## Mathematics

## 2014-15 Tuition

$29,500## Application deadlines

Fall, Jan. 4; no spring admission## Requirements summary

- all Graduate School Requirements, including the TOEFL Exam for Non-Native English Applicants
- TOEFL scores of Writing: 20, Listening: 15, Reading: 20, and Speaking: 22
- three recommendations
- GRE general test
- GRE subject test in mathematics

## Degrees

- Ph.D.

## Subjects

- Mathematics (Ph.D.)

## Major concentrations

- mathematics

All three major subdivisions of mathematics (algebra, analysis, and geometry) are well represented at Cornell. The department is also very strong in logic, probability, statistics, numerical methods for partial differential equations, and symbolic computations, topology, and Lie theory.

Candidates are expected to obtain a broad acquaintance with the basic subjects of present-day mathematics and to be able to do research in one or more branches of mathematics. A reading knowledge of French, German or Russian must be demonstrated. Candidates must obtain some teaching experience.

Students seeking a minor in mathematics should contact the Director of Graduate Studies. A course work master's degree in computer science is available to students in the Ph.D. program in mathematics. Details are available from the graduate field office.**Application:**

Applicants must have completed the work for an undergraduate degree in mathematics. That work should have included a rigorous course in advanced calculus and real variable theory that will serve as an introduction to measure theory. The student should also have some familiarity with applications of advanced calculus and should have had courses in linear algebra and modern abstract algebra at an advanced level. Applicants are required to submit GRE general and mathematics subject test scores; scores need to be reported by January 15. Non-native English speaking applicants must also submit minimum TOEFL scores of Writing: 20, Listening: 15, Reading: 20, and Speaking: 22 (Internet-based test).

**Concentrations:**mathematics;

**Research interests:**Algebra, Combinatorics, Category Theory

**Concentrations:**mathematics;

**Research interests:**representation theory of reductive Lie groups

**Concentrations:**mathematics;

**Research interests:**geometric and algebraic combinatorics

**Concentrations:**mathematics;

**Research interests:**differential geometry; geometric analysis; nonlinear parabolic equations

**Concentrations:**mathematics;

**Research interests:**commutative and noncommutative algebra; algebraic K-theory

**Concentrations:**mathematics;

**Research interests:**elliptic partial differential equations; nonlinear elasticity and analysis

**Concentrations:**mathematics;

**Research interests:**symplectic geometry; algebraic topology; algabraic geometry

**Concentrations:**mathematics;

**Research interests:**analysis; differential equations; differential geometry

**Concentrations:**

**Research interests:**topology; lie groups; representation theory; graph theory; geometric/combinatorical group theory

**Concentrations:**mathematics;

**Research interests:**Algorithms and theoretical computer science, especially economic aspects of algorithms, online learning and its applications, random processes in networks.

**Concentrations:**mathematics;

**Research interests:**Issues at the interface of networks and information, with an emphasis on the social and information networks that underpin the Web and other on-line media.

**Concentrations:**mathematics;

**Research interests:**algebraic geometry and algebraic combinations

**Concentrations:**mathematics;

**Research interests:**computational theory; computational algebra and logic

**Concentrations:**mathematics;

**Research interests:**Geometric group theory, geometric topology

**Concentrations:**mathematics;

**Research interests:**mathematical logic; recursive functions, computer science

**Concentrations:**mathematics;

**Research interests:**applied mathematics; differential equations

**Concentrations:**mathematics;

**Research interests:**computational complexity; mathematical programming

**Concentrations:**mathematics;

**Research interests:**mathematical logic; recursion theory; set theory

**Concentrations:**mathematics;

**Research interests:**nonlinear partial differential equations and probability.

**Concentrations:**mathematics;

**Research interests:**algebraic geometry; computational algebra

**Concentrations:**mathematics;

**Research interests:**harmonic analysis; partial differential equations

**Concentrations:**mathematics;

**Research interests:**nonlinear dynamics and chaos; coupled oscillators

**Concentrations:**mathematics;

**Research interests:**combinatorics and discrete geometry

**Concentrations:**

**Research interests:**Algorithm Design and Algorithmic Game Theory. Algorithmic game theory, an emerging new area of designing systems and algorithms for selfish users. My research focuses algorithms and games on graphs or networks. I am mostly interested in designing algorithms and games that provide provably close-to-optimal results.

