Mathematics Department - Graduate Course Descriptions - Spring 2017

### Theory of Functions of a Real Variable II

Text: Folland

Prerequisites: 501

Description: This course is a continuation of 640:501 from Fall 2006. The goal is to give an introduction to core topics in real and functional analysis that every professional mathematician should know.

### Theory of Functions of a Complex Variable II

Text: Notes written by the Instructor to be distributed during the lectures

Prerequisites: 501, 503

Description: This will be a continuation of Math 503. It is a fundamental course for anyone who wants to pursue their further studies in pure mathematics. The following two parts will be covered:

(1) Classical Complex Analysis and (2) Riemann surfaces.

Part 1. Classical Complex Analysis. We will discuss topics not covered in Math 503, such as the invariant metrics, Picard theorems, elliptic functions.

Part 2. Introduction to Riemann surfaces. Hyperbolic geometry, uniformization theorem, Riemann-Roch theorem, Abel theorem.

### Functional Analysis II

Text: Functional Analysis by M. Reed and B. Simon, parts I, II. Operator Theory (part 4) by B. Simon, AMS 2016.

Prerequisites: Real Analysis

Description: Topics covered will include: review of Hilbert spaces and Operators.

Spectral Theorem for bounded and unbounded Operators.

Compact Operators and Introduction to Spectral Theory: Fredholm Theorem, Weyl, Resolvent Estimates.

Duality, Distributions and Fourier transforms.

All topics will include examples from Partial Differential equations, Mathematical-Physics and Differential Geometry.

### Selected Topics in Analysis

Subtitle: Mean curvature flow

Text:

Prerequisites: 16:640:517 - Partial Differential Equations I

Description: We will introduce the mean curvature flow, which is one of well studied geometric flows in which the hypersurface in R^{n+1} evolves in the normal direction by the speed given by its mean curvature.

We will go over Husiken's paper in which he introduces the mean curvature flow and shows that every closed convex hypersurface evolving by the MCF must shrink to a point in finite time, in a spherical manner. We will introduce Huisken's monotonicity formula and talk about how it can be used to study singularities that are inevitable in the case of a closed MCF. We will also talk about classification of singularities occurring in the MCF, which is possible if extra conditions on the hypersurface are being assumed (for example, in the case we start the flow from a mean convex closed hypersurface, the property that turns out to be preserved along the flow).

We will also talk about the mean curvature flow starting with an entire graph, an example of a complete, mean curvature flow. In this case we will learn a few useful tricks, how to make a use of cut off functions to apply maximum principle and get useful curvature interior estimates. As a consequence we will see that the MCF starting at an entire graph exists forever.

If time permits we will also discuss ancient solutions to the mean curvature flow (solutions that exist from time equal to -infty and that occur as singularity models) and their classification in certain cases.

### Ordinary Differential Equations

Text: Ordinary Differential Equations: A Constructive Approach, Notes by M. Gameiro, J.-P. Lessard, J. Mireles James, K. Mischaikow

Prerequisites: 350, 411

Description: The impossibility of finding explicit analytic solutions to nonlinear systems of differential equations led Poincare to initiate a qualitative theory of dynamics. Work in the last half the 20th century, based on differential topology, led to a understanding of generic properties of solutions. However, with the advances in computing technologies, the primary tool for studying specific nonlinear systems involves numerical simulations without a guarantee of mathematical rigor.

In this introductory course on ordinary differential equations we will cover the classical results of dynamical systems: existence and uniqueness theorems; linear theory including Floquet theory and elementary bifurcations; stable and unstable manifolds; boundary value problems; and a brief introduction to chaotic dynamics. The novelty of this course is that the associated theorems are proven using constructive analytic methods. As a consequence, standard numerical methods can be easily adapted to provide rigorous proofs.

