# Mathematics

http://www.mathematics.jhu.edu/

Mathematics, more than the fundamental language and underlying analytical structure of science and technology, is a formal way of thinking—an art that ties together the abstract structure of reason and the formal development of the logic that defines the scientific method. From the study of just how arguments and theories are formed in language and technology to the framework of quantitative and qualitative models of the natural and social sciences, mathematics is based upon the development of precise expressions, logical arguments, and the search for and exposure of pattern and structure.

The undergraduate program in the Department of Mathematics is intended both for students interested in attaining the proper preparation for graduate study and research in pure mathematics, and for students interested in using mathematics to define, properly pose, and solve problems in the sciences, engineering, and other areas. With either purpose, the focus of the program is to help those who wish to understand further the logical content, geometric meaning, and abstract reasoning of mathematics itself. A flexible program involving a broad selection of courses is a department tradition. The program begins by introducing students to the basics of algebra and mathematical analysis and then gives them the choice of exploring topics in theoretical mathematics or studying applications to physics, economics, engineering, computer science, probability, statistics, or mechanics.

The graduate program is designed primarily to prepare students for research and teaching in mathematics. It is naturally centered around the research areas of the faculty, which include algebraic geometry, algebraic number theory, differential geometry, partial differential equations, topology, several complex variables, algebraic groups, and representation theory. The program can be supplemented in applied directions by courses in theoretical physics, computer science, mechanics, probability, and statistics offered in other departments of the Krieger School of Arts and Sciences and in the Department of Applied Mathematics in the Whiting School of Engineering.

##### Facilities

The Mathematics Department resides in Krieger Hall on the Keyser Quad of Homewood. Adjacent to Krieger Hall, The University’s Milton S. Eisenhower Library has an unusually extensive collection of mathematics literature, including all the major research journals, almost all of which are accessible electronically. The stacks are open to students. The department also has a useful reference library, the Philip Hartman Library. Graduate students share departmental offices, and study space can also be reserved in the university library. Graduate students may access the department’s Linux and Windows servers, as well as computers in graduate student offices. The department also hosts numerous research seminars, special lectures, and conferences throughout the academic year.

## Undergraduate Programs

### Math Course Placement and Sequencing for All Students

There are three different versions of single variable calculus offered by the Mathematics Department, including 2 versions of semester courses in Calculus I and II, roughly equivalent to Calculus AB and BC in the College Board’s Advanced Placement (AP) system, and a single semester honors version encompassing both Calculus I and II. Students should select their first course in mathematics at JHU based on their intended areas of study, prior experience and training in mathematics, and the results of an advisory Placement Exam offered to incoming freshmen. Students intending to major in mathematics, the natural sciences, or engineering, or who are interested in studying mathematics beyond a year of single variable calculus are strongly encouraged to begin with the AS.110.108 Calculus I - AS.110.109 Calculus II (For Physical Sciences and Engineering) sequence or AS.110.113 Honors Single Variable Calculus. Students majoring in other subjects, or who do not intend to continue taking mathematics courses beyond a year of calculus, may wish to take the sequence AS.110.106 Calculus I (Biology and Social Sciences) - AS.110.107 Calculus II (For Biological and Social Science). This latter sequence relates the methods of calculus to the biological and social sciences. A one-semester pre-calculus course (AS.110.105 Introduction To Calculus) is a pre-calculus course offered for students who would benefit from additional preparation in the basic tools (algebra, trigonometry and the properties of functions) used in calculus.

Entering students may receive course credit for Calculus I or Calculus I and II on the basis of the performance level on either the (AP) or International Baccalaureate (IB) exams. All students, regardless of completion of advanced placement exams previously, must take a departmental placement exam to determine their appropriate first course in mathematics. Additional placement information can be found here.

After completing a full year of calculus, the courses AS.110.201 Linear Algebra, AS.110.202 Calculus III, or AS.110.302 Differential Equations and Applications may be taken in any order. The department offers honors courses of the former 2; AS.110.212 Honors Linear Algebra and AS.110.211 Honors Multivariable Calculus.

### Requirements for the B.A. Degree

In addition to the Requirements for a Bachelor's Degree, a candidate for the Bachelor of Arts Degree in Mathematics is required to have completed the major requirements listed below. All courses used to meet these requirements must be completed with a grade of C- or better and may not be taken satisfactory/unsatisfactory (S/U) grading scheme.

AS.110.106 | Calculus I (Biology and Social Sciences) | 4 |

or AS.110.108 | Calculus I | |

AS.110.107 | Calculus II (For Biological and Social Science) ^{****} | 4 |

or AS.110.109 | Calculus II (For Physical Sciences and Engineering) | |

or AS.110.113 | Honors Single Variable Calculus | |

AS.110.202 | Calculus III ^{*} | 4 |

or AS.110.211 | Honors Multivariable Calculus | |

AS.110.201 | Linear Algebra ^{*} | 4 |

or AS.110.212 | Honors Linear Algebra | |

AS.110.401 | Introduction to Abstract Algebra | 4 |

or AS.110.411 | Honors Algebra I | |

One Additional Algebra Course From List: | ||

AS.110.304 | Elementary Number Theory ^{*} | 4 |

or AS.110.412 | Honors Algebra II | |

or AS.110.413 | Introduction To Topology | |

or AS.110.422 | Representation Theory | |

or AS.110.423 | Lie Groups for Undergraduates | |

or AS.110.435 | Introduction to Algebraic Geometry | |

AS.110.405 | Real Analysis I ^{*} | 4 |

or AS.110.415 | Honors Analysis I | |

One Additional Analysis Course From List: | ||

Methods of Complex Analysis | ||

or AS.110.406 | Real Analysis II | |

or AS.110.407 | Honors Complex Analysis | |

or AS.110.413 | Introduction To Topology | |

or AS.110.416 | Honors Analysis II | |

or AS.110.417 | Partial Differential Equations | |

or AS.110.421 | Dynamical Systems | |

or AS.110.427 | Introduction Calculus of Variations | |

or AS.110.439 | Introduction To Differential Geometry | |

or AS.110.441 | Calculus on Manifolds | |

or AS.110.443 | Fourier Analysis | |

One 300-level or higher math course | 4 | |

Two courses in any one of the approved applications of mathematics or other courses approved by the Director of Undergraduate Studies ^{**} | 8 | |

