126 courses with the subject MATH, each shown exactly as we captured it from the college's catalog, with every element we hold. Where the wording looks broken, that is our reading of the catalog, not the college's text.
MATH 0021First-Year Seminar: Math and Movies
Introduction to the creative processes through which screenwriters and mathematicians conceive and develop ideas, craft dramatic arguments and proofs, and solve narrative and mathematical problems. Using films featuring math and mathematician-based storylines for reference, we’ll develop a common language for evaluating the accuracy, success, and significance of dramatic and mathematical arguments. This is not a course in advanced math, nor is it about the nuts and bolts mechanics of screenwriting. They will learn with an instructor with experience selling screenplays to Hollywood about what the screenplay is, what it aims to accomplish, and how screenwriters approach crafting a narrative. 1 Course Unit
This class is about incidence geometry and projective geometry: how lines and planes intersect, and how to add points to euclidean spaces (the familiar R^2, R^3, etc.) to obtain a space that include points and line at infinity. It's also about art, and how classical artists used projective geometry, sometimes without knowling they were doing so, to create perspective drawings that render three-dimensional space on a canvas in a way our eyes intuitively understand. The projective geometry content will be at times pretty mathy. We use axiom systems, figure out what's true about them, prove theorems, and construct abstract spaces that are models for the axioms. The applications to art will be very hands-on. Expect to sketch a lot, to draw lines on existing pieces of art or sample drawings, and to look at physical objects and attempt to capture them in perspective drawings. If you think of yourself as a bad artist (as I do) it shouldn't matter: we're all going to take a major step forward in one technical aspect of art, namely how to get the lines right in perspective drawings. If you have no real math background, or are even a bit math- phobic, that shouldn't matter either. No math background beyond algebra and trigonometry is necessary. All that is required is willingness to try your hand at logic, to learn the structure of mathematical argument, and to harness your geometric intuition. 1 Course Unit
PHYS 091 (1 c.u.), PHYS 050 (.5 c.u.), PHYS 092 (1 c.u.), and PHYS 051 (.5 c.u.)** PHYS 091 (1 c.u.) and PHYS 050 (.5 c.u.) “Free” indicates that students PHYS 092 (1 cu) and PHYS 051 (.5 cu) receive credit without a specific PHYS 093 (1 c.u.) and PHYS 050 (.5 c.u.)** course listed on their transcript. The subjects and scores listed here PHYS 094 (1 c.u.) and PHYS 051 (.5 c.u.)** receive the equivalencies indicated. Waiver for PSYC 001 (no credit) These policies are in effect for students entering Penn during the
Multivariate calculus; optimization; multivariate probability densities. Introduction to linear algebra; introduction to differential equations. Mathematical modeling and applications to the social, economic and information sciences. of 14+ before taking MATH 1080. 1 Course Unit
Introduction to concepts and methods of calculus for students with little or no previous calculus experience. Polynomial and elementary transcendental functions and their applications, derivatives, extremum problems, curve-sketching, approximations; integrals and the s fundamental theorem of calculus.
Math 1610 is an intensive, proof based introduction to single- variable and multivariable calculus designed for students with strong mathematical ability and interest. The course emphasizes conceptual understanding, rigorous reasoning, and the development of mathematical communication skills. Students explore the foundations of real and complex numbers, limits, continuity, differentiation, and integration, with all major results presented and proved. Building on these foundations, the course extends to vector algebra, vector valued functions, differential calculus of scalar and vector fields, and the theory and applications of line, surface, and multiple integrals. Typical topics include the axiomatic development of the real numbers, a proof based recap of single variable calculus, vector algebra, differential calculus of scalar and vector fields, and integrals over lines and surfaces.
Topics from among the following: logic, sets, calculus, probability, history and philosophy of mathematics, game theory, geometry, and their relevance to contemporary science and society.