For a more detailed overview of the philosophy of the course (we will only consider ODEs as opposed to PDEs and FDEs) see http://www.ams.org/notices/201509/rnoti-p1057.pdf and the description of the AMS short course delivered at the National Meeting January 2015 http://www.ams.org/notices/201509/rnoti-p1106.pdf

### Partial Differential Equations II

Text: There is no textbook for this course, suggested books are listed below.

Prerequisites: 501, 502, 503, 517, some very rudimentary Functional analysis

Description: This will be a basic PDE course with a strong focus on basic estimates that lead to regularity and existence of solutions. I plan to cover Schauder theory of elliptic PDE, and attendant regularity theory. I hope to cover basic Sobolev spaces if it has not been covered in Math 517. I shall then also lecture on basic properties of pseudo-differential operators. The important theory of L^p regularity of solutions that is fundamental for the study of Nonlinear problems can only be covered if students have understood some Harmonic Analysis and the Calderon-Zygmund theory. If students have had some exposure to Harmonic Analysis, I will deal with the L^p theory, or else restrict myself to Schauder theory. No one book covers all this material. However, a good combination of books that has all this material is:

[1] Elliptic Partial Differential equations of Second Order, David Gilbarg and Neil S. Trudinger

[2] Introduction to Pseudodifferential and Fourier Integral Operators, Volume I, J. Francois Treves.

### Algebraic Geometry II

Text: Hartshorne, Algebraic Geometry, GTM 52

Prerequisites: Math 535. Familiarity with commutative algebra is an advantage, but is not required.

Description: This course continues the study of algebraic geometry from the fall by replacing algebraic varieties with the more general theory of schemes, which makes it possible to assign geometric meaning to an arbitrary commutative ring. One major advantage of This course continues the study of algebraic geometry from the fall by replacing algebraic varieties with the more general theory of schemes, which makes it possible to assign geometric meaning to an arbitrary commutative ring. One major advantage of schemes is the availability of a well-behaved fiber product. Combined with Grothendieck's philosophy that properties of schemes should be expressed as properties of morphisms between schemes, fiber products make the theory very flexible. In addition, schemes provide a natural context for introducing the theory of sheaf cohomology, which is a central tool in modern algebraic geometry. For example, one can use cohomological methods to give a simple proof of the classical Riemann-Roch theorem for curves. The goal of the course is to cover the basic definitions, properties, and applications of the above mentioned concepts.

### Introduction to Algebraic Topology II

Text: Hatcher, same as Fall course 640:540

Prerequisites: 640:540

Description: This course will be a continuation of Math 540, and will start where 540 leaves off. The core topics will be: basic homotopy theory, homotopy groups, generalized cohomology theories (especially oriented theories) and stable homotopy theory. To study these, several tools will be introduced: simplicial sets, CW complexes, de Rham cohomology and Poincare duality, universal coefficients for homology and cohomology, Hom and Ext, spectral sequences and/or other topics of interest to the participants such as applied and computational topology.

### Abstract Algebra II

Text: Jacobson, Basic Algebra I, II (both volumes)

Prerequisites: Any standard course in abstract algebra for undergraduates and/or Math 551

Description: This is the continuation of Math 551, aimed at a discussion of many fundamental algebraic structures. The course will cover a selection of the following topics (and perhaps some others).

1) Basic module theory and introductory homological algebra - most of Chapter 3 and part of Chapter 6 of Basic Algebra II: hom and tensor, projective and injective modules, resolutions, completely reducible modules, the Wedderburn-Artin theorem, simple algebras

2) Commutative ideal theory and Noetherian rings - part of Chapter 7 of Basic Algebra II: rings of polynomials, localization, primary decomposition theorem, Dedekind domains, Noether normalization

3) Galois Theory - Chapter 4 of Basic Algebra I and part of Chapter 8 of Basic Algebra II: algebraic and transcendental extensions, separable and normal extensions, the Galois group, solvability of equations by radicals

### Selected Topics in Algebra

Subtitle: Modular tensor categories for affine Lie algebras

Text: No textbook. The lectures will be basd on research papers.