Total Credits | 40 |

* | Majors are encouraged but not required to take honors variant. |

** | See table below for approved application courses. |

*** | AS.110.413 Introduction to Topology cannot be used for more than one requirement. |

**** | Honors Single Variable is a single 4 credit course that will count toward the major or minor in mathematics as both Calculus I and Calculus II. |

#### Approved courses in areas of application:

Physics | ||

AS.171.204 | Classical Mechanics II | 4 |

AS.171.301 | Electromagnetic Theory II | 4 |

AS.171.303 | Quantum Mechanics I | 4 |

AS.171.304 | Quantum Mechanics II | 4 |

AS.171.312 | Statistical Physics/Thermodynamics | 4 |

Chemistry | ||

AS.030.302 | Physical Chemistry II | 3 |

AS.030.345 | Chemical Applications of Group Theory | 3 |

AS.030.453 | Intermediate Quantum Chemistry | 3 |

Economics | ||

AS.180.301 | Microeconomic Theory | 4.5 |

AS.180.302 | Macroeconomic Theory | 4.5 |

AS.180.401 | Advanced Microeconomic Theory | 3 |

AS.180.334 | Econometrics | 3 |

AS.180.434 | Advanced Econometrics | 3 |

Computer Science | ||

EN.601.231 | Automata & Computation Theory | 3 |

EN.601.433 | Intro Algorithms | 3 |

EN.601.442 | Modern Cryptography | 3 |

EN.601.457 | Computer Graphics | 3 |

EN.601.461 | Computer Vision | 3 |

EN.601.464 | Artificial Intelligence | 3 |

EN.601.475 | Machine Learning | 3 |

EN.601.476 | Machine Learning: Data to Models | 3 |

EN.601.488 | Foundations of Computational Biology & Bioinformatics II | 3 |

#### Sample Program of Study

The following chart is one example of how a student might progress through the mathematics major. As potential math majors enter JHU with a wide range of prior math abilities, students should begin courses at their current level of knowledge.

Freshman | |||||
---|---|---|---|---|---|

Fall | Credits | Spring | Credits | ||

AS.110.108 | Calculus I | 4 | AS.110.109 | Calculus II (For Physical Sciences and Engineering) | 4 |

4 | 4 | ||||

Sophomore | |||||

Fall | Credits | Spring | Credits | ||

AS.110.202 or 211 | Calculus III | 4 | AS.110.201 or 212 | Linear Algebra | 4 |

AS.110.302 | Differential Equations and Applications (or other 300+ level elective) | 4 | |||

4 | 8 | ||||

Junior | |||||

Fall | Credits | Spring | Credits | ||

AS.110.405 or 415 | Real Analysis I | 4 | AS.110.406 or 416 | Real Analysis II | 4 |

Math application course | 3-4 | ||||

7-8 | 4 | ||||

Senior | |||||

Fall | Credits | Spring | Credits | ||

AS.110.401 | Introduction to Abstract Algebra | 4 | AS.110.304 | Elementary Number Theory | 4 |

Math application course | 3-4 | ||||

4 | 7-8 | ||||

Total Credits: 42-44 |

### Requirements for a Minor in Mathematics

All courses used to meet the mathematics minor requirements must be completed with a grade of C- or better and may not be taken using the S/U grading scheme. One course in the Applied Mathematics and Statistics Department (at the 300-level or above) may be substituted for one of the elective courses for the minor.

AS.110.106 | Calculus I (Biology and Social Sciences) | 4 |

or AS.110.108 | Calculus I | |

AS.110.107 | Calculus II (For Biological and Social Science) | 4 |

or AS.110.109 | Calculus II (For Physical Sciences and Engineering) | |

or AS.110.113 | Honors Single Variable Calculus | |

AS.110.202 | Calculus III | 4 |

One 200-level or above math course (excluding AS.110.202) | 4 | |

Three 300-level or above math courses | 12 | |

Total Credits | 28 |

### Honors Program in Mathematics

As a general guideline, departmental honors are awarded to recipients of the B.A. degree who have completed AS.110.411 Honors Algebra I, as well as AS.110.412 Honors Algebra II-, AS.110.415 Honors Analysis I-AS.110.416 Honors Analysis II, AS.110.407 Honors Complex Analysis and one more course at the 400-level or above with a combined grade point average of at least 3.6/4.0.

### J.J. Sylvester Prize

The J.J. Sylvester Prize in Mathematics, which carries a cash award, is given each year to the one of two top-performing graduating seniors majoring in mathematics for outstanding achievement.

### The B.A./M.A. Program

By applying some courses simultaneously toward the requirements for the Bachelor of Arts degree and a Master of Arts degree, an advanced student can qualify for both degrees during the four years of undergraduate study. Admission to the BAMA Program is by the standard graduate application form, completed during a student’s junior year of study. A current GPA of least a 3.0/4.0 is required in the 400-level mathematics courses taken while resident at the university, and at the time of application, a student must be a candidate for honors in the undergraduate degree. Students may contact the graduate program assistant for further information.