This course counts as a regular elective for both the Mathematics Major and Minor. This is a course about mathematical reasoning and the media. Embedded in many stories one finds in the media are mathematical questions as well as implicit mathematical models for how the world behaves. We will discuss ways to recognize such questions and models, and how to think about them from a mathematical perspective. A key part of the course will be about what constitutes a mathematical proof, and what passes for proof in various media contexts. The course will cover a variety of topics in logic, probability and statistics as well as how thes subjects can be used and abused.
or ESE 2030 Linear Algebra with Applications to Engineering and AI or CIS 5150 Fundamentals of Linear Algebra and Optimization One in Introduction to Proofs:
MATH 2300Introduction to Ordinary and Partial Differential Equations
This course introduces students to both ordinary and partial differential equations (ODEs and PDEs) with applications in physics, engineering, and applied sciences. Topics include first- and second-order linear ODEs, equations with constant coefficients, the Cauchy-Euler equation, boundary-value problems, the heat and wave equations, separation of variables, Fourier series, Sturm-Liouville theory, eigenfunction expansions, and solutions to PDEs in higher dimensions. Students will also study Legendre and Bessel equations and associated functions, and learn techniques for fitting initial and boundary conditions. Fall, Spring, and Summer Terms Mutually Exclusive: ENM 2510 or have a placement score of 24+ before taking 1 Course Unit
Students must take (MATH 2200, ESE 2030, or ENM 2400)
MATH 2400Calculus, Part III
Linear algebra: vectors, matrices, systems of linear equations, vector spaces, subspaces, spans, bases, and dimension, eigenvalues, and eigenvectors, maxtrix exponentials. Ordinary differential equations: higher-order homogeneous and inhomogeneous ODEs and linear systems of ODEs, phase plane analysis, non-linear systems.
MATH 2900Undergraduate Mathematics Research Course
This is a project-oriented mathematics research course that teaches students tosolve real-world problems by constructing and analyzing mathematical models. Typically the problems considered will come from mathematics, chemistry, biology, and materials science but sometimes they will also come from economics, finance, and social sciences. The research problems in the course vary from year to year.1 Course Unit
MATH 3200Computer Methods in Mathematical Science I
Students will use symbolic manipulation software and write programs to solve problems in numerical quadrature, equation-solving, linear algebra and differential equations. Theoretical and computational aspects of the methods will be discussed along with error analysis and a critical comparison of methods.
Topics will be drawn from some subjects in combinatorial analysis with applications to many other branches of math and science: graphs and networks, generating functions, permutations, posets, asymptotics. Not Offered Every Year Also Offered As: LGIC 2100 1 Course Unit
Topics will be drawn from some subjects useful in the analysis of information and computation: logic, set theory, theory of computation, number theory, probability, and basic cryptography. Also Offered As: LGIC 2200 1 Course Unit
Complex numbers, DeMoivre's theorem, complex valued functions of a complex variable, the derivative, analytic functions, the Cauchy-Riemann equations, complex integration, Cauchy's integral theorem, residues, computation of definite integrals by residues, and elementary conformal mapping.
After a rapid review of the basic techniques for solving equations, the course will discuss one or more of the following topics: stability of linear and nonlinear systems, boundary value problems and orthogonal functions, numerical techniques, Laplace transform methods.
Method of separation of variables will be applied to solve the wave, heat, and Laplace equations. In addition, one or more of the following topics will be covered: qualitative properties of solutions of various equations (characteristics, maximum principles, uniqueness theorems), Laplace and Fourier transform methods, and approximation techniques.
A mathematical approach to game theory, with an emphasis on examples of actual games. Topics will include mathematical models of games, combinatorial games, two person (zero sum and general sum) games, non-cooperating games and equilibria.
