# Fall 2017 Graduate Courses

The following is a tentative schedule of graduate courses for fall 2017. Course descriptions are included below the table. If you see any potential conflicts for yourself, please contact Mikki Klinger at mmk8@cornell.edu by Tuesday, April 4th.

**Note about course renumbering:** MATH 6150 - Partial Differential Equations (formerly MATH 6190) and MATH 6840 - Recursion Theory (formerly MATH 7840) will appear in the class roster and be available for pre-enrollment under the old course number. All systems will update to the new course number some time in July.

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

MATH 6110 | Real Analysis | Goncharuk | MWF 1:25-2:15 | MLT224 |

MATH 6150 | Partial Differential Equations (formerly MATH 6190, see note above) | Strichartz | TR 1:25-2:40 | MLT206 |

MATH 6210 | Measure Theory and Lebesgue Integration | Kudryashov | MWF 12:20-1:10 | MLT205 |

MATH 6310 | Algebra | Templier | MW 8:40-9:55 | MLT203 |

MATH 6340 | Commutative Algebra with Applications in Algebraic Geometry | Peeva | TR 1:25-2:40 | MLT230 |

MATH 6390 | Lie Groups and Lie Algebras | Speh | TR 8:40-9:55 | MLT205 |

MATH 6520 | Differentiable Manifolds | Connelly | TR 11:40-12:55 | MLT205 |

MATH 6530 | K-Theory and Characteristic Classes | Zakharevich | MWF 10:10-11:00 | MLT206 |

MATH 6640 | Hyperbolic Geometry | Manning | TR 8:40-9:55 | MLT206 |

MATH 6670 | Algebraic Geometry | Knutson | TR 11:40-12:55 | MLT206 |

MATH 6710 | Probability Theory I | Sosoe | TR 10:10-11:25 | MLT207 |

MATH 6740 | Mathematical Statistics II | Nussbaum | MWF 11:15-12:05 | MLT230 |

MATH 6840 | Recursion Theory (formerly MATH 7840, see note above) | Shore | TR 10:10-11:25 | MLT206 |

MATH 7110 | Topics in Analysis: Calculus of Variations | Healey | TR 2:55-4:10 | MLT230 |

MATH 7510 | Berstein Seminar in Topology: an introduction to Riemann Surfaces | Kassabov | MWF 11:15-12:05 | MLT206 |

MATH 7670 | Topics in Algebraic Geometry: Intersection Theory | Stillman | TR 10:10-11:25 | MLT205 |

MATH 7740 | Statistical Learning Theory: Classification, Pattern Recognition, Machine Learning |
Wegkamp | TR 2:55-4:10 | MLT205 |

MATH 7810 | Seminar in Logic | Shore | T 2:55-4:10 and W 4:00-5:15 |
MLT206 |

MATH 7850 | Topics in Logic: Dynamics of Large Groups | Solecki | MWF 12:20-1:10 | MLT206 |

### MATH 6110 - Real Analysis

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. 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: (1) Fundamentals of Measure theory and integration; (2) Lebesgue measure; (3) Properties of L^p spaces; (4) Hilbert and Banach spaces; (5) Fourier series; (6) Comparison of measures and differentiation; (7) Operations on measures: theorems of Fubini and Tonelli.

The following topics may be included in the course: change of variables formulae, notions from probability, the Fourier transform, interpolation, basics of theory of distributions.

The main course textbook is: Stein, Shakarchi - Real analysis. The course will include material also from the following textbooks: Rudin, Real and Complex analysis; Stein Shakarchi - Functional analysis; Stein, Shakarchi - Fourier Analysis.

### MATH 6150 - Partial Differential Equations

Prerequisite: MATH 4130, MATH 4140, or the equivalent, or permission of instructor. Offered alternate years.

