that covers three main topics: Knot Theory, Point Set Topology, and Surfaces. The curriculum delves into Knot Theory by examining loops in 3D space, proving the existence of knots, and introducing modern polynomial invariants. Students will also study Point Set Topology, which provides the foundational concepts of continuity, connectedness, and compactness crucial for many mathematical disciplines. The course concludes with Surface Theory, where students will learn to distinguish and classify various surfaces, such as spheres and Klein bottles. It is highly recommended for mathematically talented students, especially those considering graduate studies in mathematics or research in theoretical sciences.
Extended Description: This course explores transformations of the plane, with a focus on their applications in group theory. The curriculum covers different types of isometries, including reflections and glide reflections, and their classification. It also delves into the analysis of frieze groups and wall paper groups, providing students with the tools to analyze and understand repeating patterns. This course is especially recommended for prospective mathematics teachers.
Extended Description: This course explores the foundational concepts of differential geometry and general relativity, with a primary focus on curvature. The curriculum is designed to help students understand these concepts both geometrically and computationally. The course emphasizes the cases of curves and surfaces in Euclidean space before moving to higher-dimensional frameworks. Throughout the course, interdisciplinary applications of differential geometry will be highlighted.
Extended Description: This course explores abstract algebraic concepts, such as groups, fields, and rings, by tracing their evolution from concrete examples. The curriculum begins with group theory, which arose from the study of symmetries and is defined by three fundamental axioms. Students will learn to apply these axioms to solve problems and understand their consequences. The course then transitions to the study of fields, demonstrating how the need to solve equations—from simple linear equations to more complex algebraic equations—necessitates the creation of larger number systems, like the rational, real, and complex numbers. Finally, the course will touch upon the famous problem of why equations of degrees 2, 3, and 4 are solvable in radicals, while those of degree 5 are not, and how this relates to the symmetries of an equation's splitting field.
Extended Description: This course explores the relationship between syntax (symbols) and semantics (meaning) within formal deductive systems. It formalizes how syntax is presented and interpreted in propositional, equational, and first-order logic. The main objective is to understand the connections and distinctions between these two foundational concepts of logic.
Extended Description: This course applies algebra to reliability and secrecy of information transmission. The curriculum covers error-correcting codes, including linear, Hamming, and cyclic codes, and possibly more advanced codes like BCH or Reed-Solomon. It also delves into cryptography, focusing on symmetric-key, DES, and RSA encryption. Additional material on topics like finite fields and elliptic curve cryptography may be provided.
Extended Description: This course covers a wide range of topics in linear algebra and functional analysis. The curriculum explores real and complex vector spaces, norms, inner products, and Hilbert spaces. Key concepts include orthonormal bases, eigenvalues and eigenvectors, and various types of operators (e.g., unitary, normal). The course also delves into spectra of operators, spectral theorems, and several matrix decompositions, such as the Jordan-Holder, polar, and singular value decompositions. A brief introduction to applications, including some related to data science, will also be provided.
Extended Description: This course in numerical analysis provides scientists and engineers with essential tools for modeling and understanding various phenomena. It explores the principles and limitations of numerical approximation and floating-point arithmetic. The curriculum develops methods for solving both linear and non-linear equations (e.g., bisection, Newton’s method) and techniques for interpolation and approximation (e.g., Lagrange polynomials, least squares regression). The course also covers solving systems of linear equations using methods like Gaussian elimination and LU factorization, as well as numerical integration and the numerical solution of ordinary differential equations using methods like the Taylor series and Runge-Kutta.
Extended Description: Mathematical Models in Biology and Medicine is an introductory course for students in science, mathematics, and medicine. It is designed to provide a general overview of using mathematical tools to model biological and medical phenomena. The course reviews key mathematical concepts, including linear regression, differential equations, matrix analysis, and probability theory. The curriculum explores general approaches to modeling and computer simulation with examples spanning population dynamics, random movement, neurophysiology, biochemistry, disease pathogenesis, and specific models related to the COVID-19 pandemic. Prior knowledge of calculus and differential equations is assumed.