#### Undergraduate Teaching Assistantships

The department awards many upper-level undergraduates the opportunity to act as recitation instructors to our freshman courses. This award enables a student to practice the art of teaching and communicating mathematics in an environment where they are hired as a formal instructor to aid the professor of a regular curriculum course as a Teaching Assistant (TA). Undergraduate TAs are fully mentored and monitored, and the position provides a valuable credential and experience.

## Graduate Programs

#### Admission

Admission to the Ph.D. program is based on academic records, letters of recommendation, a statement of purpose, an optional personal statement, and Graduate Record Examination scores.

### Basic Program

Graduate study is centered around three core areas:

Analysis | ||

AS.110.605 | Real Variables | |

AS.110.607 | Complex Variables | |

AS.110.608 | Riemann Surfaces | |

AS.110.631 & AS.110.632 | Partial Differential Equations I and Partial Differential Equations II | |

AS.110.645 | Riemannian Geometry | |

Algebra | ||

AS.110.601 & AS.110.602 | Algebra and Algebra | |

AS.110.617 | Number Theory | |

AS.110.619 | Lie Groups and Lie Algebras | |

AS.110.643 & AS.110.644 | Algebraic Geometry and Algebraic Geometry | |

Topology | ||

AS.110.615 & AS.110.616 | Algebraic Topology and Algebraic Topology |

These 600-level graduate courses are preliminary to research upon the foundations of an undergraduate math major.

The 700-level courses are designed to bring students abreast of recent developments and to prepare them for research in the area of their choice.

### Requirements for the M.A. Degree

Although the Mathematics Department does not admit students seeking a terminal M.A. degree, students in the Ph.D. program may earn an M.A. degree. Advanced undergraduate students may also apply to be admitted to the accelerated B.A./M.A. program.

M.A. candidates must complete:

- Four graduate courses given by the Johns Hopkins Mathematics Department;
- Two additional courses at the graduate or 400-level, other than AS.110.401, AS.110.405, and AS.110.415, given by the Johns Hopkins Mathematics Department, or, with the permission of the graduate program director, graduate mathematics courses given by other departments or universities.

All courses used to satisfy the requirements must be completed with a grade of B- or better. (Advanced graduate courses completed with a grade of P can also be used to satisfy the requirements.)

### Requirements for the Ph.D. Degree

The departmental requirements for the Ph.D. degree are:

- Candidates must show satisfactory work in Algebra (AS.110.601-AS.110.602), Real Variables (AS.110.605), Complex Variables (AS.110.607), Algebraic Topology (AS.110.615), and one additional mathematics graduate course in their first year. The seminars and qualifying exam preparation course cannot be used to fulfill this requirement. The algebra and analysis requirements can be satisfied by passing the corresponding written qualifying exam in September of the first year; these students must complete at least two courses each semester. Students having sufficient background in topology can substitute an advanced topology course for AS.110.615, with the permission of the instructor.
- Candidates must pass written qualifying exams by the beginning of their second year in Analysis (Real and Complex) and in Algebra. Exams are scheduled for September and May of each academic year.
- Candidates must show satisfactory work in at least two mathematics graduate courses each semester of their second year, and, if they have not passed their oral qualifying exam, in the first semester of their third year.
- Candidates must pass a departmental oral qualifying examination in the student's chosen area of research by April 8th of the third year. The topic of the exam is chosen in consultation with the faculty member who has agreed (provisionally) to be the student's thesis advisor, who will also be involved in administering the exam.
- There is no longer a Mathematics Department foreign language requirement. With the vast majority of articles written in English, the importance of having the capability of reading another language has diminished. However, important earlier literature in certain areas of mathematics may be written in French, German, or Russian. Moreover, some articles are still being written in French. It is now at the discretion of the student's thesis advisor whether to impose a language requirement.
- Candidates must produce a dissertation based upon independent and original research.
- Candidates will gain teaching experience in mathematics as a teaching assistant for undergraduate courses. The student will be under the supervision of both the faculty member teaching the course and the director of undergraduate studies. First year students are given a reduced TA workload in the spring semester (this is related to item #2).
- After completion of the thesis research the student will defend their dissertation by means of the Graduate Board Oral Exam. The exam must be held at least three weeks before the Graduate Board deadline the candidate wishes to meet.

#### Financial Aid

Students admitted to the Ph.D. program receive teaching assistantships and full tuition fellowships. Exceptional applicants become candidates for one of the university's George E. Owen Fellowships.

### William Kelso Morrill Award

The William Kelso Morrill Award for excellence in the teaching of mathematics is awarded every spring to the graduate student who best exemplifies the traits of Kelso Morrill: a love of mathematics, a love of teaching, and a concern for students.

#### Excellence in Teaching Awards

Three awards are given each year to a junior faculty member and graduate student teaching assistants who have demonstrated exceptional ability and commitment to undergraduate education.

For current course information and registration go to https://sis.jhu.edu/classes/

### Courses

**AS.110.105. Introduction To Calculus. 4.0 Credits.**

This course starts from scratch and provides students with all the background necessary for the study of calculus. It includes a review of algebra, trigonometry, exponential and logarithmic functions, coordinates and graphs. Each of these tools will be introduced in its cultural and historical context. The concept of the rate of change of a function will be introduced. Not open to students who have studied calculus in high school.

Instructor(s): C. VanBlargan

Area: Quantitative and Mathematical Sciences.

**AS.110.106. Calculus I (Biology and Social Sciences). 4.0 Credits.**

Differential and integral calculus. Includes analytic geometry, functions, limits, integrals and derivatives, introduction to differential equations, functions of several variables, linear systems, applications for systems of linear differential equations, probability distributions. Many applications to the biological and social sciences will be discussed.

Instructor(s): B. Smithling; X. Zheng

Area: Quantitative and Mathematical Sciences.