We will mostly follow the book "Partial Differential Equations" by L.C. Evans. I intend to cover in detail representation formulas for linear PDEs [Chapter 2] and the basic theory for first-order non-linear equations (starting with the method of characteristics and concentrating on Hamilton-Jacobi PDEs and conservation laws) [Chapter 3]. Time permitting, we will also discuss assorted topics from Chapter 4 (similarity solutions, transform methods, asymptotics, power series, homogenization), Chapter 8 (calculus of variations), and/or Chapter 10 (control theoretic interpretation of Hamilton-Jacobi PDEs). There will be class discussions based on readings from the book and questions.

### 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.

The Riemann integral familiar from undergraduate calculus has poor convergence properties and does not behave well in higher dimensions. A much more convenient and flexible theory of integration, based on the notion of a countably additive measure, was developed by Henri Lebesgue. In this course we develop Lebesgue's theory from the ground up.

### 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

Prerequisites: A good background in abstract algebra.

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, free resolutions. The lectures will emphasize connections between commutative algebra and algebraic geometry.

### 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.

MATH 6510-MATH 6520 are the core topology courses in the mathematics graduate program. MATH 6520 covers smooth manifolds and PL manifolds and functions, tangent and cotangent bundle, submanifolds and normal bundles, embedding and approximation, flows, Lie derivative, foliations and Frobenius theorem, tensors, differential forms, exterior derivative, integration, de Rham cohomology.

### MATH 6530 - K-Theory and Characteristic Classes

Prerequisite: MATH 6510, or permission of instructor.

An introduction to topological K-theory and characteristic classes. Topological K-theory is a generalized cohomology theory which is surprisingly simple and useful for computation while still containing enough structure for proving interesting results. The class will begin with the definition of K-theory, Chern classes, and the Chern character. Additional topics may include the Hopf invariant 1 problem, the J-homomorphism, Stiefel-Whitney classes and Pontrjagin classes, cobordism groups and the construction of exotic spheres, and the Atiyah-Singer Index Theorem.

### MATH 6640 - Hyperbolic Geometry

Prerequisite: MATH 6510 or permission of instructor.

An introduction to the topology and geometry of hyperbolic manifolds. The class will begin with the geometry of hyperbolic $n$-space, including the upper half-space, Poincaré disc, and Lorentzian models. Particular attention will be paid to the cases $n=2$ and $n=3$. Hyperbolic structures on surfaces will be parametrized using Teichmüller space, and discrete groups of isometries of hyperbolic space will be discussed. Other possible topics include the topology of hyperbolic manifolds and orbifolds; Mostow rigidity; hyperbolic Dehn filling; deformation theory of Kleinian groups; complex and quaternionic hyperbolic geometry; and convex projective structures on manifolds.

### MATH 6670 - Algebraic Geometry

Prerequisite: MATH 6310 or MATH 6340, or equivalent.

A first course in algebraic geometry. Affine and projective varieties. The Nullstellensatz. Schemes and morphisms between schemes. Dimension theory. Potential topics include normalization, Hilbert schemes, curves and surfaces, and other choices of the instructor.

### 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.

Classical topics in probability theory, from a measure theoretic point of view: law of large numbers, central limit theorem, random walks, Markov chains and, time permitting, martingales. The emphasis will be on a rigorous development of the basics, but we will attempt when possible to make contact with applications like statistical mechanics, and statistics.

We will not closely follow any textbook. The main references for the material are R. Durrett, Probability: Theory and Examples and D. Williams: Probability with Martingales. Texts for further reading include: J. Norris: Markov Chains, and. D. Stroock: Probability: An Analytic View.

### MATH 6740 - Mathematical Statistics II

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 6840 - Recursion Theory

A first course in the theory of computability. We will assume some background in logic. MATH 6810 or CS 6820 should be more than sufficient.

The pace and content of the course will depend on the background of the students. Plausible outlines are as follows:

We will begin with a brief discussion of the basic properties of a reasonable model of computability: universal machines, the enumeration, s-m-n and recursion theorems, r.e. (effectively or computably enumerable) sets and the halting problem. Next will come the notions of relative computability, the Turing jump operator and the arithmetical hierarchy. Then there will be some development of construction procedures for non-r.e. sets, in particular, the Kleene-Post finite extension method (really Cohen forcing in arithmetic). An example or two of other forcing type constructions such as with trees (perfect set forcing) to construct a minimal degree may also be presented later.