**Concentrations:**mathematics;

**Research interests:**number theory, automorphic forms, arithmetic geometry, ergodic theory, quantum chaos.

**Concentrations:**mathematics;

**Research interests:**applied mathematics; numerical methods; dynamic systems; nonlinear PDEs; control theory

*(Minor Member)*--

**Concentrations:**mathematics;

**Research interests:**Classification, Copula Modeling, Learning Theory, Empirical Process Theory, Machine Learning, Matrix Estimation and Completion, Model Selection and Aggregation, Nonparametric estimation

**Concentrations:**mathematics;

**Research interests:**geometric topology; infinite-dimensional topology

**Concentrations:**mathematics;

**Research interests:**Number theory, arithmetic geometry My research is in number theory and more specifically in arithmetic geometry. I like to study the family of compatible Galois representations associated to various arithmetic objects (eg. abelian varieties, modular forms, Drinfeld modules); these representations encode much, if not all, of the arithmetic of the originally object. Galois representations can also be used to study the absolute the Galois group of the rational numbers, and I have recently been playing with some applications to the Inverse Galois Problem.

### Graduate School Professors (emeritus)

**Concentrations:**mathematics;

**Research interests:**homological algebra; algebraic number theory

**Concentrations:**mathematics;

**Research interests:**complex variables; Teichmuller spaces

**Concentrations:**mathematics;

**Research interests:**functional analysis; analysis of path spaces

**Concentrations:**mathematics;

**Research interests:**statistics; confidence set theory

**Concentrations:**

**Research interests:**mathematical logic; model theory

## Learning Goals and Assessment

### Learning Goals

The PhD program in mathematics teaches you to create and communicate mathematics. The chief requirement for the doctoral degree is to complete under the guidance of an advisor a dissertation which makes an original and substantial contribution to its subject matter. You will be expected to disseminate the main results of your dissertation in the form of journal articles and conference presentations.

We do not require you to choose a specific research area or a dissertation advisor at the outset of your graduate education. The best way to make an informed choice of a research area and to make headway in it is to gain knowledge in a number of areas of mathematics. As a beginning student you will be taking various required core courses in basic subjects. Furthermore, aside from providing research training we prepare our students for careers as professional mathematicians in a variety of settings including academia, business and government. Our aim therefore is to educate flexible and broadly knowledgeable mathematicians, and to this end we offer besides the core courses a wide selection of advanced courses and seminars.

We will help you develop the oral and written communication skills expected of a professional mathematician. You will acquire these skills in part through courses and dissertation work. Active participation in our many seminars, several of which are targeted to students, is another way to improve your presentation skills and will also ease your transition from a learner to a researcher of mathematics. In addition, many practising mathematicians are involved in teaching at some level, and that is why we require every student to undergo our teaching assistant training program and to participate in the teaching mission of the mathematics department.

See also the statement on doctoral proficiencies by the Cornell Graduate School.

### Assessment

Our program has no qualifying exam; instead we assess performance in the core courses through homework, written exams, oral presentations, or term papers. Performance in the teaching program is assessed by means of student teaching evaluations.

In your first two years in the program we expect you to choose your area of specialization and to find a permanent advisor, who will chair your special committee and be your principal guide on the path to the dissertation. Before the end of your third year you will take the examination for admission to candidacy or A-exam. This is an oral exam administered by the special committee, which tests your readiness to embark on a dissertation project, your mastery of the relevant mathematics, and your oral presentation skills. After the exam your special committee will provide an assessment of your performance to the Director of Graduate Studies and you will turn in a progress report on your experiences and accomplishments in the program.

The final examination for the PhD degree, or B-exam, is an oral exam at which you present the results of your dissertation research and respond to the queries of your special committee and any other graduate faculty members who may be present. After the exam your special committee will report to the Director of Graduate Studies on the quality and originality of the research and of the written and oral presentation. You will be asked for a final report containing such items as a publication list, your first job placement, and any other feedback you may care to offer.