#### Fall 2019 Courses

## Description of the courses for Fall 2019 are posted here

https://www.ma.utexas.edu/academics/courses/course-descriptions#graduate-courses

#### Spring 2020 courses

## M380D *Algebra*

It is assumed that students know the basic material from an undergraduate course in linear algebra and an undergraduate abstract algebra course. The first part of the Prelim examination will cover sections 1 and 2 below. The second part of the Prelim examination will deal with section 3 below.

**1. Groups: **Finite groups, including Sylow theorems, *p*-groups, direct products and sums, semi-direct products, permutation groups, simple groups, finite Abelian groups; infinite groups, including normal and composition series, solvable and nilpotent groups, Jordan-Holder theorem, free groups.

**References:** Goldhaber Ehrlich, Ch. I except 14; Hungerford, Ch. I, II; Rotman, Ch. I-VI, VII (first three sections).

**2. Rings and modules:** Unique factorization domains, principal ideal domains, modules over principal ideal domains (including finitely generated Abelian groups), canonical forms of matrices (including Jordan form and rational canonical form), free and projective modules, tensor products, exact sequences, Wedderburn-Artin theorem, Noetherian rings, Hilbert basis theorem.

**References: **Goldhaber Ehrlich, Ch. II, III 1,2,4, IV, VII, VIII; Hungerford, Ch. III except 4,6, IV 1,2,3,5,6, VIII 1,4,6.

**3. Fields:** Algebraic and transcendental extensions, separable extensions, Galois theory of finite extensions, finite fields, cyclotomic fields, solvability by radicals.

**References:** Goldhaber Ehrlich, Ch. V except 6; Hungerford, Ch. V, VI; Kaplansky, Part I.

**References:**

Goldhaber Ehrlich, *Algebra*, reprint with corrections, Krieger, 1980.

Hungerford, *Algebra*, reprint with corrections, Springer, 1989.

Isaacs, *Algebra, a Graduate Course*, Wadsworth, 1994.

Kaplansky, *Fields and Rings*, 2nd Edition, University of Chicago Press, 1972.

Rotman, *An Introduction to the Theory of Groups*, 4th Edition, W.C. Brown, 1995.

## M381D *Complex Analysis*

The objective of this syllabus is to aid students in attaining a broad understanding of analysis techniques that are the basic stepping stones to contemporary research. The prelim exam normally consists of eight to ten problems, and the topics listed below should provide useful guidelines and strategy for their solution. It is assumed that students are familiar with the subject matter of the undergraduate analysis courses M365C and M361. The first part of the Prelim examination will cover Real Analysis. The second part of the prelim examination will cover Complex Analysis.

**1. Measure Theory and the Lebesgue Integral**

Basic properties of Lebesgue measure and the Lebesgue integral on **R**^{n} (see [5], Ch. 1-4) and general measure and integration theory in an abstract measure space (see [5], Ch. 11-12; and especially [6], Ch. 1-2). L^{p} spaces (see [6], Ch. 3); convergence almost everywhere, in norm and measure; approximation in L^{p}-norm and L^{p}-L^{q} duality; integration in product spaces (see [6], Ch. 8) and convolution on **R**^{n}; and the concept of a Banach space, Hilbert space, dual space and the Riesz representation theorem.

**2. Holomorphic Functions and Contour Integration**

Basic properties of analytic functions of one complex variable (see [1], Ch. 4-5; [2], Ch. 4-7; [4], Ch. 4-8; or [6], Ch. 10-12 and 15). Integration over paths, the local and global forms of Cauchy's Theorem, winding number and residue theorem, harmonic functions, Schwarz's Lemma and the Maximum Modulus theorem, isolated singularites, entire and meromorphic functions, Laurent series, infinite products, Weierstrass factorization, conformal mapping, Riemann mapping theorem, analytic continuation, "little" Picard theorem.

**3. Differentiation**

The relationship between differentiation and the Lebesgue integral on a real interval (see [5], Ch. 5), derivatives of measures (see [6], Ch. 5), absolutely continuous functions and absolute continuity between measures, functions of bounded variation.