At this point there are two likely scenarios.

One will concentrate on the recursively (computably) enumerable sets and degrees. The primary text will then the old Recursively Enumerable Sets and Degrees by R. I. Soare or the new version Turing Computability, Theory and Applications of Computability. The heart of the course will then be the development of various kinds of priority arguments for the construction of r.e. sets including finite and infinite injury as well as tree arguments. We will use these methods to analyze the structure of the (Turing) degrees of r.e. sets and something of their set theoretic structure as well.

The second scenario will instead study the structure of the Turing degrees of all sets and functions as well as important substructures such as the degrees below 0' (the Halting problem) and the degrees of the arithmetic sets (those definable in first order arithmetic or equivalently computable form some finite iteration of the Turing jump). The primary techniques will be forcing arguments in the setting of arithmetic rather than set theory. We begin with the development of construction procedures such as the Kleene-Post finite extension method (now seen as Cohen forcing in arithmetic) and minimal degree constrictions by forcing with trees (perfect set forcing), forcing with Pi-0-1 classes (closed sets) and others.

Relations with rates of growth and the jump hierarchy will be explored. We will prove the basic results about the complexity of theories of these structures such as that the theories of the degrees and the degrees below 0' are of the same complexity as second order arithmetic and first order arithmetic, respectively. We will also study the restrictions on possible automorphisms of the structures and definability results: which apparently external (but natural) relations on the structures can be defined internally. In particular, we may reach the proof that the Turing jump which captures quantification in arithmetic is definable in terms of relative computability alone.

In either case, connections between degree theoretic and other properties such as types of approximations, rates of growth and complexity of definition will be considered. Notes will also be provided.

### MATH 7110 - Topics in Analysis: Calculus of Variations

Prerequisite: MATH 6160 (formerly MATH 6200), or equivalent.

As time permits: The direct method for scalar and vector-valued minimization problems, constraints, existence theorems in nonlinear elasticity, Gamma convergence with applications. We will use books by Evans, Dacorogna, Ciarlet and Attouch et. al., as sources.

### MATH 7510 - Berstein Seminar in Topology: An Introduction to Riemann Surfaces

This Berstein seminar will cover an introduction to Riemann surfaces and will cover the interplay between analysis and topology. The goal of the course is to explain the main results in the compact case, i.e., Riemann-Roch and Hodge decompositions. The students will be expected to give presentations during the course.

### MATH 7670 - Topics in Algebraic Geometry

Prerequisites: some knowledge of algebraic geometry, preferably some schemes, sheaves, cohomology.

This is an intermediate course in algebraic geometry, centered around enumerative geometry and intersection theory of mostly non-singular varieties. Many examples, and applications of the general apparatus of algebraic geometry, will be given. In some ways, the central value of this course is seeing algebraic geometry in action. On the other hand, enumerative geometry is a beautiful field, full of many interesting examples and techniques. (Many of these methods are implemented in Macaulay2 as well, so more in depth examples are possible, and we will include such examples).

Textbook: *3264 and all that, Intersection theory in Algebraic Geometry*, by Eisenbud and Harris. This is a wonderful book on intersection theory and intermediate algebraic geometry. Each Chapter has a wealth of examples, applications, and exercises.

Potential topics include (these are roughly the chapters of Eisenbud-Harris):

1. Chow groups -- rational equivalence, proper pushforward

2. First examples, including intersections on smooth surfaces.

3. Introduction to Grassmannians and lines in projective 3-space

4. Grassmannians in general, Schubert classes

5. Vector bundles and Chern classes -- linearizing non-linear behavior

6. Example: Lines on hypersurfaces (e.g. 27 lines on a cubic surface)

7. Singular elements of a linear system (applications include Pluecker formulas)

8. Parameter spaces, e.g.: complete conics, Kontsevich space.

9. Chow rings and projective bundles

10. Segre classes and varieties of linear spaces

11. Contact bundles: Tangent bundles of projective bundles (application: contact bundles)

12. Degeneracy loci of maps of vector bundles, Porteous' formula

13. Excess intersection, Chow ring of a blowup

14. Groethendieck-Riemann-Roch theorem and applications

The book (and also this course) is organized such that there are interesting explicit enumerative geometry problems that will be solved in each chapter. The amount that we cover of these topics will depend on the background and interests of those taking the course. We cannot cover all of these topics, we will pick and choose.

