# Fall 2018 Graduate Courses

The following graduate courses are expected to be offered in fall 2018. Times being considered for each course are included in the table with course descriptions below (scroll down). If you are considering taking one or more of these courses, your input on scheduling is welcome. Send questions, comments, and suggestions for avoiding conflicts to Michelle Klinger by Wednesday, April 4th. The schedule will be finalized soon after that.

Course # | Course Title | Instructor | Time |
---|---|---|---|

MATH 6110 | Real Analysis | Uraltsev, Gennady | TR 8:40-9:55 |

MATH 6210 | Measure Theory and Lebesgue Integration | Strichartz, Robert S. | TR 10:10-11:25 |

MATH 6260 | Dynamical Systems | Goncharuk, Nataliya | MWF 12:20-1:10 |

MATH 6310 | Algebra | Knutson, Allen | MWF 1:25-2:15 |

MATH 6340 | Commutative Algebra | Peeva, Irena | TR 1:25-2:40 |

MATH 6390 | Lie Groups and Lie Algebras | Kassabov, Martin | MWF 9:05-9:55 |

MATH 6520 | Differentiable Manifolds | Sjamaar, Reyer | MWF 11:15-12:05 |

MATH 6540 | Homotopy Theory | Zakharevich, Inna | TR 11:40-12:55 |

MATH 6710 | Probability Theory I | Noack, Christian | TR 1:25-2:40 |

MATH 6740 | Mathematical Statistics II | Nussbaum, Michael | MWF 1:25-2:15 |

MATH 6810 | Logic | Shore, Richard A. | TR 10:10-11:25 |

MATH 7150 | Fourier Analysis | Muscalu, Camil | TR 11:40-12:55 |

MATH 7310 | Topics in Algebra - Generation questions for finite groups |
Dennis, R. Keith | MWF 10:10-11:00 |

MATH 7350 | Topics in Homological Algebra - Simplicial methods in algebra and geometry |
Berest, Yuri | MWF 12:20-1:10 |

MATH 7410 | Topics in Combinatorics - Geometric and topological combinatorics |
Swartz, Edward | MWF 10:10-11:00 |

MATH 7510 | Berstein Seminar in Topology - Curves on surfaces | Manning, Jason | TR 8:40-9:55 |

MATH 7710 | Topics in Probability Theory - Mathematical statistical mechanics |
Sosoe, Philippe | TR 2:55-4:10 |

MATH 7740 | Statistical Learning Theory | Wegkamp, Marten | TR 2:55-4:10 |

MATH 7810 | Seminar in Logic | Nerode, Anil | T 2:55-4:10 W 4:00-5:15 |

## Course Descriptions

### MATH 6110 - Real Analysis

Forbidden Overlap: Due to an overlap in content, students will not receive credit for both MATH 6110 and MATH 6210.

Prerequisite: Strong performance in an undergraduate analysis course at the level of MATH 4140, or permission of instructor.

MATH 6110-6120 are the core analysis courses in the mathematics graduate program. MATH 6110 aims to provide a base knowledge of real analysis and integration theory. We will introduce some notions of functional analysis and probability to the extent that they are useful for a command of the main topics.

Main topics:

- Fundamentals of Measure theory and integration
- Lebesgue measure
- Properties of $L^p$ spaces
- Hilbert and Banach spaces
- Fourier series
- Comparison of measures and differentiation
- Operations on measures: theorems of Fubini and Tonelli

The following additional topics may be included in the course: Change of variables formulae; Notions from probability; The Fourier transform; Interpolation; Basics of theory of distributions; $L^p$ spaces on topological groups; Dyadic martingales.

### MATH 6210 - Measure Theory and Lebesgue Integration

Forbidden Overlap: Due to an overlap in content, students will not receive credit for both MATH 6110 and MATH 6210.

Covers measure theory, integration, and Lp spaces.