**4. Specific Important Theorems**

Students should be familiar with Monotone and Dominated Convergence theorems, Fatou's lemma, Egorov's theorem, Lusin's theorem, Radon-Nikodym theorem, Fubini-Tonelli theorems about product measures and integration on product spaces, Cauchy's theorem and integral formulas, Maximum Modulus theorem, Rouche's theorem, Residue theorem, and Fundamental Theorem of Calculus for Lebesgue Integrals. Students should be familiar with Minkowski's Inequality, Holder's Inequality, Jensen's Inequality, and Bessel's Inequality.

## M382D *Differential Topology*

It is assumed that students have a working knowledge of the equivalent of a one semester course in general topology (for example, see the appended syllabus for the undergraduate course M367K). For the semester in differential topology, it will also be assumed that students know the basic material from an undergraduate linear algebra course. The first part of the Prelim examination will deal with Algebraic Topology and the second part will deal with Differential Topology.

**Algebraic Topology**

**1. Manifolds: **Identification (quotient) spaces and identification (quotient) maps; topological *n*-manifolds, including surfaces, ** S^{n}**,

**,**

*RP*^{n}**, and lens spaces.**

*CP*^{n}**2. Triangulated manifolds: **Representation of triangulated, closed 2-manifolds as connected sums of tori or projective planes.

**3. Fundamental group and covering spaces: **Fundamental group, functoriality, retract, deformation retract; Van Kampen's Theorem, classification of surfaces by abelianizing the fundamental group, covering spaces, path lifting, homotopy lifting, uniqueness of lifts, general lifting theorem for maps, covering transformations, regular covers, correspondence between subgroups of the fundamental group and covering spaces, computing the fundamental group of the circle, ** RP^{n}**, lens spaces via covering spaces.

**4. Simplicial homology: **Homology groups, functoriality, topological invariance, Mayer-Vietoris sequence; applications, including Euler characteristic, classification of closed triangulated surfaces via homology and via Euler characteristic and orientability; degree of a map between oriented manifolds, Lefschetz number, Brouwer Fixed Point Theorem.

** References:**

Armstrong, *Basic Topology,* Springer, 1983 (principal text).

Greenberg, *Lectures on Algebraic Topology,* W.A. Benjamin, 1967.

Massey, *Algebraic Topology, an Introduction,* 4th corrected printing, Springer, 1977.

Munkres, *Elements of Algebraic Topology*, Addison-Wesley, 1984.

**Differential Topology**

**1. Smooth mappings: **Inverse Function Theorem, Local Submersion Theorem (Implicit Function Theorem).

**2. Differentiable manifolds:** Differentiable manifolds and submanifolds; examples, including surfaces,** S^{n}**,

**,**

*RP*^{n}**and lens spaces; tangent bundles; Sard's Theorem and its applications; differentiable transversality; orientation.**

*CP*^{n}**3. Vector fields and differential forms:** Integrating vector fields; degree of a map, Brouwer Fixed Point Theorem, No Retraction Theorem, Poincare-Hopf Theorem; differential forms, Stokes Theorem.

**References:**

Guillemin Pollack, *Differential Topology*, Prentice-Hall, 1974 (basic reference).

Hirsch, *Differential Topology,* Springer, 1976.

Milnor, *Topology from the Differentiable Viewpoint,* University of Virginia Press, 1965.

Spivak, *Calculus on Manifolds*, Benjamin, 1965 (differentiation, Inverse Function Theorem, Stokes Theorem).

For the examples indicated we refer to the books of Greenberg, Hirsch and Munkres.

## M383D *Methods of Applied Mathematics*

It is assumed that students are familiar with the subject matter of the undergraduate analysis course M365C (see the Analysis section for a syllabus of that course) and an undergraduate course in linear algebra.

