## Math 610: Algebra I

Introduction: This course is the first of two courses that together provide a foundation in abstract algebra for Ph.D. Students. We cover in this first half the theory of groups, rings, and modules, and introduce the language of category theory. This material is absolutely essential for any student of graduate-level mathematics.

Topics: Group Theory: Group Actions, Sylow's theorems, Free Groups and Relations, Composition Series, Vector Spaces, Bilinear Forms, GLn, classification of symmetric and antisymmetric bilinear forms. Modules and Linear Algebra: Free Modules, Algebras, Tensor Product, Multilinear Algebra, Structure Theory of modules over PID, Jordan Canonical forms, Rational Canonical Forms.

References:

1. Dummit & Foote: Abstract Algebra.
2. Hungerford: Algebra.
3. Herstein: Abstract Algebra.
4. Artin: Algebra.
5. Lang: Algebra.
6. Bourbaki: Algebra.
7. Alperin & Bell: Groups and Representations.

## Math 611: Analysis I

Introduction: This course is the first of two courses that together form Analysis for PhD students. The course material is absolutely essential for a student wanting to specialize in any subject that needs Real and/or Functional Analysis.

Topics: Integration w.r.t. abstract measures, Convergence of Integrals, signed measures, Radon-Nikodym Theorem, product measures, Fubini's theorem and Tonelli's theorem, Properties of bounded linear operators on normed linear spaces, Hahn-Banach theorems, Banach spaces, Lp spaces, Approximation in Lp, Riesz representation theorem for Lp spaces on finite measure spaces and continuous functions on compact spaces, open mapping theorem, closed graph theorem, Topological vector spaces, weak and weak* topologies, Banach Alaoglu theorem, spectral theorem for bounded linear operators, Hilbert spaces, complete orthonormal system.

References:

1. K.B. Athreya and S.N Lahiri: Measure Theory and Probability Theory.
2. J.B. Conway: A Course in Functional Analysis.
3. G.B. Folland: Real Analysis.
4. C. Goffman and G. Pedrick: First course in Functional Analysis.
5. E. Kreyszig: Introductory Functional Analysis with Applications.
6. B.V. Limaye: Functional Analysis.
7. H.L. Royden: Real Analysis.
8. W. Rudin: Functional Analysis.
9. W. Rudin: Real and Complex Analysis.
10. E.M. Stein and R. Shakarchi: Real Analysis.

Prerequisites: Fundamental concepts of analysis such as metric spaces, uniform continuity, differentiation, Riemann integral and uniform convergence as for example in Chapters 1 to 7 of Rudin's book "Principles of Mathematical Analysis" should be a sufficient preparation.

## Math 612: Topology I

Introduction: This course is the first of two courses that together forms an introduction to Algebraic Topology for PhD students. The course material is absolutely essential for a student wanting to specialize in any subject that needs Topology and/or Geometry. In these courses, for the underlying spaces, the emphasis will be on smooth manifolds. The functorial nature of the subject will be stressed throughout.

Topics: Manifolds, Tangent Spaces, Lie bracket, Lie Groups, Sard's Theorem, degree map, Vector Bundles/Connections, Curvature/geodesics, Homotopy Theory, Covering Spaces, Seifel Van Kampen Theorem, Group Actions, proper discontinuous, Higher homotopy groups: Definition & Hopf fibration, Applications: Brauer Fixed Point Thm, Borsuk-Ulam

References:

1. Guilleman & Pollack: Differential Topology.
2. Milnor's Diff Theory: Topology from a Differential Viewpoint.
3. Massey: Basic Course in Algebraic Topology.
4. A. Hatcher: Algebraic Topology.
5. Hicks: Notes on Differential Geometry.
6. Lee (several books)
7. Milnor/Stasheff: Characteristic Classes (first three chapters).
8. Madsen/Tornehave: From Calculus to Cohomology.
9. Bott/Tu.
10. Warner.
11. G. Bredon: Geometry and Topology.
12. J. Vick: Homology Theory.
13. J. Munkres: Elements of Algebraic Topology.

Prerequisites: The basics of point set topology and differentiable manifolds as for example in Chapters 1 and 2 of Bredon's book should be a sufficient preparation.