Point set topology: metric spaces and topological spaces, compactness, connectedness, continuity, extension theorems, separation axioms, quotient spaces, topologies on function spaces, Tychonoff theorem. Fundamental groups and covering spaces, and related topics. Not Offered Every Year 1 Course Unit
Differential geometry of curves in the plane and in 3-space;n gauge theories Surfaces in 3-space; The geometry of the Gauss map;ons. The language of Intrinsic geometry of surfaces; Geodesics; Moving frames; of vector bundles, The Gauss-Bonnet Theorem; Assorted additional topics. Not Offered Every Year Mutually Exclusive: MATH 5010 MATH 5140) 1 Course Unit
Point set topology: metric spaces and topological spaces, compactness, connectedness, continuity, extension theorems, separation axioms, quotient spaces, topologies on function spaces, Tychonoff theorem. Fundamental groups and covering spaces, and related topics. Not Offered Every Year 1 Course Unit
The course moves from a study of extrinsic geometry (curves and surfaces in n-space) to the intrinsic geometry of manifolds. After a revie of vector calculus and a section on tensor algebra, we study manifolds and their intrinsic geometry, including metrics, connections, geodesics, and the Riemann curvature tensor. Topics include Eulerian curvature and Euler's theorems, the Gauss map and first/second fundamental forms, the Theorema Egregium, minimal surfaces in n-space; other topics as time permits. Not Offered Every Year Mutually Exclusive: MATH 4650 MATH 5140) 1 Course Unit
An introduction to groups, rings, fields and other abstract algebraic systems, elementary Galois Theory, and linear algebra -- a more theoretical course than Math 3700.
This course focuses on problems from Algebra (especially linear algebra and multilinear algebra) and Analysis (especially multivariable calculus through vector fields, multiple integrals and Stokes theorem). The material is presented through student solving of problems. In addition there will be a selection of advanced topics which will be accessible via this material.
This course focuses on problems from Algebra (especially linear algebra and multilinear algebra) and Analysis (especially multivariable calculus through vector fields, multiple integrals and Stokes theorem). The material is presented through student solving of problems. In addition there will be a selection of advanced topics which will be accessible via this material.
Continuation of Math 5080. The Arzela-Ascoli theorem. Introduction to the topology of metric spaces with an emphasis on higher dimensional Euclidean spaces. The contraction mapping principle. Inverse and implicit function theorems. Rigorous treatment of higher dimensional differential calculus. Introduction to Fourier analysis and asymptotic methods.
A number of important and interesting problems in a wide range of disciplines within computer science are solved by recourse to techniques from linear algebra. The goal of this course will be to introduce students to some of the most important and widely used algorithms in matrix computation and to illustrate how they are actually used in various settings. Motivating applications will include: the solution of systems of linear equations, applications matrix computations to modeling geometric transformations in graphics, applications of the Discrete Fourier Transform and related techniques in digital signal processing, the solution of linear least squares optimization problems and the analysis of systems of linear differential equations. The course will cover the theoretical underpinnings of these problems and the numerical algorithms that are used to perform important matrixcomputations such as Gaussian Elimination, LU Decomposition and Singular Value Decomposition. Mutually Exclusive: MATH 3130 1 Course Unit
Topics will include: Vector spaces, Basis and dimension, quotients; Linear maps and matrices; Determinants, Dual spaces and maps; Invariant subspaces, Cononical forms; Scalar products; Euclidean, unitary and symplectic spaces; Orthogonal and Unitary operators; Tensor products and polylinear maps; Symmetric and skew-symmetric tensors and exterior algebra. Also Offered As: AMCS 5141 Mutually Exclusive: MATH 3140 1 Course Unit 2026-27 Catalog | Generated 08/03/26
MATH 5400Selections from Classical and Functional Analysis
Informal introduction to such subjects as compact operators and Fredholm theory, Banach algebras, harmonic analysis, differential equations, nonlinear functional analysis, and Riemann surfaces. Not Offered Every Year 1 Course Unit
The required background is (1) enough math background to understand proof techniques in real analysis (closed sets, uniform covergence, fouri series, etc.) and (2) some exposure to probability theory at an intuitive level (a course at the level of Ross's probability text or some exposure probability in a statistics class).