### MATH 6260 - Dynamical Systems

Topics include existence and uniqueness theorems for ODEs; Poincaré-Bendixon theorem and global properties of two dimensional flows; limit sets, nonwandering sets, chain recurrence, pseudo-orbits and structural stability; linearization at equilibrium points: stable manifold theorem and the Hartman-Grobman theorem; and generic properties: transversality theorem and the Kupka-Smale theorem. Examples include expanding maps and Anosov diffeomorphisms; hyperbolicity: the horseshoe and the Birkhoff-Smale theorem on transversal homoclinic orbits; rotation numbers; Herman's theorem; and characterization of structurally stable systems.

### MATH 6310 - Algebra

Prerequisite: strong performance in an undergraduate abstract algebra course at the level of MATH 4340, or permission of instructor.

MATH 6310-6320 are the core algebra courses in the mathematics graduate program. MATH 6310 covers group theory, especially finite groups; rings and modules; ideal theory in commutative rings; arithmetic and factorization in principal ideal domains and unique factorization domains; introduction to field theory; tensor products and multilinear algebra. (Optional topic: introduction to affine algebraic geometry.)

### MATH 6340 - Commutative Algebra with Applications in Algebraic Geometry

Commutative Algebra is the theory of commutative rings and their modules. We will cover several basic topics: localization, primary decomposition, dimension theory, integral extensions, Hilbert functions and polynomials, free resolutions. The lectures will emphasize the connections between commutative algebra and algebraic geometry.

Prerequisites: a good background in abstract algebra.

### MATH 6390 - Lie Groups and Lie Algebras

Prerequisite: an advanced course in linear algebra at the level of MATH 4310 and a course in differentiable manifolds.

The course is an introduction to Lie groups and Lie algebras and covers the basics of Lie groups and Lie algebras. Topics include real and complex Lie groups, relations between Lie groups and Lie algebras, exponential map, homogeneous manifolds and the classification of simple Lie algebras.

### MATH 6520 - Differentiable Manifolds

Prerequisite: strong performance in analysis (e.g., MATH 4130 and/or MATH 4140), linear algebra (e.g., MATH 4310), and point-set topology (e.g., MATH 4530), or permission of instructor.

This course is an introduction to geometry and topology from a differentiable viewpoint, suitable for beginning graduate students. I will largely follow the standard syllabus. The objects of study are manifolds and differentiable maps. The collection of all tangent vectors to a manifold forms the tangent bundle, and a section of the tangent bundle is a vector field. Alternatively, vector fields can be viewed as first-order differential operators. We will study flows of vector fields and prove the Frobenius integrability theorem. In the presence of a Riemannian metric, the notions of parallel transport, curvature, and geodesics are developed. We will examine the tensor calculus and the exterior differential calculus and prove Stokes' theorem. If time permits, de Rham cohomology, Morse theory, or other optional topics will be covered.

This is one of the core courses in the Cornell Ph.D. program in Mathematics, and can be used to satisfy the core course requirement in the program. The course is also suitable for students in many other Cornell Ph.D. programs, such as Physics or the Center for Applied Mathematics.

### MATH 6540 - Homotopy Theory

Prerequisite: MATH 6510 or permission of instructor.

This course is an introduction to some of the fundamentals of homotopy theory. Homotopy theory studies spaces up to homotopy equivalence, not just up to homeomorphism. This allows for a variety of algebraic techniques which are not available when working up to homeomorphism. This class studies the fundamentals and tools of homotopy theory past homology and cohomology. Topics may include computations of higher homotopy groups, simplicial sets, model categories, spectral sequences, and rational homotopy theory.

### MATH 6710 - Probability Theory I

Prerequisite: knowledge of Lebesgue integration theory, at least on the real line. Students can learn this material by taking parts of MATH 4130-4140 or MATH 6210.

A mathematically rigorous development of probability theory from a measure-theoretic perspective. Topics include: independence, the law of large numbers, Poisson and central limit theorems, and random walks.