## Math 613: Discrete Mathematics I

Introduction: This course is a graduate-level introduction to the fundamental ideas and results of discrete mathematics. This course is essential for students wanting to specialize in combinatorics, graph theory, and algorithms. The course is offered in two parts, namely Discrete Mathematics I & II. Discrete Mathematics I covers set theory, combinatorics, and part of graph theory.

Topics: Set Systems: Representing Sets, Hall's Theorem, Sperner systems, Intersecting set systems, Helly families, Relations, Partially ordered sets, Dilworth's Theorem. Combinatorics: Permutation and Combinations: Permutation of multi sets, Combination of multi sets, The Pigeonhole Principle (strong form), Theory of Ramsey, Generating permutation and combinations, The Inclusion and Exclusion Principles and Applications, Mobius Inversion, Recurrence Relations and (exponential) Generating Functions, Polya Counting. Graph Theory:Eulerian Cycle, Hamiltonian Graph, Matching and Covers, Maximum Bipartite Matching Algorithms, Matching in General Graphs, Tutte's 1-factor Theorem.

References:

1. B. Bollobas: Combinatorics.
2. R. Brualdi: Introductory Combinatorics.
3. D.B. West: Introduction to Graph Theory.
4. F. Harary: Graph Theory.

Prerequisites: Nil

## Math 614: Differential Equations I

Introduction: This course is the first of two courses that together forms an introduction to Differential Equations for PhD students. The course material is an essential core for pursuing several branches of both pure and applied mathematics, such as optimization problems, mathematical physics, dynamical systems, financial mathematics, mathematical biology.

Topics: Existence and uniqueness of initial value problems. Linear equations. Boundary value problems and Sturm-Liouville theory.Asymptotic behavior of nonlinear systems.Perturbation theory and Poincare-Bendixson theorems. Numerical methods. Introductory bifurcation theory.

References:

1. G. Teschl: Ordinary Differential Equations and Dynamical Systems.
2. Coddington & Levinson: Theory of Ordinary Differential Equations.
3. Hirsch, Smale and Devaney: Differential Equations, Dynamical Systems, and an Introduction to Chaos.
4. V.I. Arnold: Geometrical Methods in the Theory of Ordinary Differential Equations.
5. C. Chicone: Ordinary Differential Equations.
6. J.D. Meiss: Differential Dynamical Systems.

Prerequisites: Familiarity with analysis, multivariable calculus and linear algebra.

## MTH 620: Algebra II

Introduction: This course is the second of two courses that together provide a foundation in abstract algebra for PhD Students. It will cover the theory of abstract linear algebra and fields, as well as additional topics such as advanced ring theory and homological algebra, according to time and the instructor's preference. This material is ideal for a student wanting to specializing in algebra or any subject with a strong algebraic flavor.

Topics: Field Theory: Separable and Normal extensions, Finite Fields, Algebraic and Transcendental extensions, Galois' theorem, Infinite Galois Groups, Hilbert 90. Commutative Algebra: Localization, DVR, Noetherian, Hilbert Basis Theorem, Dedekind domains, Integral extensions, Hilbert's Nullstellensatz, spec of a ring. Homological Algebra: Projective and Injective Modules, Ext and Tor functors, connecting homomorphism.

References:

1. All the references mentioned in MTH 610; and
2. Atiyah & MacDonald: Introduction to Commutative Algebra.
3. Bourbaki: Commutative Algebra.
4. Weibel: Introduction to Homological Algebra.

Prerequisites: Math 610.

## MTH 621: Analysis II

Introduction: This course is the second of two courses that together form Analysis for PhD students. The course material is absolutely essential for a student wanting to specialize in any subject that needs Complex Analysis.

Topics: Compact Operators, Spectral theorem for compact self-adjoint operators, Fourier transforms and Plancherel theorem, Cauchy Riemann Equations, Analytic functions, Exponential Map, Trigonometric maps, Cauchy's Theorem, Cauchy's Integral Formula, Calculus of Residue, Maximum Modulus Principle, Conformal Mappings, Linear fractional transformation, Riemann mapping theorem, Zeroes of a holomorphic function, Weirstrass' factorization, Analytic continuation, Sheaves and Riemann surfaces.