**AS.110.107. Calculus II (For Biological and Social Science). 4.0 Credits.**

Differential and integral Calculus. Includes analytic geometry, functions, limits, integrals and derivatives, introduction to differential equations, functions of several variables, linear systems, applications for systems of linear differential equations, probability distributions. Applications to the biological and social sciences will be discussed, and the courses are designed to meet the needs of students in these disciplines.

Instructor(s): L. Sun

Area: Quantitative and Mathematical Sciences.

**AS.110.108. Calculus I. 4.0 Credits.**

Differential and integral calculus. Includes analytic geometry, functions, limits, integrals and derivatives, polar coordinates, parametric equations, Taylor's theorem and applications, infinite sequences and series. Some applications to the physical sciences and engineering will be discussed, and the courses are designed to meet the needs of students in these disciplines.

Instructor(s): Staff

Area: Quantitative and Mathematical Sciences.

**AS.110.109. Calculus II (For Physical Sciences and Engineering). 4.0 Credits.**

Differential and integral calculus. Includes analytic geometry, functions, limits, integrals and derivatives, polar coordinates, parametric equations, Taylor's theorem and applications, infinite sequences and series. Some applications to the physical sciences and engineering will be discussed, and the courses are designed to meet the needs of students in these disciplines.

Instructor(s): Y. Wang

Area: Quantitative and Mathematical Sciences.

**AS.110.113. Honors Single Variable Calculus. 4.0 Credits.**

This is an honors alternative to the Calculus sequences AS.110.106-AS.110.107 or AS.110.108-AS.110.109 and meets the general requirement for both Calculus I and Calculus II (although the credit hours count for only one course). It is a more theoretical treatment of one variable differential and integral calculus and is based on our modern understanding of the real number system as explained by Cantor, Dedekind, and Weierstrass. Students who want to know the "why's and how's" of Calculus will find this course rewarding. Previous background in Calculus is not assumed. Students will learn differential Calculus (derivatives, differentiation, chain rule, optimization, related rates, etc), the theory of integration, the fundamental theorem(s) of Calculus, applications of integration, and Taylor series. Students should have a strong ability to learn mathematics quickly and on a higher level than that of the regular Calculus sequences.

Instructor(s): A. Nakade

Area: Quantitative and Mathematical Sciences.

**AS.110.160. The Mathematics of Infinity. 1.0 Credit.**

An interdisciplinary introduction to the history of infinity in mathematics, from Zeno's paradox to the development of calculus to the crisis in the foundations of mathematics in the early 20th century. We will read about history, discuss philosophy, and learn some mathematics (including a crash course in mathematical logic and proof, building up to the rigorous definition of limits). A previous course in calculus is not required, but some mathematical maturity will be necessary.

Instructor(s): V. Lorman.

**AS.110.201. Linear Algebra. 4.0 Credits.**

Vector spaces, matrices, and linear transformations. Solutions of systems of linear equations. Eigenvalues, eigenvectors, and diagonalization of matrices. Applications to differential equations.**Prerequisites: Grade of C- or better in AS.110.107 or AS.110.109 or AS.110.113 or AS.110.202 or AS.110.302, or a 5 on the AP BC exam.**

Instructor(s): S. Tang

Area: Quantitative and Mathematical Sciences.

**AS.110.202. Calculus III. 4.0 Credits.**

Calculus of functions of more than one variable: partial derivatives, and applications; multiple integrals, line and surface integrals; Green's Theorem, Stokes' Theorem, and Gauss' Divergence Theorem.**Prerequisites: Grade of C- or better in AS.110.107 OR AS.110.109 OR AS.110.113 OR AS.110.201 OR AS.110.212 OR AS.110.302, or a 5 or better on the AP BC exam.**

Instructor(s): E. Riehl

Area: Quantitative and Mathematical Sciences.

**AS.110.211. Honors Multivariable Calculus. 4.0 Credits.**

This course includes the material in AS.110.202 with some additional applications and theory. Recommended for mathematically able students majoring in physical science, engineering, or especially mathematics. AS.110.211-AS.110.212 used to be an integrated yearlong course, but now the two are independent courses and can be taken in either order.**Prerequisites: Grade of C- or better in (AS.110.201 or AS.110.212)**

Instructor(s): X. Zheng

Area: Quantitative and Mathematical Sciences.

**AS.110.212. Honors Linear Algebra. 4.0 Credits.**

This course includes the material in AS.110.201 with additional applications and theory, and is recommended only for mathematically able students majoring in physical science, engineering, or mathematics who are interested in a proof-based version of linear algebra. This course can serve as an Introduction to Proofs (IP) course.
Prerequisites: Grade of B+ or better in 110.107 or 110.109 or 110.113, or a 5 on the AP BC exam.
Area: Quantitative and Mathematical Sciences.**Prerequisites: Grade of B+ or better in AS.110.107 or AS.110.109 or AS.110.113 or AS.110.202, or AS.110.302, or a 5 on the AP BC exam.**

Instructor(s): W. Wilson

Area: Quantitative and Mathematical Sciences.

**AS.110.225. Problem Solving Lab. 2.0 Credits.**

This course is an introduction to mathematical reason and formalism in the context of mathematical problem solving, such as induction, invariants, inequalities and generating functions. This course does not satisfy any major requirement, and may be taken more than once for credit It is primarily used as training for the William Lowell Putnam Mathematics Competition.
Area: Quantitative and Mathematical Sciences.

Instructor(s): Staff

Area: Quantitative and Mathematical Sciences.