Informal introduction to such subjects as homology and homotopy theory, classical differential geometry, dynamical systems, and knot theory. Not Offered Every Year 1 Course Unit
Informal introduction to such subjects as homology and homotopy theory, classical differential geometry, dynamical systems, and knot theory. Not Offered Every Year 1 Course Unit
The course focuses on topics drawn from the central areas of mathematical logic: model theory, proof theory, set theory, and computability theory. Not Offered Every Year Also Offered As: PHIL 6721 1 Course Unit
Standard tools of enumerative combinatorics including partitions and compositions of integers, set partitions, generating functions, permutations with restricted positions, inclusion-exclusion, partially ordered sets. Permission of the instructor required to enroll. Not Offered Every Year 1 Course Unit
Variable topics connected to current research in combinatorial theory. Recent topics include algebraic combinatorics and symmetric functions, analytic combinatorics and discrete probability. Not Offered Every Year 1 Course Unit
MATH 5840The Mathematics of Medical Imaging and Measurement
The last several decades have seen major revolutions in both medical and non-medical and imaging technologies. Underlying all of these advances are sophisticated mathematical tools to model the measurement process and reconstruct images. This course begins with an introduction of the mathematical models and then proceeds to discuss the integral transforms that underlie these models: the Fourier transform, the Radon transform and the Laplace transform. We discuss how each of these er transforms is inverted, both in theory and in practice. Along the way we study interpolation, sampling, approximation theory, filtering and noise to analysis. This course assumes a thorough knowledge of linear algebra and a knowledge of analysis at the undergraduate level (MATH 3140 and MATH 3600 and MATH 3610, or MATH 5080 and MATH 5090). Not Offered Every Year Also Offered As: AMCS 5840, BE 5840 (MATH 3610 OR MATH 5090) 1 Course Unit
This course will cover various mathematical models and tools that are used to study modern biological problems. Mathematical models may be drawn from cell biology, physiology, population genetics, or ecology. Tools in dynamical systems or stochastic processes will be introduced as necessary. No prior knowledge of biology is needed to take this course, but some familiarity with differential equations and probability will be assumed.
A discussion of those concepts and techniques of classical analysis employed inphysical theories. Topics include complex analysis. Fourier series and transforms, ordinary and partial equations, Hilbert spaces, among others.
Differentiable functions, inverse and implicit function theorems. Theory of manifolds: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields and differential forms: Frobenius' theorem, integration on manifolds, Stokes' theorem in n dimensions, de Rham cohomology. Introduction to Lie groups and Lie group actions.
Covering spaces and fundamental groups, van Kampen's theorem and classification of surfaces. Basics of homology and cohomology, singular and cellular; isomorphism with de Rham cohomology. Brouwer fixed point theorem, CW complexes, cup and cap products, Poincare duality, Kunneth and universal coefficient theorems, Alexander duality, Lefschetz fixed point theorem.
Group theory: permutation groups, symmetry groups, linear algebraic groups, Jordan-Holder and Sylow theorems, finite abelian groups, solvable and nilpotent groups, p-groups, group extensions. Ring theory: Prime and maximal ideals, localization, Hilbert basis theorem, integral extensions, Dedekind domains, primary decomposition, rings associated to affine varieties, semisimple rings, Wedderburn's theorem, elementary representation theory. Linear algebra: Diagonalization and canonical form of matrices, elementary representation theory, bilinear forms, quotient spaces, dual spaces, tensor products, exact sequences, exterior and symmetric algebras. Module theory: Tensor products, flat and projective modules, introduction to homological algebra, Nakayama's Lemma. Field theory: separable and normal extensions, cyclic extensions, fundamental theorem of Galois theory, solvability of equations.
This is a seminar for first year Mathematics graduate students, supervised by faculty. Students give talks on topics from all areas of mathematics at a level appropriate for first year graduate students. Attendance and preparation will be expected by all participants, and learning how to present mathematics effectively is an important part of the seminar.