Prerequisites: First year graduate algebra and analysis courses. Basic knowledge in Lie algebras will be very helpful but is not required.

Description: Affine Lie algebras are one of the most important classes of infinite-dimensional Lie algebras. Suitable module categories for affine Lie algebras have structures of modular tensor categories. Moreover, these modular tensor categories are equivalent to the modular tensor categories constructed from suitable module categories for the corresponding quantum groups at the corresponding roots of unity. These modular tensor categories give the quantum invariants of knots and three-manifolds and play an important role in topological quantum computing.

In this course, I will discuss the construction of these modular tensor categories. Below are the detailed topics to be covered in this course:

1. Finite-dimensional Lie algebras and modules.

2. Affine Lie algebras and modules.

3. The categories generated by integrable highest weight modules for affine Lie algebras.

4. Vertex operator algebras and modules associated to affine Lie algebras.

5. Intertwining operators and operator product expansion.

6. Ribbon braided tensor category structures.

7. Modular invariance of intertwining operators.

8. Moore-Seiberg equations and the Verlinde formula

9. Rigidity and nondegeneracy properties.

### Representation Theory

Text: None

Description: This will be a highly individualized course covering topics in representation theory with "low-hanging" fruit, i.e. areas in which a semester's work might conceivably lead to publishable results. I will help each student choose a topic appropriate to his/her background and taste, sketch the main results in the topic, discuss possible generalizations, and suggest lines of attack. Students can work on one or more topic, either individually or in teams. In particular, if you took this course last time, there will be no overlap! If you are interested in this course, please send me an e-mail.

### Selected Topics in Logic

Subtitle: Introduction to Borel Equivalence Relations

Text: None

Prerequisites: None

Description: We will study Borel equivalence relations.

We will study number of dichotomy results in descriptive set theory. The basic fundamental question is the following. Given an equivalence relation E on some natural space of mathematical objects, when can we effectively choose representatives from each equivalence class. Surprisingly, often this cannot be done. But this is well known, take for instance the Vitali equivalence relation. Any set of representatives for it is not Lebesgue measurable, and hence, it cannot be Borel or analytic. Another such equivalence relation is the following. Given two infinite sequences of integers x and y, say x is related to y if they are eventually equivalent. Call it E_0. It is known that again we cannot choose a Borel set of representatives of E_0 classes. Moreover, given any other equivalence relation E that is itself Borel, we can either choose representative in a Borel way or we can embed E_0 into E in a Borel way.

It turns out that many equivalence relations of mathematical interest can be represented as subsets of some Polish space. For instance, this can be done for the set of infinite groups. One can then study the natural equivalence relations on these objects, such as the isomorphism relation. We will study basic techniques that have been used to prove that such equivalence relation are complex. This area of set theory contains many open problems.

We will use the top-down approach. The goal would be to cover most of the material in the following papers.

1. Harrington, Marker and Shelah, Borel Orderings

2. Harrington, Kechris and Louveau, A Glimm-Effros dichotomy for Borle equivale relations.

3. Jackson, Kechris, and Louveau, Countable Borel equivalence relations

4. Kechris, New directions in descriptive set theory

### Special Topics in Number Theory

Subtitle: Prime Numbers

Text: H.Iwaniec and E.Kowalski, Analytic Number Theory, AMS Colloquium Publications, Vol 53

Prerequisites: Complex Function Theory

Description: This will be a one semester course about prime numbers. In particular I will present various techniques and results concerning the distribution of prime numbers in arithmetic progressions. Of course, connections with the zeros of L-functions and the Riemann hypothesis will be the central topic. Moreover I will show selected results about primes represented by polynomials using sieve methods and the bilinear forms techniques. I will also give an account of the recent results by Y. Zhang and J. Maynard about bounded gaps between prime numbers. Some parts of the course will be quite advanced, nevertheless the participants are not assumed to be familiar with any special mathematical subjects. A good skill in the complex function theory will be helpful.