**AS.110.302. Differential Equations and Applications. 4.0 Credits.**

This is a course in ordinary differential equations (ODEs), equations involving an unknown function of one independent variable and some of its derivatives, and is primarily a course in the study of the structure of and techniques for solving ODEs as mathematical models. Specific topics include first and second ODEs of various types, systems of linear differential equations, autonomous systems, and the qualitative and quantitative analysis of nonlinear systems of first-order ODEs. Laplace transforms, series solutions and the basics of numerical solutions are included as extra topics.
Prerequisites: Grade of C- or better in 110.107 or 110.109 or 110.113, or a 5 on the AP BC exam.
Area: Quantitative and Mathematical Sciences.**Prerequisites: Grade of C- or better in AS.110.107 or AS.110.109 or AS.110.113 or AS.110.201 or AS.110.202 or AS.110.211 or AS.110.212, or a 5 on the AP BC exam.**

Instructor(s): R. Brown

Area: Quantitative and Mathematical Sciences.

**AS.110.304. Elementary Number Theory. 4.0 Credits.**

The student is provided with many historical examples of topics, each of which serves as an illustration of and provides a background for many years of current research in number theory. Primes and prime factorization, congruences, Euler's function, quadratic reciprocity, primitive roots, solutions to polynomial congruences (Chevalley's theorem), Diophantine equations including the Pythagorean and Pell equations, Gaussian integers, Dirichlet's theorem on primes.**Prerequisites: Grade of C- or better in (AS.110.201 or AS.110.212)**

Instructor(s): J. Kong

Area: Quantitative and Mathematical Sciences.

**AS.110.311. Methods of Complex Analysis. 4.0 Credits.**

This course is an introduction to the theory of functions of one complex variable. Its emphasis is on techniques and applications, and it serves as a basis for more advanced courses. Functions of a complex variable and their derivatives; power series and Laurent expansions; Cauchy integral theorem and formula; calculus of residues and contour integrals; harmonic functions.**Prerequisites: Grade of C- or better in 110.202 or 110.211**

Instructor(s): C. Mese

Area: Quantitative and Mathematical Sciences.

**AS.110.328. Non-Euclidean Geometry. 4.0 Credits.**

For 2,000 years, Euclidean geometry was the geometry. In the 19th century, new, equally consistent but very different geometries were discovered. This course will delve into these geometries on an elementary but mathematically rigorous level.

Instructor(s): M. Merling

Area: Quantitative and Mathematical Sciences.

**AS.110.401. Introduction to Abstract Algebra. 4.0 Credits.**

An introduction to the basic notions of modern abstract algebra and can serve as as Introduction to Proofs (IP) course. This course is an introduction to group theory, with an emphasis on concrete examples, and especially on geometric symmetry groups. The course will introduce basic notions (groups, subgroups, homomorphisms, quotients) and prove foundational results (Lagrange's theorem, Cauchy's theorem, orbit-counting techniques, the classification of finite abelian groups). Examples to be discussed include permutation groups, dihedral groups, matrix groups, and finite rotation groups, culminating in the classification of the wallpaper groups.
Prerequisites: Grade of C- or better in 110.201 or 110.212
Area: Quantitative and Mathematical Sciences.**Prerequisites: Grade of C- or better in (AS.110.201 or AS.110.212)**

Instructor(s): Staff

Area: Quantitative and Mathematical Sciences.

**AS.110.405. Real Analysis I. 4.0 Credits.**

This course is designed to give a firm grounding in the basic tools of analysis. It is recommended as preparation (but may not be a prerequisite) for other advanced analysis courses. Real and complex number systems, topology of metric spaces, limits, continuity, infinite sequences and series, differentiation, Riemann-Stieltjes integration.**Prerequisites: Grade of C- or better in (AS.110.201 OR AS.110.212) AND (AS.110.202 OR AS.110.211)**

Instructor(s): Staff

Area: Quantitative and Mathematical Sciences.

**AS.110.406. Real Analysis II. 4.0 Credits.**

This course continues AS.110.405 with an emphasis on the fundamental notions of modern analysis. Sequences and series of functions, Fourier series, equicontinuity and the Arzela-Ascoli theorem, the Stone-Weierstrass theorem, functions of several variables, the inverse and implicit function theorems, introduction to the Lebesgue integral.

Instructor(s): J. Spruck

Area: Quantitative and Mathematical Sciences.

**AS.110.407. Honors Complex Analysis. 4.0 Credits.**

AS.110.407. Honors Complex Analysis. 4.00 Credits.
This course is an introduction to the theory of functions of one complex variable for honors students. Its emphasis is on techniques and applications, and can serve as an Introduction to Proofs (IP) course. Topics will include functions of a complex variable and their derivatives; power series and Laurent expansions; Cauchy integral theorem and formula; calculus of residues and contour integrals; harmonic functions, as well as applications to number theory and harmonic analysis.
Area: Quantitative and Mathematical Sciences.
This is not an Introduction to Proofs course (IP) and may not be taken as a first proof-based mathematics course except at the discretion of the instructor. This course satisfies a core requirement of the mathematics major as a second analysis course, and is a core requirement for honors in the major.**Prerequisites: AS.110.405 OR AS.110.415**

Instructor(s): B. Shiffman

Area: Quantitative and Mathematical Sciences.

**AS.110.411. Honors Algebra I. 4.0 Credits.**

An introduction to the basic notions of modern algebra for students with some prior acquaintance with abstract mathematics. Elements of group theory: groups, subgroups, normal subgroups, quotients, homomorphisms. Generators and relations, free groups, products, abelian groups, finite groups. Groups acting on sets, the Sylow theorems. Definition and examples of rings and ideals.**Prerequisites: Grade of C- or better in AS.110.212 OR AS.110.304 OR AS.110.113 OR AS.110.405 OR AS.110.415 OR AS.110.407 OR AS.110.413 OR AS.110.421**

Instructor(s): J. Specter

Area: Quantitative and Mathematical Sciences.