Prerequisites: The textbook best describes the prerequisites, but here is a short version: (a) An undergraduate course in algebraic geometry (e.g. "An invitation to algebraic geometry" (Smith et al, 2000)). (b) An acquaintance with the language of schemes (e.g. first 3 chapters of "Geometry of Schemes" (Eisenbud and Harris, 2000)). (c) An acquaintance with coherent sheaves and their cohomology.

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

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

The course aims to present the developing interface between machine learning theory and statistics. Topics include an introduction to classification and pattern recognition; the connection to nonparametric regression is emphasized throughout. Some classical statistical methodology is reviewed, like discriminant analysis and logistic regression, as well as the notion of perception which played a key role in the development of machine learning theory. The empirical risk minimization principle is introduced, as well as its justification by Vapnik-Chervonenkis bounds. In addition, convex majoring loss functions and margin conditions that ensure fast rates and computable algorithms are discussed. Today's active high-dimensional statistical research topics such as oracle inequalities in the context of model selection and aggregation, lasso-type estimators, low rank regression and other types of estimation problems of sparse objects in high-dimensional spaces are presented.

### MATH 7810 - Seminar in Logic

The logic seminar will meet twice a week for 75 minutes each time. One meeting will, generally, be devoted to participants and visitors speaking about their current reading or research. The other day will be devoted to participants lecturing on a single topic.

We expect the topic for this semester will be Computable Structure Theory and we will use a draft of a new book being written by Antonio Montalbán. It represents a new view of the subject and makes connections with several different areas.

A description by the author: ”The objective of this book is to describe some of the main ideas and techniques of the field so that graduate students and researchers can use it for their own research. The author has worked on the subject for many years, and along the years he has not only obtained new results, but also developed new frameworks for presenting old ideas. The author's real motivation is to have these frameworks presented in a single monograph where the reader can see how they all fit together currently they are all scattered around different papers, using different notations, and even different definitions. Having the right framework to explain an idea or a construction should help to understand what is underneath it; this was the main objective of the author when developing his way of exposing the subject. Here are a few examples of these new frameworks: a new definition of Scott rank which interacts better with notions of categoricity, Borel complexity, and descriptive complexity; a new way of doing Ash-style -priority arguments using true stages; a new way of a dealing with computations among relations on a structure, including jumps on relations (these are new expositions of old ideas by Moschovakis, Ershov, and Soksov); a new way of dealing with generic presentations of structures (similar, but not equivalent, to that developed by Knight and Slaman and Manasse); the treatment of "natural" behaviors by considering properties "on a cone" (Martin measure); etc.”

Which parts we cover will be based on what is written by fall and the interests and background of the participants.

### MATH 7850 - Topics in Logic: Dynamics of Large Groups

The course will focus on continuous actions of non-locally compact groups that are equipped with a complete separable topology (Polish groups). This class of groups includes the unitary group of the separable Hilbert space, homeomorphism groups of compact metric spaces, automorphism groups of countable structures, and the group of measure preserving transformations, among others. The emphasis of the course will be on the structure of orbit equivalence relations and on the structure of continuous actions on compact spaces (flows). In addition to the general introduction to the subject, we will cover the following topics: (1) The Kechris-Louveau theorem identifying a fundamental obstacle for an equivalence relation to be an orbit equivalence relation. (2) Effros-type theorems on embedding of the Vitali equivalence relation into orbit equivalence relations. (3) Universal minimal flows.