The Applied Math Prelim divides into these six areas. The first three are discussed in M383C and will be covered in the first part of the Prelim examination:

**1. Banach spaces:**

Normed linear spaces and convexity; convergence, completeness, and Banach spaces; continuity, open sets, and closed sets; continuous linear transformations; Hahn-Banach Extension Theorem; linear functionals, dual and reflexive spaces, and weak convergence; the Baire Theorem and uniform boundedness; Open Mapping and Closed Graph Theorems; Closed Range Theorem; compact sets and Ascoli-Arzelà Theorem; compact operators and the Fredholm alternative.

**2. Hilbert spaces:** Basic geometry, orthogonality, bases, projections, and examples; Bessel’s inequality and the Parseval Theorem; the Riesz Representation Theorem; compact and Hilbert-Schmidt operators; spectral theory for compact, self-adjoint and normal operators; Sturm-Liouville Theory.

**3. Distributions:** Seminorms and locally convex spaces; test functions and distributions; calculus with distributions.

These three areas are discussed in M383D and will be covered in the second part of the Prelim examination:

**4. The Fourier Transform and Sobolev Spaces:** The Schwartz space and tempered distributions; the Fourier transform; the Plancherel Theorem; convolutions; fundamental solutions of PDE’s; Sobolev spaces; Imbedding Theorems; the Trace Theorems for H^{s}.

**5. Variational Boundary Value Problems (BVP):** Weak solutions to elliptic BVP’s; variational forms; Lax-Milgram Theorem; Green’s functions.

**6. Differential Calculus in Banach Spaces and Calculus of Variations:** The Fréchet derivative; the Chain Rule and Mean Value Theorems; Banach’s Contraction Mapping Theorem and Newton’s Method; Inverse and Implicit Function Theorems, and applications to nonlinear functional equations; extremum problems, Lagrange multipliers, and problems with constraints; the Euler-Lagrange equation.

## M385D *Theory of Probability*

**(THE FIRST PART OF THE PRELIM EXAM WILL DEAL WITH THE MATERIAL COVERED IN M385C AND THE SECOND PART OF THE PRELIM EXAM WILL DEAL WITH THE MATERIAL COVERED IN M385D)**

#### 1. THEORY OF PROBABILITY I - M385C

- Prerequisites:
- Real Analysis (M365C or equivalent),
- Linear Algebra (M341 or equivalent),
- Probability (M362K or equivalent).

- R. Durrett, Probability: theory and examples, third ed., Duxbury Press, Belmont, CA, 1996. (required)
- D. Williams, Probability with martingales, Cambridge University Press, Cambridge, 1991. (recommended)

Literature:
- Syllabus:

(Note: all references are to Durrett's book)**Foundations of Probability:**- Random variables (Sections 1.1, 1.2): probability spaces, σ-algebras, measurability, continuity of probabilities, product spaces, random variables, distribution functions, Lebesgue-Stieltjes measures (without proof), random vectors, generation, a.s.-convergence
- Expected value (Section 1.3): abstract Lebesgue integration (without proofs), inequalities (Jensen, Cauchy-Schwarz, Chebyshev, Markov, Hölder, Minkowski), limit theorems (Fatou's lemma, monotone convergence and dominated convergence theorems), change-of-variables formula,
- Dependence (Section 1.4): independence, pairwise independence, Dynkin's - theorem, convolution of measures, Fubini's theorem, Kolmogorov's extension theorem (without proof)

**Classical Theorems:**

- Weak laws of large numbers (Sections 1.5, 1.6): the L
^{2}-weak law of large numbers, triangular arrays, Borel-Cantelli lemmas, modes of convergence, inequalities (Markov, Chebyshev, Jensen, Hölder), the weak law of large numbers - Central limit theorems (Sections 2.2, 2.3a, 2.3b, 2.3c, 2.4a, 2.9part ): weak convergence of distributions, the continuous mapping theorem, Helly's selection theorem, tightness, characteristic functions, the inversion theorem, continuity theorem, the central limit theorem, multivariate normal distributions

**Discrete-Time Martingale Theory:**- Conditional expectation (Sections 4.1a, 4.1b): Radon-Nikodym theorem (without proof), conditional expectation, filtrations, predictability and adaptivity
- Martingales (Sections 4.2, 4.4, 4.5, 4.6part , 4.7): martingale transforms, the optional sampling the- orem, the upcrossing inequality, Doob's decomposition, Doob's inequality, Lp -convergence, maxi- mum inequalities, L
^{2}-theory, uniform integrability, backwards martingales and the strong law of large numbers.