**AS.110.412. Honors Algebra II. 4.0 Credits.**

This is a continuation of 110.411 Honors Algebra I. Topics studies include principal ideal domains, structure of finitely generated modules over them. Introduction to field theory. Linear algebra over a field. Field extensions, constructible polygons, non-trisectability. Splitting field of a polynomial, algebraic closure of a field. Galois theory: correspondence between subgroups and subfields. Solvability of polynomial equations by radicals. Prerequisites: Grade of C- or better in 110.201 or 110.212
Area: Quantitative and Mathematical Sciences.**Prerequisites: C- or better in AS.110.411**

Instructor(s): C. Consani

Area: Quantitative and Mathematical Sciences.

**AS.110.413. Introduction To Topology. 4.0 Credits.**

Topological spaces, connectedness, compactness, quotient spaces, metric spaces, function spaces. An introduction to algebraic topology: covering spaces, the fundamental group, and other topics as time permits.**Prerequisites: Grade of C- or better in (AS.110.202 OR AS.110.211)**

Instructor(s): M. Rovelli

Area: Quantitative and Mathematical Sciences.

**AS.110.415. Honors Analysis I. 4.0 Credits.**

This highly theoretical sequence in analysis is reserved for the most able students. The sequence covers the real number system, metric spaces, basic functional analysis, the Lebesgue integral, and other topics.

Instructor(s): H. Xu

Area: Quantitative and Mathematical Sciences.

**AS.110.416. Honors Analysis II. 4.0 Credits.**

Lebesgue integration and differentiation. Elementary Hilbert and Banach space theory. Baire category theorem. Continuation of AS.110.415, introduction to real analysis.**Prerequisites: Grade of C- or better in AS.110.415**

Instructor(s): J. Bernstein

Area: Quantitative and Mathematical Sciences.

**AS.110.417. Partial Differential Equations. 4.0 Credits.**

Characteristics. classification of second order equations, well-posed problems. separation of
variables and expansions of solutions. The wave equation: Cauchy problem, Poisson's solution,
energy inequalities, domains of influence and dependence. Laplace's equation: Poisson's formula, maximum principles, Green's functions, potential theory Dirichlet and Neumann problems, eigenvalue problems. The heat equation: fundamental solutions, maximum principles.
Recommended Course Background: AS.110.405 or AS.110.415

Instructor(s): J. Luehrmann

Area: Quantitative and Mathematical Sciences.

**AS.110.421. Dynamical Systems. 4.0 Credits.**

This is a course in the modern theory of Dynamical Systems. Topic include both discrete (iterated maps) and continuous (differential equations) dynamical systems and focuses on the qualitative structure of the system in developing properties of solutions. Topics include contractions, interval and planar maps, linear and nonlinear ODE systems including bifurcation theory, recurrence, transitivity and mixing, phase volume preservation as well as chaos theory, fractional dimension and topological entropy. May be taken as an Introduction to Proofs (IP) course.
Prerequisites: Grade of C- or better in 110.201 or 110.212 OR 110.202 or 110.211 and 110.302
Area: Quantitative and Mathematical Sciences**Prerequisites: Grade of C- or better in (AS.110.201 OR AS.110.212) AND (AS.110.202 or AS.110.211) AND 110.302**

Instructor(s): R. Brown

Area: Quantitative and Mathematical Sciences.

**AS.110.422. Representation Theory. 4.0 Credits.**

This course will focus on the basic theory of representations of finite groups in characteristic zero: Schur's Lemma, Mashcke's Theorem and complete reducibility, character tables and orthogonality, direct sums and tensor products. The main examples we will try to understand are the representation theory of the symmetric group and the general linear group over a finite field. If time permits, the theory of Brauer characters and modular representations will be introduced.**Prerequisites: Grade of C- or better in (AS.110.201 OR AS.110.212) AND (AS.110.401 OR AS.110.411)**

Instructor(s): J. Morava

Area: Quantitative and Mathematical Sciences.

**AS.110.423. Lie Groups for Undergraduates. 4.0 Credits.**

This course is an introduction to Lie Groups and their representations at the upper undergraduate level. It will cover basic Lie Groups such as SU (2), U(n) , the Euclidean Motion Group and Lorentz Group. This course is useful for students who want a working knowledge of group representations. Some aspects of the role of symmetry groups in particle physics such as some of the formal aspects of the electroweak and the strong interactions will also be discussed. Recommended Course Background: AS.110.202; prior knowledge of group theory (AS.110.401) would be helpful.

Instructor(s): S. Zucker

Area: Quantitative and Mathematical Sciences.

**AS.110.427. Introduction Calculus of Variations. 4.0 Credits.**

The calculus of variations is concerned with finding optimal solutions (shapes, functions, etc.) where optimality is measured by minimizing a functional (usually an integral involving the unknown functions) possibly with constraints. Applications include mostly one-dimensional (often geometric) problems: brachistochrone, geodesics, minimum surface area of revolution, isoperimetric problem, curvature flows, and some differential geometry of curves and surfaces.
Recommended Course Background: Calculus III**Prerequisites: Grade of B+ or better in AS.110.201 and AS.110.202.**

Instructor(s): Y. Zhang

Area: Quantitative and Mathematical Sciences.

**AS.110.431. Knot Theory. 4.0 Credits.**

The theory of knots and links is a facet of modern topology. The course will be mostly self-contained, but a good working knowledge of groups will be helpful. Topics include braids, knots and links, the fundamental group of a knot or link complement, spanning surfaces, and low dimensional homology groups.

Instructor(s): C. McTague

Area: Quantitative and Mathematical Sciences.