### MATH 6740 - Mathematical Statistics II

(also STSCI 6740)

Prerequisite: MATH 6710 (measure theoretic probability) and STSCI 6730/MATH 6730, or permission of instructor.

Some familiarity with basic statistical theory is assumed, i.e. with point estimation, hypothesis testing and confidence intervals, as well as with the concepts of Bayesian and minimax decisions. The course is intended as an introduction to some modern nonparametric and Bayesian methods. The following topics will be treated: (1) a recap of Bayesian estimation, (2) empirical Bayes and shrinkage estimators; (3) unbiased estimation of risk in nonparametric estimation; (4) adaptive estimation, leading up to the study of oracle inequalities, a powerful concept which has also found applications in the related area of classification and machine learning; (5) Bayesian inference: modeling and computation, which will touch upon the use of Markov random fields for image restoration; (6) asymptotic optimality of estimators, discussing the concepts of contiguity and local asymptotic normality of statistical models.

### MATH 6810 - Logic

This course will be a basic introduction to mathematical logic. The content will, to some extent, depend on the background and interests of the students. As a common starting point, we will describe and develop a formal syntax for mathematical discourse along with precise semantics by defining both the notions of a formula in a given language and a structure for the language. The next step in our analysis is to give precise definitions of proofs and provability and to establish Gödel's Completeness theorem: A sentence is provable (from given axioms) iff it is true in every structure (which satisfies the axioms).

We will next develop some of the basic results of model theory such as the Compactness Theorem: A set *S* of sentences is consistent (i.e. does not prove a contradiction) iff it has a model (a structure in which every one of the sentences is true) iff every finite subset of *S* has a model. (Corollary: If a sentence of the appropriate language is true in all fields of characteristic 0, it is true in all fields of sufficiently large characteristic.) The Skolem-Löwenheim Theorem: If a countable set of sentences *S* has an infinite model then it has a countable one. We will also develop other connections between the forms of axioms and properties of models. Sample: If a sentence is true in one algebraically closed field of characteristic 0, e.g. in the complexes, then it is true in every algebraically closed field of characteristic 0.

The time devoted to these basic topics will depend on the background of the students. Other topics as time permits and student interest dictates will include some of the following:

- A brief development of the basic facts of recursion theory (computability theory) to the point that we can prove the undecidability of the halting problem as well as various mathematical theories especially by Interpreting one theory in another. Tarksi's result on the undefinability of truth. Church's result on the undecidability of of first order logic (validity).
- Decidability or undecidability of some mathematical theories, i.e. algorithms for determining of every sentence if it is a theorem or not.
- Gödel's Incompleteness Theorem: Given any reasonable, consistent theory
*T*containing arithmetic, there is a true sentence of arithmetic which is not provable in*T*(nor, of course, is its negation). - The basics of axiomatic set theory to ordinals and cardinals and their arithmetic.

The text for the course will possibly be Mathematical Logic by Ebbinghaus, Flum and Thomas plus supplementary material.

### MATH 7150 - Fourier Analysis

An introduction to (mostly Euclidean) harmonic analysis. Topics usually include convergence of Fourier series, harmonic functions and their conjugates, Hilbert transform, Calderon-Zygmund theory, Littlewood-Paley theory, pseudo-differential operators, restriction theory of the Fourier transform, connections to PDE. Applications to number theory and/or probability theory may also be discussed, as well as Fourier analysis on groups.

### MATH 7310 - Topics in Algebra - Generation questions for finite groups

Prerequisites: Basic group theory, e.g., MATH 4340 or MATH 6310.

About 25 years ago, Persi Diaconis asked the following question: Is it possible to develop a "K-theory of finite groups using finite generating sequences" in a fashion analogous to what is done in the classical situation?

New properties and questions about finite groups that arise from studying various approaches to answering this question will be developed. Natural connections with older work of Tarski, the Neumanns, and others will appear. Many unsolved problems about finite groups, both old and new, appear in a natural way. Diaconis' question seems to generated many interesting problems and gives an interpretation to how some of the current work in finite group theory might fit into a larger picture.