Extended Description: This course delves into advanced topics in probability theory beyond introductory statistics. The curriculum covers a selection of in-depth subjects, including combinatorics, various probability models for random variables, and limit theorems like the Law of Large Numbers. A significant portion of the course is dedicated to stochastic processes, with a particular focus on discrete Markov processes. The course equips students with the mathematical tools to solve challenging problems, such as those related to probability of random selections, matches in a matchbox, and the barefoot runner problem.
Extended Description: This course covers fundamental topics in advanced statistical theory. The curriculum includes distribution theory, parameter estimation, hypothesis testing, statistical inference, analysis of variance, and regression. The course also discusses concrete applications of these theoretical tools in various fields, such as engineering, actuarial sciences, and econometrics. Students will learn to analyze real-world problems, such as the effects of different sugar solutions on bacterial growth, differences in sentencing severity for criminal defendants, and whether a grader assigns scores randomly.
Extended Description: This course covers the theory of interest, focusing on the time value of money, annuities, rates of return, yield curves, and immunization. It also teaches the pricing of financial instruments, such as annuities, bonds, stocks, and derivatives, using arbitrage-free assumptions. This course is foundational for students pursuing a career in actuarial science.
Extended Description: This course provides a probabilistic analysis of insurance, covering single-life models and incorporating the time-value of benefits, life annuities, and premiums. The curriculum also explores multiple-decrement and multiple-life models. Key probabilistic topics include Markov chains and Poisson processes. The course focuses on applying these actuarial (probabilistic) models to various insurance scenarios and covers a subset of the topics on the SOA ALTAM exam.
Extended Description: This course focuses on mathematical modeling, with an emphasis on models used in economics. The curriculum covers key topics such as demand functions, profit maximization, Nash equilibrium, and the analysis of mergers. It also delves into auction models, option valuation, present value of income streams, and interest rate risk. Additionally, the course introduces statistical models and estimation techniques, and provides detailed instruction on computing these models using Mathematica.
Extended Description: The curriculum provides a detailed exploration of linear methods for regression and classification, which are foundational techniques in machine learning. It delves into the critical concept of the bias-variance tradeoff, helping students understand how to balance model complexity to achieve optimal performance. The course also covers basis expansions, a method for transforming data to better fit a model. Students will then advance to more sophisticated techniques, including kernel methods and support vector machines, which are powerful tools for complex pattern recognition. Additionally, the course addresses dimension reduction to handle high-dimensional data efficiently, and introduces various clustering algorithms for unsupervised learning.
Extended Description: This course focuses on combinatorics, developing and applying various counting techniques to solve complex problems. The curriculum covers a range of challenging scenarios, from arranging books and dealing bridge hands to distributing doughnuts and people in hotel rooms, requiring a solid understanding of combinatorial principles. The course also includes a brief introduction to graph theory, where these counting techniques are applied to solve problems related to networks. The primary objective is to equip students with the skills to tackle intricate counting and arrangement questions.
Extended Description: This course covers the fundamentals of game theory. It begins with zero-sum games and the Minimax Theorem, then progresses to general-sum games and the existence of Nash equilibria. The curriculum also explores extensive games with incomplete and imperfect information, repeated games with discounting, and cooperative games, including Shapley's Theorem and Nash bargaining. A key focus of the course is on the construction and verification of strategies for these various game types.
Extended Description: Number Theory is an introductory course that explores the fundamental properties of integers, delving into a subject with a rich history and a wealth of unsolved problems. The curriculum covers core concepts such as the Fundamental Theorem of Arithmetic, Fermat's Little Theorem, quadratic reciprocity, and Hensel's lemma. It also examines intriguing topics like Pythagorean triples and sums of squares. The course highlights the relevance of number theory by laying the groundwork for understanding key applications, particularly in modern cryptography and web security.