**AS.110.433. Introduction to Harmonic Analysis and Its Applications. 4.0 Credits.**

The course is an introduction to methods in harmonic analysis, in particular Fourier series, Fourier integrals, and wavelets. These methods will be introduced rigorously, together with their motivations and applications to the analysis of basic partial differential equations and integral kernels, signal processing, inverse problems, and statistical/machine learning.
Part I : Fourier Analysis
- Normed vectors spaces. Completeness. Function spaces. Hilbert spaces.
- Fourier theory in L^2(T): The Fourier basis, Fourier series, Parseval’s theorem. Smoothness and Fourier, Sobolev spaces. Convolutions. Translation-invariant kernels. Fourier multipliers. The heat equation, and its solution with Fourier series.
_ Fourier theory in L^2(R): Riemann-Lebesgue Lemma, smoothness and Fourier.
- Local/windowed Fourier analysis, local power spectrum. Applications to sound compression, equalization and denoising.
- Fourier analysis in 2-D and higher-D. Applications to imaging: compression and denoising.
Part II : Wavelet and Multiscale Analysis
- Haar wavelet basis, multiscale decompositions. Multi-Resolution Analysis; the time-frequency plane. Construction, filters, regular wavelets.
- Linear and nonlinear approximation spaces for wavelets; approximation of singularities, local regularity.
- Wavelets in 2-D and higher-D.
- Wavelet Scattering Transform and its applications.
- Applications to sound and image compression, denoising.
Part III: Extensions and Applications
- The Fast Fourier Transform; the Fast Wavelet Transform. Compression of integral operators. Basics of finite element methods with wavelets.
- Fourier and Wavelet analysis in imaging. Denoising, a statistical analysis, Donoho-Johnstone thresholding. Sparse representations. Beyond wavelets: curvelets; adaptive dictionaries.
- Inverse problems in imaging. Regularization (and connections with sparsity). Radon transform.
- Connections with Statistical Learning Theory: Regression, curse of dimensionality, least squares, regularization, nonparametric methods. Fourier and multiscale methods.
Co-listed with EN.553.485 and EN.553.685**Prerequisites: (AS.110.201 OR AS.110.212 OR EN.550.291 OR EN.553.291) AND (AS.110.202 OR AS.110.211) AND (AS.110.405 OR AS.110.415)**

Instructor(s): M. Maggioni

Area: Quantitative and Mathematical Sciences.

**AS.110.435. Introduction to Algebraic Geometry. 4.0 Credits.**

Algebraic geometry studies zeros of polynomials in several variables and is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometric problems about these sets of zeros. The fundamental objects of study are algebraic varieties which are the geometric manifestations of solutions of systems of polynomial equations. Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with diverse fields such as complex analysis, topology and number theory.
This course aims to provide to an undergraduate student majoring in mathematics the fundamental background to approach the study of algebraic geometry by providing the needed abstract knowledge also complemented by several examples and applications.

Instructor(s): C. Consani

Area: Quantitative and Mathematical Sciences.

**AS.110.439. Introduction To Differential Geometry. 4.0 Credits.**

Theory of curves and surfaces in Euclidean space: Frenet equations, fundamental forms, curvatures of a surface, theorems of Gauss and Mainardi-Codazzi, curves on a surface; introduction to tensor analysis and Riemannian geometry; theorema egregium; elementary global theorems.**Prerequisites: Grade of C- or better in (AS.110.201 or AS.110.212) and (AS.110.202 or AS.110.211)**

Instructor(s): H. Lindblad

Area: Quantitative and Mathematical Sciences.

**AS.110.441. Calculus on Manifolds. 4.0 Credits.**

This course provides the tools for classical three-dimensional physics and mechanics. This course extends these techniques to the general locally Euclidean spaces (manifolds) needed for an understanding of such things as Maxwell's equations or optimization in higher dimensional contexts, eg. in economics. The course will cover the theory of differential forms and integration. Specific topics include Maxwell's equations in terms of 4D Lorentz geometry, vector (in particular, tangent) bundles, an introduction to de Rham theory, and Sard's theorem on the density of regular values of smooth functions. The course is intended to be useful to mathematics students interested in analysis, differential geometry, and topology, as well as to students in physics and economics.

Instructor(s): J. Morava

Area: Quantitative and Mathematical Sciences.

**AS.110.443. Fourier Analysis. 4.0 Credits.**

An introduction to the Fourier transform and the construction of fundamental solutions of linear partial differential equations. Homogeneous distributions on the real line: the Dirac delta function, the Heaviside step function. Operations with distributions: convolution, differentiation, Fourier transform. Construction of fundamental solutions of the wave, heat, Laplace and Schrödinger equations. Singularities of fundamental solutions and their physical interpretations (e.g., wave fronts). Fourier analysis of singularities, oscillatory integrals, method of stationary phase.**Prerequisites: Grade of C- or better in (AS.110.201 OR AS.110.212 ) AND ( AS.110.202 OR AS.110.211)**

Instructor(s): S. Tang

Area: Quantitative and Mathematical Sciences.

