6630    Mathematical Analysis of Computer Algorithms  Discrete algorithms (number-theoretic, graph-theoretic, combinatorial, and algebraic) with an emphasis on techniques for their mathematical analysis. Topics include proofs, induction, the metric structure of the reals, the Bolzano-Weierstrass theorem, and the diagonalization theorem. We appreciate your financial support. 8800   Directed Reading  A directed reading and/or project at the doctoral level. 8260    Differential Geometry II  Riemannian geometry:  connections, curvature, first and second variation; geometry of submanifolds. Online Graduate Math Courses Overview. The Department offers the following wide range of graduate courses in most of the main areas of mathematics. 6780    Mathematical Biology  Mathematical models in the biological sciences: compartmental flow models, dynamic system models, discrete and continuous models, deterministic and stochastic models. 8150    Complex Variables I  The Cauchy-Riemann Equations, linear fractional transformations and elementary conformal mappings, Cauchy's theorems and its consequences including: Morera's theorem, Taylor and Laurent expansions, maximum principle, residue theorem, argument principle, residue theorem, argument principle, Rouche's theorem and Liouville's theorem. 8080    Lie Algebras  Nilpotent and solvable Lie algebras, structure and classification of semisimple Lie algebras, roots, weights, finite-dimensional representations. The following is the standard list from which grad courses are normally chosen: Basic Courses - required for the Ph.D. (offered every year): David Rittenhouse Lab.209 South 33rd StreetPhiladelphia, PA 19104-6395Email: math@math.upenn.eduPhone: (215) 898-8178 & 898-8627Fax: (215) 573-4063, © 2020 The Trustees of the University of Pennsylvania, Description of undergraduate math courses, Math 524/525 - Topics in Modern Applied Algebra, Math 540/541 - Selections from Classical and Functional Analysis, Math 560/561 - Selections from Geometry and Topology, Math 570/571(Phil506) - Introduction to Logic and Computability, Math 574/575 - Mathematical Theory of Computation, Math 580/581 - Combinatorial Analysis and Graph Theory, Math 582/583 - Applied Mathematics and Computation, Math 584/585 - The Mathematics of Medical Imaging and Measurement, Math 590/591 - Advanced Applied Mathematics, Math 594(Phys500) - Advanced Methods in Applied Mathematics, Math 504/505 - Graduate Proseminar in Mathematics, Math 600/601 - Topology and Geometric Analysis, Math 618 - Algebraic Topology, first semester (fall), Math 619 - Algebraic Topology, second semester (spring), Math 622/623 - Complex Algebraic Geometry, Math 638/639 - Algebraic Topology, Part II, Math 640/641 - Ordinary Differential Equations, Math 644/645 - Partial Differential Equations, Math 656/657 - Representation of Continuous Groups, Math 676 - Advanced Geometric Methods in Computer Science, Math 690/691 - Topics in Mathematical Foundations of Program Semantics, Math 694/695(Phys654/655) - Mathematical Foundations of Theoretical Physics, Math 724/725 - Topics in Algebraic Geometry, Math 730/731 - Topics in Algebraic and Differential Topology, Math 748/749 - Topics in Classical Analysis, Math 750/751 - Topics in Functional Analysis, Math 760/761 - Topics in Differential Geometry, Math 824/825 - Seminar in Algebra, Algebraic Geometry, Number Theory, Math 844/845 - Seminar in Partial Differential Equations, Math 850/851 - Seminar in Functional Analysis, Math 860/861 - Seminar in Reimannian Geometry, Math 872/873 - Seminar in Logic and Computation. 8400    Algebraic/Analytic Number Theory I  The core material of algebraic number theory: number fields, rings of integers, discriminants, ideal class groups, Dirichlet's unit theorem, splitting of primes; p-adic fields, Hensel's lemma, adeles and ideles, the strong approximation theorem. This is MATH 2410H for graduate students in Mathematics Education. 6670    Combinatorics  Basic counting principles: permutations, combinations, probability, occupancy problems, and binomial coefficients. 8250    Differential Geometry I  Differentiable manifolds, vector bundles, tensors, flows, and Frobenius' theorem. 8620    Stochastic Processes  Conditional expectation, Markov processes, martingales and convergence theorems, stationary processes, introduction to stochastic integration. 8430    Topics in Arithmetic Geometry  Topics in Algebraic number theory and Arithmetic geometry, such as class field theory, Iwasawa theory, elliptic curves, complex multiplication, cohomology theories, Arakelov theory, diophantine geometry, automorphic forms, L-functions, representation theory. Topics include the Cauchy integral formula, power series and Laurent series, and the residue theorem. 6700    Qualitative Ordinary Differential Equations  Transform methods, linear and nonlinear systems of ordinary differential equations, stability, and chaos. These courses generally carry three hours of credit per semester. 6720    Introduction to Partial Differential Equations  The basic partial differential equations of mathematical physics: Laplace's equation, the wave equation, and the heat equation. 7050   Basic Ideas of Calculus II  A continuation of Basic Ideas of Calculus I focusing on functions of several variables. 8550    Special Topics in Numerical Analysis  Special topics in numerical analysis, including iterative methods for large linear systems, computer aided geometric design, multivariate splines, numerical solutions for pde's, numerical quadrature and cubature, numerical optimization, wavelet analysis for numerical imaging. 8630    Stochastic Analysis  Conditional expectation, Brownian motion, semimartingales, stochastic calculus, stochastic differential equations, stochastic control, stochastic filtering. 8110    Real Analysis II  Topics including:  Haar Integral, change of variable formula, Hahn-Banach theorem for Hilbert spaces, Banach spaces and Fourier theory (series, transform, Gelfand-Fourier homomorphism). Must be a high school graduate or have completed GED, Undergraduate applicants must be a high school graduate or have completed GED and completed some college. 6200    Point Set Topology  Topological spaces, continuity; connectedness, compactness; separation axioms and Tietze extension theorem; function spaces.