Extended Description: This course is an independent study where a student works directly with a faculty member. Students must have completed four mathematics courses at the 2800 level or above prior to enrolling, and instructor and Director of Undergraduate Studies consent is also required.
Extended Description: This course covers selected advanced topics in mathematics which vary from semester to semester.
4000
Extended Description: This graduate-level course, also suitable for advanced undergraduates, offers a rigorous introduction to point-set topology. We'll delve into fundamental properties of topological spaces, exploring connectedness and compactness to describe crucial characteristics of sets. The course covers various countability and separation axioms that distinguish different types of topological spaces. A significant portion is dedicated to complete metric spaces and their unique properties, concluding with an examination of function spaces, vital for analysis. This course establishes solid foundations for further graduate study in analysis and geometry, cultivating mathematical maturity consistent with beginning graduate students in top programs. Prior completion of MATH 3200 or 3300 is highly recommended.
Extended Description: This graduate-level course, a continuation of Math 4200, is primarily for mathematics graduate students, including advanced undergraduates. It builds on foundational concepts to introduce algebraic topology, using algebraic structures to study topological spaces. We'll explore the fundamental group and covering spaces, key concepts capturing "holes" and connectivity. The course then applies these ideas to the topology of surfaces, classifying their intrinsic properties. We'll move on to simplicial complexes and homology theory, a fundamental invariant revealing deeper structural information. Finally, we'll touch upon homotopy theory, providing a glimpse into deformation properties. This course further prepares you for graduate study, especially in Geometric and Algebraic Topology, enhancing your ability to solve increasingly sophisticated problems.
Extended Description: This course explores group theory, a cornerstone of abstract algebra, providing a deep understanding of algebraic structures and their symmetries. We'll begin by developing the necessary foundations, including subgroups, normal subgroups, and quotient groups, alongside group homomorphisms and isomorphisms. This theoretical groundwork will enable us to rigorously prove and apply the powerful Sylow theorems, which offer profound insights into the structure of finite groups. Additionally, we'll cover the fundamental theorem of finitely generated abelian groups, a key result that completely characterizes these important algebraic structures. This course provides a strong foundation in the theory and application of groups, essential for further study in algebra and other mathematical disciplines.
Extended Description: This course is a direct continuation of MATH 4300. We'll begin by reviewing the fundamentals of field theory where the previous course concluded, including concepts like field extensions, algebraic and transcendental elements, and splitting fields. We'll then delve directly into the powerful concepts of Galois theory, exploring the fundamental theorem of Galois theory and its applications to the solvability of polynomials by radicals. Following this, our next major focus will be module theory, understanding modules as generalizations of vector spaces and abelian groups. If time allows, we'll explore additional topics of interest, such as an introduction to basic algebraic number theory or Wedderburn's Theorem. For the core content, we'll roughly follow the outline of Chapters 10-14 of Dummit and Foote. This course is designed for students building on their foundational knowledge in abstract algebra.
Extended Description: This course provides a comprehensive and rigorous treatment of essential mathematical methods widely used in physics and engineering. The curriculum begins with a thorough exploration of linear operators on vector spaces and matrix theory, building to the fundamental concept of Hilbert spaces, crucial for quantum mechanics and functional analysis. The course then transitions to the study of functions of a complex variable, emphasizing applications relevant to physics, including contour integration and the powerful calculus of residues. Finally, we delve into ordinary and partial differential equations (ODEs and PDEs) of mathematical physics, including detailed analysis of boundary value problems and the properties of special functions that frequently arise as solutions in diverse physical contexts.