### MATH 7350 - Topics in Homological Algebra - Simplicial Methods in Algebra and Geometry

This course will be a survey of simplicial techniques in algebra and (algebraic/differential) geometry, both old and new. In the first part, we will review some of the classical topics in simplicial homological algebra (the Dold-Kan correspondence, the Eilenberg-Zilber theorem, canonical resolutions, Barr-Beck (co)triple (co)homology). In the second part, we will turn our attention to specific algebraic categories, such as simplicial groups and simplicial (commutative) algebras. (Topics may include the Kan loop group construction and an introduction to the Andre-Quillen homology). The third (main) part of the course will focus on more recent developments: we will try to give a gentle introduction to simplicial categories and DG categories which are the main technical tools in several areas of modern mathematics (derived algebraic geometry, modern representation theory and noncommutaive geometry).

Depending on the audience preferences, topics may include a survey of \infinity-categories, simplicial presheaves on manifolds, homotopy theory of DG categories and derived Hall algebras. As for prerequisites, basic knowledge of homological algebra (at the level of MATH 6350), algebraic geometry and algebraic topology (at the level of MATH 6510) will be helpful.

### MATH 7410 - Topics in Combinatorics: Geometric and Topological Combinatorics

Specific topics will be chosen the first week of the course in consultation with the class.

Past topics have ranged from recent uses of Stanley-Reisner theory, including local cohomology, to a simple introduction and applications of discrete Morse theory. Many other topics are possible, including (but not limited to) matroids, hyperplane arrangements, combinatorial aspects of cell complexes and posets, and polytopes.

Prerequisites will depend on the specific topics chosen. A minimum of a semester of undergraduate topology, algebra and combinatorics will always be necessary. Frequently additional background will be required.

### MATH 7510 - Berstein Seminar in Topology

What can we say about (or using) immersions of 1-manifolds into 2-manifolds? Perhaps surprisingly, quite a lot! Specific directions will be negotiated between the instructor and the participants, but may include:

- The geometry and topology of the curve complex, including applications to understanding the mapping class group of a surface.
- Gabai's proof of the Simple Loop Conjecture for maps between surfaces, and other forms of the Simple Loop Conjecture.
- Calegari's work on stable commutator length.
- Determining minimal intersection and self-intersection of curves on surfaces.

Since this is a Berstein seminar, almost all the presentations will be given by the participants. A student who has taken the first year courses in topology and algebra will be well-prepared for this course.

### MATH 7710 - Topics in Probability Theory - Mathematical Statistical Mechanics

In this course, several mathematical models of statistical mechanics will be studied, with particular attention given to phase transitions. We will begin by studying percolation and the Ising model, describing the phase transition in both cases, as well as giving qualitative descriptions of the ordered and disordered phases. Time permitting, we will also study further models like the random cluster model.

### MATH 7740 - Statistical Learning Theory: Classification, Pattern Recognition, Machine Learning

Prerequisite: basic mathematical statistics (MATH 6730 or equivalent) and measure theoretic probability (MATH 6710).

Learning theory has become an important topic in modern statistics. I intend to give an overview of various topics in classification, starting with Stone's (1977) stunning result that there are classifiers that are universally consistent.

Other topics include classification, plug-in methods (k-nearest neighbors), reject option, empirical risk minimization, VC theory, fast rates via Mammen and Tsybakov's margin condition, convex majorizing loss functions and support vector machines, lasso type estimators, low rank multivariate response regression, random matrix theory and current topics in high dimensional statistics.

### MATH 7810 - Seminar in Logic

A twice weekly seminar in logic. Typically, a topic is selected for each semester, and at least half of the meetings of the course are devoted to this topic with presentations primarily by students. Opportunities are also provided for students and others to present their own work and other topics of interest.