Text:

Prerequisites:

Description:

### Methods of Applied Mathematics II

Text: Michael D. Greenberg, Advanced Engineering Mathematics (second Edition; 0-13-221431-1), Prentice-Hall, 1998.

Prerequisites: Math 527, or permission of the instructor

Description: A second semester graduate course intended primarily for students in mechanical and aerospace engineering, biomedical engineering, and other engineering programs. There will be three parts:

1. Complex variable theory, with topics chosen from the differential and integral calculus of functions of a complex variable, conformal mapping, Taylor series, Laurent series, the residue theorem, and applications to partial differential equations and to fluid mechanics.

2. Calculus of variations, including the motivation of variational principles from physical laws, derivation of Euler-Lagrange equations, stability criterion, and linearization.

3. Perturbation methods, including applications to ode systems, examples of boundary layer, multiple-scale problems, and eigenvalue problems.

### Introduction to Mathematical Physics I

Text: Prof. Kiessling's lecture notes

Prerequisites: Working knowledge of basic ODEs and the linear wave equation. Some exposure to analysis at the level of the "baby Rudin," better yet: math 501. Basic knowledge of Euclidean geometry. Curiosity about physics. N.B.: The physics department's mathematical physics course 511 has very little in common with this course. It is not a prerequisite.

Description: Summary: The course introduces the student to a rigorous mathematical treatment of the classical theories of our physical world: "matter made of point particles in space(-)time which interact via gravity and electromagnetism." The emphasis is on both, mathematical rigor and conceptual clarity of the physical theories.

Topics:

1. The Newtonian universe (Galileian space and time, point particles, Newton's law of motion, Newton's law of gravitational force, Coulomb's law of electrical force; symmetries and conservation laws; other formulations of Newton's mechanics: Hamilton, Hamilton-Jacobi, and Lagrange).

2. Einstein's universe (Minkowski's spacetime, Maxwell's electromagnetic field equations, electromagnetic waves, relativistic energy and momentum; and in *very brief* outline also: Lorentzian manifolds, Einstein's gravitational field equations, geodesics, black holes, gravitational waves)

3. Limits of validity of the classical theories (the joint Cauchy problem for fields and point particles, the problem of self-interactions; energy and momentum laws; the dawn of quantum theory.)

### Numerical Analysis II

Text: Lecture Notes

Prerequisites: 642:573

Description: This is the second part, independent of the first, of a general survey of the basic topics in numerical analysis – the study and analysis of numerical algorithms for approximating the solution of a variety of generic problems which occur in applications. In 642:574, the topics are: the numerical solution of linear systems of equations, the approximation of matrix eigenvalues and eigenvectors, the numerical solution of nonlinear systems of equations, numerical techniques for unconstrained function minimization, finite difference and finite element methods for two-point boundary value problems, and finite difference methods for some model problems in partial differential equations.

In 642:573, the topics are: the approximation of functions by polynomials and piecewise polynomials, numerical integration, and the numerical solution of initial value problems for ordinary differential equations.

Despite the many solution techniques presented in elementary calculus and differential equations courses, mathematical models used in applications often do not have the simple forms required for using these methods. Hence, a quantitative understanding of the models requires the use of numerical approximation schemes. This course provides the mathematical background for understanding how such schemes are derived and when they are likely to work.

To illustrate the theory, in addition to the usual pencil and paper problems, some short computer programs will be assigned. To minimize the effort involved, however, the use of Matlab will be encouraged. This program has many built-in features which make programming easy, even for those with very little prior programming experience.

### Numerical Solutions of Partial Differential Equations

Text: No required textbook; lecture notes provided

Prerequisites: Some knowledge of partial differential equations and linear algebra

Description: In this course, we study finite difference and finite element methods for the numerical solution of elliptic, parabolic, and hyperbolic partial differential equations and variational inequalities, and methods for the efficient solution of the algebraic equations obtained from the discretization procedures. The course will concentrate on the key ideas underlying the derivation of numerical schemes and a study of their stability and accuracy. Students will have the opportunity to gain computational experience with numerical methods with a minimal of programming by the use of various software packages.