Convexity and the Hahn Banach Theorem. Hilbert Spaces, Banach Spaces, and examples: Sobolev spaces, Holder spaces. The uniform bounded principle, Baire category theorem, bounded operators, open mapping theorem, closed graph theorem and applications. The concepts of duality and dual spaces. The Riesz theory of compact operators and Fredholm theory. Functional calculus and elementary Spectral Theory. Interpolation theorems. Applications to partial differential equations and approximation theory. 1 Course Unit
Dedekind domains, local fields, basic ramification theory, product formula, Dirichlet unit theory, finiteness of class numbers, Hensel's Lemma, quadratic and cyclotomic fields, quadratic reciprocity, abelian extensions, zeta and L-functions, functional equations, introduction to local and global class field theory. Other topics may include: Diophantine equations, continued fractions, approximation of irrational numbers by rationals, Poisson summation, Hasse principle for binary quadratic forms, modular functions and forms, theta functions. Not Offered Every Year 1 Course Unit
Algebraic geometry over the complex numbers, using ideas from topology, complex variable theory, and differential geometry. Topics include: Complex algebraic varieties, cohomology theories, line bundles, vanishing theorems, Riemann surfaces, Abel's theorem, linear systems, complex tori and abelian varieties, Jacobian varieties, currents, algebraic surfaces, adjunction formula, rational surfaces, residues. Not Offered Every Year 1 Course Unit
Algebraic geometry over algebraically closed fields, using ideas from commutative algebra. Topics include: Affine and projective algebraic varieties, morphisms and rational maps, singularities and blowing up, rings of functions, algebraic curves, Riemann Roch theorem, elliptic curves, Jacobian varieties, sheaves, schemes, divisors, line bundles, cohomology of varieties, classification of surfaces. Not Offered Every Year 1 Course Unit
Topics in commutative algebra taken from the literature. Material will vary from year to year depending upon the instructor's interests. Not Offered Every Year 1 Course Unit 2026-27 Catalog | Generated 08/03/26
Complexes and exact sequences, homology, categories, derived functors (especially Ext and Tor). Homology and cohomology arising from complexes in algebra and geometry, e.g. simplicial and singular theories, Cech cohomology, de Rham cohomology, group cohomology, Hochschild cohomology. Projective resolutions, cohomological dimension, derived categories, spectral sequences. Other topics may include: Lie algebra cohomology, Galois and etale cohomology, cyclic cohomology, l-adic cohomology. Algebraic deformation theory, quantum groups, Brauer groups, descent theory. Not Offered Every Year 1 Course Unit
Fundamentals of smooth manifolds, Sard's theorem, Whitney's embedding theorem, transversality theorem, piecewise linear and topological manifolds, knot theory. The instructor may elect to cover other topics such as Morse Theory, h-cobordism theorem, characteristic classes, cobordism theories. 1 Course Unit
Problems in differential geometry, as well as those in physics and engineering, inevitable involve partial derivatives. This course will be an introduction to these problems and techniques. We will use P.D.E. as a tool. Some of the applications will be small, some large. The proof of the Hodge Theorem will be a small application. Discussion of the Yamabe problem and Ricci flow (used to prove the Poincare Conjecture) will be larger. 1 Course Unit
Subject matter varies from year to year. Some topics are: the classical theory of the wave and Laplace equations, general hyperbolic and elliptic equations, theory of equations with constant coefficients, pseudo- differential operators, and non-linear problems. Sobolev spaces and the theory of distributions will bedeveloped as needed. Not Offered Every Year 1 Course Unit
Subject matter varies from year to year. Some topics are: the classical theory of the wave and Laplace equations, general hyperbolic and elliptic equations, theory of equations with constant coefficients, pseudo- differential operators, and nonlinear problems. Sobolev spaces and the theory of distributions will be developed as needed. Not Offered Every Year 1 Course Unit
Measure theoretic foundations, laws of large numbers, large deviations, distributional limit theorems, Poisson processes, random walks, stopping times.