References:

1. L.V. Ahlfors: Complex Analysis.
2. J.B. Conway: Functions of one Complex Variable.
3. T.W. Gamelin: Complex Analysis.
4. R.Greene and S.G. Krantz: Function Theory of one Complex Variable.
5. E. Kreyszig: Introductory Functional Analysis with Applications.
6. B.V. Limaye: Functional Analysis.
7. H.L. Royden: Real Analysis.
8. W.Rudin: Real and Complex Analysis.
9. E.M. Stein and R.Shakarchi: Complex Analysis.

Prerequisites: MTH 611

## MTH 622: Topology II

Introduction: This course is the second of two courses that together form an introduction to Algebraic Topology for PhD students. A student who has mastered these two courses should be able to compute fundamental groups, homology and cohomology groups of several standard examples like spheres, euclidean spaces, projective spaces, some Lie groups, and should be totally at ease with homological machinery. Such a student will be well-prepared to study other cohomology theories or higher homotopy theory.

Topics: Simplicial/Singular Homology, CW-complexes/cellular homology, Eilenberg-Steenrod Axioms: statement and uniqueness, Differential forms and Stokes' Theorem, DeRham Cohomology, DeRham Theorem, Ext and Tor, Universal Coefficient Theorem, Kunneth formula, Cup and cap products, and Poincare Duality, Applications: Jordan Curve Theorem, Lefschetz FPT, multiplication structures on R^n.

References:

1. All the references mentioned in MTH 612; and
2. S. Maclane: Homology.
3. Greenberg/Harper.
4. J. Vick: Homology Theory.
5. C. Weibel: Introduction to Homological Algebra.

Prerequisites: MTH 612

## MTH 623: Discrete Mathematics II

Introduction: Discrete Mathematics II covers algorithms and further graph theory. Discrete Mathematics I & II together provide a thorough introduction to Combinatorics, graph theory and algorithms at the graduate level.

Topics: Graph Theory:Cuts and Connectivity: 2-connected Graphs, Menger's Theorem. Planar Graphs: drawing, Euler's Formula, Kuratowski's theorem, plane duality; Coloring: coloring maps and planar graphs, coloring vertices, coloring edges. Algorithms: Asymptotic order of growth: big O notation and its relatives,Divide and Conquer and Recurrences: The master theorem, application to the complexity of recursive algorithms.Data Structures: Priority queues, heaps, queues, stacks, Union-Find. Basic Algorithms: breadth first search, depth first search, DAGs (directed acyclic graphs) and topological ordering, strongly connected components. Greedy Algorithms: interval scheduling, Dijkstra's algorithm for finding shortest paths in a graph, minimum spanning trees, Huffman codes for data compression. Dynamic Programming: weighted interval scheduling, subset sums and knapsacks. Network Flow: Max-Flow Min-Cut and the Ford-Fulkerson algorithm.

References: 1. J. Kleinberg and E. Tardos: Algorithm Design. 2. H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein: Introduction to Algorithms. 3. D.B. West: Introduction to Graph Theory. 4. F. Harary: Graph Theory.

Prerequisites: MTH 613

## MTH 624: Differential Equations II

Introduction: This course is the second of two courses that together forms an introduction to Differential Equations for PhD students, covering Partial Differential Equations. The course material is an essential core for pursuing several branches of both pure and applied mathematics, such as optimization problems, mathematical physics, dynamical systems, financial mathematics, and mathematical biology.

Topics: First-order equations, methods of characteristics, conservation laws, weak solutions, wave equations, the heat equation, the fundamental solution, diffusion and Brownian motion; Laplace's equation, maximum principle, fundamental solutions, Dirichlet and Neumann problems; Fourier methods.

References: 1. L. Evans: Partial differential equations. 2. G. Folland: Introduction to Partial Differential Equations. 3. F. John: Partial Differential Equations. 4. M. Renardy and R. Rogers: An Introduction to Partial Differential Equations.

Pre-requisites: MTH 614