# Syllabi for Basic First-year Courses

## Syllabi for Basic First-year Courses

### Algebra I, II

The examinable material for the graduate algebra candidacy exam is 1 through the first part of 3 below (up to but not including categories), though Algebra I will usually cover more than this. Topics labeled *, and perhaps additional topics not mentioned, may be covered at the discretion of the instructor.

1. Groups

Groups. Cyclic groups, permutation groups symmetric and alternating groups, matrix groups. Subgroups, quotient groups, direct products, homomorphism theorems. Automorphisms, conjugacy. Cosets, Lagrange's theorem. Group actions, G-sets, Sylow theorems, free groups and presentations.

2. Rings

Rings. Polynomial rings, matrix rings. Ideals, quotient rings, homomorphism theorems. Prime and maximal ideals. Principal ideal domains, Euclidean domains, unique factorization

domains. Localization of rings, field of fractions.

3. Modules

Finite dimensional vector spaces. Basis, dimension, linear transformations, matrices. Modules. Submodules, quotient modules, homomorphism theorems, structure of finitely generated modules over a principal ideal domain. Language of categories and functors. Direct sums and products, free, projective and injective modules. Duality, multilinear forms, determinants.

4. Canonical forms of matrices of linear transformations

Jordan, rational and primary rational canonical forms. Invariant factors and elementary divisors.

5. Fields

Fields, algebraic and transcendental field extensions, degree, transcendence degree, algebraic closure. Fundamental theorem of Galois theory, separability, normality. Finite fields. *Cyclotomic extensions, *cyclic extensions, *solvable and nilpotent groups, *Impossibility proofs (trisecting angles, etc), *solvability of polynomial equations by radicals

6. Tensor products

Tensor products, *algebras, *(the tensor, symmetric and exterior algebras)

7. Chain conditions

Artinian and Noetherian rings and modules, Hilbert basis theorem. *Simple and semisimple modules.

8. Commutative algebra

Localization of modules. Integral ring extensions. *Localization and spec of a ring.

*Nakayama's lemma, *Noether normalization lemma. *Nullstellensatz. *Affine algebraic varieties and commutative rings.

References

T. Hungerford, Algebra, Graduate Texts in

Mathematics 73.

N. Jacobson, Basic algebra I, II. S. Lang, Algebra.

### Real Analysis I, II

Basic Real Analysis I:

Measures and outer measures, measurable functions, integration, monotone convergence theorem, Fatou's lemma, Lebesgue's dominated convergence theorem, convergence in measure, convergence in $L^1$, theorems of Egorov and Lusin, product measures, Fubini-Tonelli theorem, Lebesgue integral in $\mathbb{R}^n$, comparison with Riemann integral, signed and complex measures, Radon-Nikodym theorem, Hardy-Littlewood maximal function, Lebesgue's differentiation theorem, functions of bounded variation, absolutely continuous functions, Banach spaces, Hahn-Banach theorem, Baire category theorem, open mapping and closed graph theorems, Banach-Steinhaus theorem, weak and weak* topologies, Alaoglu theorem, Hilbert spaces, Frechet spaces, Arzela-Ascoli and Stone-Weierstrass theorems;

Basic Real Analysis II:

$L^p$ spaces and their duals, distribution functions and weak $L^p$ spaces, inequalities of Holder, Minkowski and Jensen, Riesz-Thorin and Marcinkiewicz interpolation theorems, Radon measures and Riesz representation theorem, the dual of $C_0$, Fourier transform in $L^1$, the Schwarz class, convolutions and approximation, Fourier inversion formula and Plancherel theorem, Fourier transform in $L^p$ with $1 \leq p \leq 2$, Hausdorff-Young and Young inequalities, tempered distributions, operations on distributions, convergence and summability of Fourier series and integrals, summability kernels (Dirichlet, Fejer, Poisson and Gauss-Weierstrass), Sobolev spaces;

Optional material (depending on instructor) may include further topics from:

harmonic analysis on euclidean spaces, spectral theory, applications to PDE, applications to probability;

Textbooks:

Folland, Real Analysis

Rudin, Real and Complex Analysis

Stein and Shakarchi, Princeton Lectures in Analysis

Wheeden and Zygmund, Measure and Integral

### Complex Analysis I, II

I. Winding number, integral along curves. Various definitions of a holomorphic function. Connection with harmonic functions. Cauchy Integral Theorems and Cauchy Integral Formula for closed curves in a domain, and for the boundary of a domain, Poisson Formula. The integral of a holomorphic function and its dependence on the path of integration. Open Mapping Theorem, Inverse Function Theorem, maximum and minimum principle, Liouville’s Theorem. Uniform convergence of holomorphic functions. Normal families of holomorphic functions. Montel and Vitali Theorems, Picard’s Theorem. Power series, Laurent series. Residues and classification of isolated singularities, meromorphic functions. Divisor of a meromorphic function. Residue Theorem, argument principle, Rouché’s Theorem, computation of integrals. Riemann Mapping Theorem, argument principle. Möbius maps. Schwartz Lemma. Theorems of Mittag-Leffler and Weierstrass. Gamma Function, Riemann Zeta Function, Weierstrass ℘-function.

II. Definition of complex manifolds and examples. Riemann surfaces. The concepts of divisors, line bundles, differential forms and Chern forms. The Riemann-Roch Theorem. The Dirichlet problem for harmonic functions.  The concept of genus of a Riemann surface.

References

Ahlfors, Complex Analysis.

Burchel, An Introduction to Classical Complex Analysis I.

Conway, Functions of One Complex Variable.

Forster, Lectures on Riemann Surfaces.

Gunning, Lectures on Riemann Surfaces.

Knopp, Theory of Functions I, II, and Problem Books.

Complex Analysis I covers approximately Chapters 1-6 of Ahlfors.  Complex Analysis II covers the remaining material.

### Basic Geometry and Topology (Fall Semester)

1. Point-set topology (a quick review):  topological spaces, subspaces, quotients and products. Properties of topological spaces:  Hausdorff, compact, connected. Examples:  spheres projective spaces, homogeneous spaces, CW-complexes.

2. Basic algebraic topology:  fundamental group, covering spaces, the van Kampen Theorem, the fundamendal group of a surface.

3. Smooth manifolds:  tangent bundle and cotangent bundle, vector bundles, constructions with vector bundles (sums, products symmetric and exterior powers), sections of vector bundles, differential forms, the de Rham complex, inverse and implicit function theorems, transversality.

References

Munkres, Topology.

Hatcher, Algebraic Topology (Chapter 1).

Lee, Introduction to Smooth Manifolds.

### Basic Topology II (Spring Semester)

1. Homology:  singular homology, the Eilenberg-Steenrod axioms, homology group of spheres, the degree of a map between spheres, homology calculations via CW complexes, proof of homotopy invariance, proof of excision, universal coefficient and Kunneth Theorem.

2. Cohomology:  the cup product, the cohomology ring of projective spaces.

3. Poincare duality.

Reference

Hatcher, Algebraic Topology

### Basic Differential Geometry (Spring Semester)

1. Connections in vector bundles:  Covariant derivative, parallel transport, orientability, curvature, baby Chern-Weil.

2. Riemannian geometry:  Levi Cevita connection, exponential map, Jacobi fields, arc length variation formulas, fundamental equations for metric immersions and submersions, space forms, Hopf-Rinow, Hadamard-Cartan, Bonnet-Myers (Gauss-Bonnet, Bochner technique).

3. Other geometric structures:  (one or more of Kahler manifolds, symplectic manifolds, contact manifolds).

References

Chavel, Riemannian Geometry:  A modern introduction.

Grove, Riemannian geometry:  A metric entrance.

Petersen, Riemannian Geometry.

Gallot, Hulin, and Lafontaine, Riemannian geometry.

Cullen, Introduction to General Topology.

Dugundji, Topology.

Kelley, General Topology. Munkres, Topology.

Steen, Counterexamples in Topology.