Connection of Lie groups with Lie algebras, Lie subgroups, exponential map. Algebraic Lie groups, compact and complex Lie groups, solvable and nilpotent groups. Other topics may include relations with symplectic geometry, the orbit method, moment map, symplectic reduction, geometric quantization, Poisson-Lie and quantum groups. Not Offered Every Year MATH 6030 1 Course Unit
Possible topics: harmonic analysis on locally compact abelian groups; almost periodic functions; direct integral decomposition theory, Types I, II and III: induced representations, representation theory of semisimple groups. Not Offered Every Year 1 Course Unit
Riemannian metrics and connections, geodesics, completeness, Hopf- Rinow theorem, sectional curvature, Ricci curvature, scalar curvature, Jacobi fields, second fundamental form and Gauss equations, manifolds of constant curvature, first and second variation formulas, Bonnet- Myers theorem, comparison theorems, Morse index theorem, Hadamard theorem, Preissmann theorem, and further topics such as sphere theorems, critical points of distance functions, the soul theorem, Gromov- Hausdorff convergence. Not Offered Every Year MATH 6030 1 Course Unit University of Pennsylvania Catalog 2287
This graduate course focuses on topics drawn from the central areas of mathematical logic: model theory, proof theory, set theory, and computability theory. Not Offered Every Year Also Offered As: PHIL 6720 1 Course Unit
An opportunity for graduate students to serve as instructors for their own course, supported by faculty and peer mentoring. The practicum integrates teaching experience with pedagogical readings, discussions of teaching practice, and peer classroom observations. 1 Course Unit
MATH 6940Mathematical Foundations of Theoretical Physics
Selected topics in mathematical physics, such as mathematical methods of classical mechanics, electrodynamics, relativity, quantum mechanics and quantum field theory. Not Offered Every Year 1 Course Unit
MATH 6950Mathematical Foundations of Theoretical Physics
Selected topics in mathematical physics, such as mathematical methods of classical mechanics, electrodynamics, relativity, quantum mechanics and quantum field theory. Not Offered Every Year 1 Course Unit
Possible topics: harmonic analysis on locally compact abelian groups; almost periodic functions; direct integral decomposition theory, Types I, II and III: induced representations, representation theory of semisimple groups. Not Offered Every Year 1 Course Unit
Topics from the literature. The specific subjects will vary from year to year. Not Offered Every Year 1 Course Unit 2026-27 Catalog | Generated 08/03/26
Ramification theory, adeles and ideles, Tate's thesis, group cohomology and Galois cohomology, class field theory in terms of ideles and cohomology, Lubin-Tate formal groups, Artin and Swan conductors, central simple algebras over local and global fields, general Hasse principles. Other topics may include the following: zero-dimensional Arakelov theory, Tate duality, introduction to arithmetic of elliptic curve local and global epsilon factors in functional equations, p-adic L- functions and Iwasawa theory, modular forms and functions and modular curves. Not Offered Every Year 1 Course Unit
Harmonic analysis in Euclidean space, Riemann surfaces, Discontinuous groups and harmonic analysis in hyperbolic space, Pseudodifferential operators and index theorems, Variational methods in non-linear PDE, Hyperbolic equations and conservation laws, Probability and stochastic processes, Geometric measure theory, Applications of analysis to problems in differential geometry. The specific subjects will vary from year to year. Not Offered Every Year 1 Course Unit
Topics from the literature. The specific subjects will vary from year to year. Not Offered Every Year 0.5,1 Course Unit 2026-27 Catalog | Generated 08/03/26
Reading of mathematical literature under the direction of a faculty member in a group of students. Hours and syllabus to be arranged with the supervising faculty member 1 Course Unit
Total Course Units 14-17 At least four courses must be taken in the Penn Math Department. Thesis: Each student must write, under the supervision of a Mathematics Department faculty member, a satisfactory M.Phil. thesis, which is typically expository in nature, but may also be a research paper. The preparation of this thesis should involve the mastery of some area of mathematics beyond the curriculum of the courses that the student has taken. (To obtain an M.Phil. degree, it is not necessary to have previously written a masters thesis. But a student in the M.Phil. program who previously submitted a masters thesis cannot resubmit that thesis toward the M.Phil.)