### Graph Theory

Text: Douglas B. West: Introduction to Graph Theory

Prerequisites: While I will not assume that students have prior knowledge of graph theory, I will assume a basic knowledge of proofs and discrete probability theory.

Description: This course will serve as an advanced introduction to graph theory. We will work mainly from the textbook of D.B. West, with some excursions outside the book.

### Combinatorics II

Text: There is no text, but there are several books that we will be referring to -Some of the relevant books include: The Probabilistic Method (Alon & Spencer), Extremal Combinatorics (Stasys Jukna), Linear Algebra Methods in Combinatorics (Babai & Frankl)

Prerequisites: 16:642:582 or permission of instructor

Description: This is the second part of a two-semester course surveying basic topics in combinatorics. Topics for the full year should include most of:

• Enumeration (basics, generating functions, recurrence relations, inclusion-exclusion, asymptotics)

• Matching theory, polyhedral and fractional issues

• Partially ordered sets and lattices, Mobius functions

• Theory of finite sets, hypergraphs, combinatorial discrepancy, Ramsey theory, correlation inequalities

• Probabilistic methods

• Algebraic and Fourier methods • Entropy methods

### Selected Topics in Discrete Mathematics

Subtitle: Probabilistic Methods in Combinatorics

Text: Alon-Spencer, The Probabiistic Method (optional but useful)

Prerequisites: I will try to make the course self-contained except for basic combinatorics and very basic probability. See me if in doubt.

Description: We will discuss applications of probabilistic ideas to problems in combinatorics and related areas (e.g. geometry, graph theory, complexity theory). We will also at least touch on topics, such as percolation and mixing rates for Markov chains, which are interesting from both combinatorics/TCS and purely probabilistic viewpoints.

### Topics in Probability and Ergodic Theory I

Subtitle: Thermodynamics and Number Theory

Text: None

Prerequisites: First year Algebra, Real/Complex Analysis

Description: We will study recent applications of ideas from thermodynamic to problems number theory. No prior knowledge of either topic will be assumed.

### Selected Topics in Applied Mathematics

Subtitle: Obstacle Problems and Applications

Text: (1) P. Feehan, M. Poghosyan, and H. Shahgholian, Variational Inequalities, Obstacle Problems, and Free Boundary Problems in Economics, Finance, and Engineering", book in preparation.

2. J-F. Rodrigues, "Obstacle Problems in Mathematical Physics, North-Holland, New York, 1987.

Prerequisites: (1) An undergraduate (or higher-level) course on real analysis covering basic integration theory and the concepts of Hilbert spaces and Banach spaces;

(2) A one-semester undergraduate (or higher-level) course on partial differential equations (for example, based on the text by Walter Strauss).

Description: We introduce graduate students to the theory required to understand variational inequalities, obstacle problems, and free boundary problems. They have many applications in applied mathematics, economics, engineering, and quantitative finance, including the study of constrained heating, economics, elasto-plasticity, fluid filtration in porous media, game theory, minimal surfaces, physics, optimal control, and valuation of American-style options. The intended audience includes second and higher-year doctoral and master’s level students in mathematics, engineering, mathematical finance, and materials science.

### Mathematical Finance II

This course is part of the Mathematical Finance Master's Degree Program.

### Computational Finance

This course is part of the Mathematical Finance Master's Degree Program.

### Fixed Income Securities and Derivative Modeling

This course is part of the Mathematical Finance Master's Degree Program.

### Special Research Projects in Mathematical Finance

This course is part of the Mathematical Finance Master's Degree Program.

### Seminar in Mathematical Finance

This course is part of the Mathematical Finance Master's Degree Program.

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