### Logic I, II

1. Model Theory

First order predicate logic. Structures and theories. Compactness theorem. Ultraproducts. Löwenheim-Skolem theorems. Saturation and homogeneity. Quantifier elimination: methods, examples, and applications. Types and type spaces. Prime and Saturated models. Countable models and countable categoricity. Henkin constructions should be seen, for example in the omitting types theorem. If time permits, more advanced topics may be included.

References

C. C. Chang and H. J. Keisler, Model Theory.

D. Marker, Model theory, an introduction.

K. Tent and M. Ziegler, A course in model theory.

2. Computability Theory

Turing machines. Primitive recursive functions, partial recursive functions, equivalence of Turing and Kleene definitions. Recursive sets, computably enumerable sets. The Recursion Theorem. Index sets and Rice's Theorem. Strong reducibilities (m-reducibility, 1-reducibility). Relative computability, Turing degrees, jumps, the Kleene-Post Theorem. The arithmetical hierarchy. Computably enumerable degrees, the Friedberg-Muchnik Theorem, the Low Basis Theorem.

If time permits, further topics may be included.

References

R. I. Soare, Recursively Enumerable Sets and Degrees.

H. Rogers, Theory of Recursive Functions and Effective Computability.

S. B. Cooper, Computability Theory

3. Set Theory

Axioms of ZFC, Schröder-Bernstein Theorem, ordinals, ordinal arithmetic, proof and definition by recursion. The Von Neumann hierarchy. Cardinals, cardinal arithmetic. Special kinds of cardinals—regular and singular, successor and limit, inaccessible, equivalent versions of the Axiom of Choice, the constructible hierarchy. If time permits, further topics may be included.

References

T. Jech, Set Theory.

K. Kunen, Set Theory.

P. Cohen, Set Theory and the Continuum Hypothesis.

Logic I covers the material on model theory and part of the material on set theory, including ordinal and cardinal arithmetic, and definitions by recursion. Logic II covers the material on computability, plus more set theory.

### Discrete Mathematics

This course provides an introduction to the questions of existence, structure and enumeration of discrete mathematical objects.  Topics include:

1. Enumeration — basic counting principles (including permutations, combinations, compositions, pigeon-hole principle and inclusion-exclusion), basic counting sequences (such as binomial coefficients, Catalan numbers and Stirling numbers), and recurrence relations and generating functions.

2. Structure and existence — Graphs (including trees, connectivity, Euler trails and Hamilton cycles, matching and coloring), partially ordered sets and lattices, basic Ramsey theory, error detecting and correcting codes, combinatorial designs, and techniques from probability and linear algebra.

Other topics chosen by the instructor may be included if time permits.

The course will be at the level of the following books:

Stasys Jukna, Extremal combinatorics (with applications to computer science)

Peter Cameron, Combinatorics (topics, techniques and algorithms)

J.H. van Lint and R.M. Wilson, A course in Combinatorics

### Optimization

Convex sets. Caratheodory and Radon's theorems.  Helly's Theorem. Facial stricture of convex sets.  Extreme points. Krein-Milman Theorem. Separation Theorem. Optimality conditions for convex programming problems. Introduction to subdifferential calculus. Chebyshev approximations.

References

Barvinok, A Course in Convexity.

R. Webster, Convexity.

### Basic PDE

The goal of this course is to teach the basics of Partial Differential Equations (PDE), linear and nonlinear. It begins by providing a list of the most important PDE and systems arising in mathematics and physics and outlines strategies for their “solving.” Then, it focusses on the solving of the four important linear PDE: The transport, the Laplace, the heat, and the wave equations, all in several space dimensions. In addition to deriving solution formulas, it also introduces “energy” methods involving L2-norms, which foreshadow modern theoretical developments for studying PDE. Next, it continues with the solving of nonlinear first-order equations (like the Burgers equation) by using the method of characteristics. Also, it defines weak solutions, introduces several nonlinear concepts and phenomena, and demonstrates the point that nonlinear PDE are much more difficult to solve than linear PDE. The first (and major) part of the course concludes with the presentation of other ways for representing solutions, including the separation of variables, traveling waves and solitons, similarity and scaling, Fourier, Radon and Laplace transform methods, converting nonlinear PDE to linear by using transformations like the Cole-Hopf, asymptotics, power series methods and the Cauchy-Kovalevskaya Theorem. The second part of the course introduces function spaces like Hӧlder and Sobolev spaces together with their properties including, approximation, extensions, traces, important inequalities, and compactness. These spaces are used for designing appropriate solution spaces for a given PDE problem (initial or boundary value) inside which the solution resides. Proving this may require sophisticated theories that could involve many branches of mathematics as well as physics. Such theories are presented in the later part of the textbook, for which there is no time available in this course. However, this course will prepare the interested student to continue learning more about PDE and their applications to mathematics, the sciences, engineering and economics.