Extended Description: This course offers a foundational introduction to the theory and practice of linear optimization, a fundamental challenge across science, engineering, and humanities. The initial half focuses on the theoretical understanding of linear programming, developed alongside a detailed exploration of the classic Simplex Method. We'll cover modeling real-world scenarios, feasibility and boundedness, and both basic and two-phase Simplex versions. Key theoretical aspects like duality, complementary slackness, the Dual Simplex Method, and sensitivity analysis will be central. The second half explores advanced methods like the Ellipsoid Method and Interior Point Methods, dedicating a significant portion to applications in combinatorial optimization, particularly network problems (e.g., shortest path, maximum flow). We'll cover as many advanced topics as time permits, such as integer programming, total unimodularity, cutting planes, and the branch-and-bound algorithm. Prerequisites include linear algebra and a computer programming course.
Extended Description: This course offers a comprehensive introduction to the theory and practice of nonlinear optimization, focusing on the most general types of continuous optimization problems prevalent in engineering. We'll begin by mastering the art of translating intricate real-world scenarios into precise mathematical optimization models. The course dives deep into unconstrained optimization, introducing crucial concepts like convexity and various methods for solving one-dimensional problems (Newton’s method, bisection, Golden Section). Building on this, we'll advance to n-dimensional unconstrained optimization, exploring methods leveraging first and second derivatives, and even derivative-free approaches, alongside line search and trust regions. The curriculum then transitions to constrained optimization, with significant emphasis on the indispensable Karush-Kuhn-Tucker (KKT) conditions. Finally, you'll learn powerful methods for constrained problems, including Sequential Quadratic Programming (SQP) and various barrier and penalty function methods. Prerequisites include multivariable calculus, linear algebra, and a computer programming course.
Extended Description: This course offers a rigorous introduction to the theory of stochastic processes, with a strong emphasis on their applications to financial economics. We will develop core mathematical foundations, beginning with Brownian motion for modeling asset prices. Key concepts such as martingales and Itô's Lemma will be introduced, leading to a deep understanding of stochastic integration. The course will also cover practical computational techniques, including Monte Carlo simulations and variance reduction methods, crucial for efficient valuation. Through these tools, we will explore significant applications in financial markets, such as discrete-time option pricing and delta hedging strategies. This course is designed for students seeking a solid quantitative foundation in modern financial modeling.
Extended Description: This course explores the application of statistical methods to the evaluation and selection of actuarial models, essential for quantifying financial risks in insurance. Students will gain a deep understanding of severity, frequency, and aggregate models, along with various measures of risk. The course introduces Bayesian analysis as applied to credibility theory, providing robust tools for updating risk assessments. Finally, we'll examine modern computational approaches, including simulation and bootstrap methods, for model validation, parameter estimation, and comprehensive analysis, ensuring practical skills for actuarial practice.
Extended Description: This course provides an introduction to combinatorics, often described as the art of counting. Beyond its inherent appeal as an area of pure mathematics, combinatorics finds practical applications in the analysis of algorithms within computer science and in foundational probability theory. The techniques explored, particularly generating functions, are also valuable tools in other mathematical disciplines, such as statistics. There are surprising links to mathematical analysis (through formal power series) and algebra (in counting objects with symmetries using group actions), although no particular background in these areas is assumed. This course will explore five fundamental counting techniques: permutations and combinations, generating functions, recurrence relations, the principle of inclusion-exclusion, and the application of group theory to enumerate objects with symmetry using Polya's Theorem. This course is suitable for graduate students and upper-level undergraduates.
Extended Description: A graph is a collection of points (vertices) connected by lines (edges)—a versatile framework for understanding networks. Graphs appear in diverse fields from electrical engineering to computer science, offering broad practical applications. In this course, we'll delve into various properties of graphs. We'll explore their resilience to disconnection, analyzing their robustness. We'll investigate conditions for Hamiltonian cycles (closed paths visiting every vertex exactly once), famously linked to the Traveling Salesman Problem. Additionally, we'll examine graph coloring, determining the minimum number of colors needed for vertices such that no two adjacent vertices share the same color—a generalization of the Four Color Theorem. This course is designed for upper-level undergraduate and graduate students in mathematics and engineering.