#### 2. THEORY OF PROBABILITY II - M385D

- Prerequisites:
- Graduate-level probability (M385C or equivalent).

- Literature:
- I. Karatzas and S. Shreve, Brownian motion and stochastic processes, second ed., Springer, 1991 (required)
- D. Revuz and M. Yor, Continuous martingales and stochastic processes, third ed., Springer, 1999 (recommended)

- Syllabus:

(Note: all references are to the book of Karatzas and Shreve)**Continuous-Time Martingale Theory:**- General theory of processes (Sections 1.1, 1.2) : Continuous-time processes and filtrations, types of measurability (optional, predictable, progressive), continuous stopping/optional times
- Path regularity of martingales (Section 1.3 A): existence of RCLL modifications, usual conditions for filtrations
- Convergence and optional sampling (Section 1.3 A-C): martingale inequalities, convergence theorems, optional sampling, uniform integrability and martingale with a last element
- Quadratic variation (Section 1.5 or Section IV.1 in Revuz-Yor): quadratic variation for continuous martingales, local martingales and localization, spaces of martingales
- Doob-Meyer decomposition (Section 1.4): no proof

**Brownian Motion:**- Definition, construction and basic properties (Sections 2.1, 2.2): construction via Kolomogorov extension theorem, Hölder regularity of paths (Kolmogorov-Centsov), Gaussian processes
- The canonical space (Section 2.4): weak convergence on C[0, infinity), invariance principle, Wiener measure
- Markov and strong Markov property of Brownian motion (Sections 2.5-2.8, selected topics): reflexion principle, density of hitting times, Brownian filtrations, Blumenthal zero-one law

**Stochastic Integration:**- Construction of the Stochastic Integral (Sections 3.1, 3.2): stochastic integration with respect to continuous local martingales, quadratic variation and Itô isometry
- Itô formula (Section 3.3): Itô formula, exponential martingales, linear stochastic differential equations

**Applications (and extensions) of Itô's formula:**- Paul Léavy's characterization of Brownian motion (Section 3.3 B):
- Changes of measure (Section 3.5): Girsanov theorem, Brownian motion with drift Representations of martingales (Section 3.4): predictable representation property and Kunita-Watanabe decomposition, time-changed Brownian motions (Dambis-Dubins-Schwarz), Knight's theorem on orthogonal martingales
- Local time (Sections 3.6, 3.7): local time for Brownian motion and continuous semimartingales, Tanaka's formula, generalized Itô's formula for convex functions.

## M387D *Numerical Analysis: Algebra and Approximations*

The Prelim sequence is M387C and M387D. The first part of the Prelim examination will cover algebra and approximation and the second part of the Prelim examination will cover diferential equations.

Principles of discretization of differential equations:

- ODEs: Stability and convergence theory, Stiff problems,Symplectic integrators
- FEM (finite element method) and FDM (finite difference method) for boundary value problems
- FEM for PDEs (main focus on elliptic problems): Basic theory, weak formulations, Lax-Milgram theorem, finite element spaces, approximation theory, a priori and a posteriori error estimates, practical algorithms, extensions, mixed methods etc.
- FDM for PDEs (main focus on hyperbolic and parabolic problems): Lax equivalence theorem, Von Neumann and other stability analysis, nonlinear conservation laws, shocks, entropy, practical algorithms

Brief survey of other methods for PDEs:

- FVM, DG, Spectral and particle methods
- Applications: Elasticity (FEM), Fluids (FVM), and Waves (FDM)
- Solution of linear and nonlinear equations
- Solution of integral equations
- Eigenvalues
- Optimization
- Monte Carlo methods
- Fast Fourier, wavelet transforms, approximation theory
- Basic undergraduate numerical methods
- Interpolation, fixed point iterations, Newton's method for root finding
- Direct and iterative methods for solving linear equations
- Quadratures