Prerequisites: Basic Analysis I and concurrently enrollment in Basic Analysis II. Some familiarity with Ordinary Differential Equations is desirable but not a prerequisite. Exposure to an undergraduate PDE course is helpful but not required.

Textbook: Lawrence C. Evans, Partial Differential Equations: Second Edition. American Mathematical Society, Graduate Studies in Mathematics, Vol. 19, Providence RI, 2010

References

Fritz John, Partial Differential Equations

David Gilbarg and Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order

Lars Hӧrmander, The Analysis of Linear Partial Differential Operators (4 volumes)

Felipe Linares and Gustavo Ponce, Introduction to nonlinear dispersive equations

Ivan Georgievich Petrovsky, Lectures on Partial Differential Equations

Walter Strauss, Partial Differential Equations: An Introduction

Michael E. Taylor, Partial Differential Equations (3 volumes)

### Probability

(one semester, graduate level, intermediate or basic, prerequisite: first sem. of Real Analysis)

1. Preliminaries:

Kolmogorov's axioms, examples of probability spaces, independence, random variables,

probability distributions, expectation, variance, Kolmogorov's consistency theorem

2. Sums of Independent Random Variables:

basic inequalities and modes of convergence (almost surely, in probability, L^p), Rademacher functions,

more inequalities (Chernoff, Bernstein, Hoeffding, Bennett, Khinchine, Marcinkiewicz-Zygmund)

3. Laws of Large Numbers:

weak Laws of Large Numbers, uniform integrability, Kolmogorov's inequality, strong Laws of Large Numbers, Levy's Refection principle, Law of the Iterated Logarithm

4. Central Limit Theorem:

Lindeberg-Levy condition, Fourier transform and characteristic functions, weak* convergence of

probability measures

5. Discrete Time Martingales:

conditional expectation, submartingales and supermartingales, Doob's inequality, stopping times,

Hunt's theorem, martingale convergence theorem

(Optional - time permitting):

6. Large Deviations

7. Markov processes

7. Continuous Martingales

8. Wiener Measure, applications to PDE

9. Marcinkiewicz-Zygmund and Burkholder inequalities, applications to harmonic analysis

10. Elements of Ergodic Theory

11. Metrics on probability measures, applications to statistics

12. Kantorovich-Wasserstein distance and Optimal Transport

References

D. Stroock, Probability Theory: an analytic view

A. Borovkov, Probability Theory

W. Feller, An Introduction to Probability Theory and its Applications 2 vols.

## Further Introductory Courses

### Introduction to Algebraic Geometry (two-semester course sequence).

This is an introduction to algebraic varieties, schemes, coherent sheaves, curves, and surfaces.

### Lie groups and Lie algebras

This course is an introduction to Lie theory and covers material related to Lie groups, Lie algebras, and their applications. Specific topics will be drawn from Lie groups, the Lie algebra of a Lie group, the exponential map, subgroups, the representation theory of Lie groups and Lie algebras, the structure of Lie algebras, root systems, the Weyl group, enveloping algebras, and infinite dimensional Lie algebras.

### Linear Control

Introduction to linear system theory. Linear-quadratic control, H-infinity control, introduction to robust control based on matrix cube theorem, linear matrix inequalities and interior-point algorithms.

References

A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization.

P. Lancaster and L. Rodman, Algebraic Riccati Equations.

S. Boyd et. al. Linear Matrix Inequalities in System and Control Theory.

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