**AS.110.446. Introduction to Statistical Learning, Data Analysis and Signal Processing. 4.0 Credits.**

Introduction to high dimensional data sets: key problems in statistical and machine learning. Geometric aspects. Principal component analysis, linear dimension reduction, random projections. Concentration phenomena: examples and basic inequalities. Metric spaces and embeddings thereof. Kernel methods. Nonlinear dimension reduction, manifold models.
Regression. Vector spaces of functions, linear operators, projections. Orthonormal bases; Fourier and wavelet bases, and their use in signal processing and time series analysis. Basic approximation theory. Linear models, least squares. Bias and variance tradeoffs, regularization. Sparsity and compressed sensing. Multiscale methods.
Graphs and networks. Random walks on graphs, diffusions, page rank. Block models. Spectral clustering, classification, semi-supervised learning. Algorithmic and computational aspects of the above will be consistently in focus, as will be computational experiments on synthetic and real data.
Linear algebra will be used throughout the course, as will multivariable calculus and basic probability (discrete random variables). Basic experience in programming in C or MATLAB or R or Octave.
7Recommended Course Background: More than basic programming experience in Matlab or R; some more advanced probability (e.g. continuous random variables), some signal processing (e.g. Fourier transform, discrete and continuous).
Co-listed with EN.553.416**Prerequisites: AS.110.201**

Instructor(s): M. Maggioni

Area: Quantitative and Mathematical Sciences.

**AS.110.503. Undergraduate Research in Mathematics. 0.0 - 4.0 Credits.**

Instructor(s): R. Brown.

**AS.110.586. Independent Study. 0.0 - 4.0 Credits.**

Instructor(s): J. Morava.

**AS.110.587. DRP Independent Study. 1.0 Credit.**

Directed Reading Program (DRP) Independent Study

Instructor(s): D. Fuentes-Keuthan

Area: Quantitative and Mathematical Sciences.

**AS.110.595. Internship. 1.0 Credit.**

**AS.110.601. Algebra. **

An introductory graduate course on fundamental topics in algebra to provide the student with the foundations for number theory, algebraic geometry, and other advanced courses. Topics include group theory, commutative algebra, Noetherian rings, local rings, modules, rudiments of category theory, homological algebra, field theory, Galois theory, and non-commutative algebras.

Instructor(s): V. Shokurov

Area: Quantitative and Mathematical Sciences.

**AS.110.602. Algebra. **

An introductory graduate course on fundamental topics in algebra to provide the student with the foundations for Number Theory, Algebraic Geometry, and other advanced courses. Topics include group theory, commutative algebra, Noetherian rings, local rings, modules, and rudiments of category theory, homological algebra, field theory, Galois theory, and non-commutative algebras. Recommended Course Background: AS.110.401-AS.110.402

Instructor(s): C. Consani

Area: Quantitative and Mathematical Sciences.

**AS.110.605. Real Variables. **

Measure and integration on abstract and locally compact spaces (extension of measures, decompositions of measures, product measures, the Lebesgue integral, differentiation, Lp-spaces); introduction to functional analysis; integration on groups; Fourier transforms.

Instructor(s): J. Bernstein

Area: Quantitative and Mathematical Sciences.

**AS.110.607. Complex Variables. **

Analytic functions of one complex variable. Topics include Mittag-Leffler Theorem, Weierstrass factorization theorem, elliptic functions, Riemann-Roch theorem, Picard theorem, and Nevanlinna theory. Recommended Course Background: AS.110.311, AS.110.405

Instructor(s): H. Xu.

**AS.110.608. Riemann Surfaces. **

Abstract Riemann surfaces. Examples: algebraic curves, elliptic curves and functions on them. Holomorphic and meromorphic functions and differential forms, divisors and the Mittag-Leffler problem. The analytic genus. Bezout's theorem and applications. Introduction to sheaf theory, with applications to constructing linear series of meromorphic functions. Serre duality, the existence of meromorphic functions on Riemann surfaces, the equality of the topological and analytic genera, the equivalence of algebraic curves and compact Riemann surfaces, the Riemann-Roch theorem. Period matrices and the Abel-Jacobi mapping, Jacobi inversion, the Torelli theorem. Uniformization (time permitting).

Instructor(s): H. Xu.

**AS.110.612. Complex geometry. **

Instructor(s): B. Shiffman.

**AS.110.615. Algebraic Topology. **

Polyhedra, simplicial and singular homology theory, Lefschetz fixed-point theorem, cohomology and products, homological algebra, Künneth and universal coefficient theorems, Poincaré and Alexander duality theorems.

Instructor(s): N. Kitchloo.

**AS.110.616. Algebraic Topology. **

Polyhedra, simplicial and singular homology theory, Lefschetz fixed-point theorem, cohomology and products, homological algebra, Künneth and universal coefficient theorems, Poincar&ecute; and Alexander duality theorems.

Instructor(s): E. Riehl

Area: Quantitative and Mathematical Sciences.

**AS.110.617. Number Theory. **

Topics in advanced algebra and number theory, including local fields and adeles, Iwasawa-Tate theory of zeta functions and connections with Hecke’s treatment, semisimple algebras over local and number fields, adeles geometry.

Instructor(s): C. Consani

Area: Quantitative and Mathematical Sciences.

**AS.110.618. Number Theory. **

Topics in advanced algebra and number theory, including local fields and adeles, Iwasawa-Tate theory of zeta-functions and connections with Hecke's treatment, semi-simple algebras over local and number fields, adele geometry.

Instructor(s): D. Savitt.

**AS.110.619. Lie Groups and Lie Algebras. **

Lie groups and Lie algebras, classification of complex semi-simple Lie algebras, compact forms, representations and Weyl formulas, symmetric Riemannian spaces.**Prerequisites: AS.110.402**

Instructor(s): C. Mese; J. Morava

Area: Quantitative and Mathematical Sciences.

**AS.110.631. Partial Differential Equations I. **

An introductory graduate course in partial differential equations. Classical topics include first order equations and characteristics, the Cauchy-Kowalewski theorem, Laplace’s equations, heat equation, wave equation, fundamental solutions, weak solutions, Sobolev spaces, maximum principles.**Prerequisites: Grade of C- or better in AS.110.605**

Instructor(s): Y. Sire.

**AS.110.632. Partial Differential Equations II. **

An introductory graduate course in partial differential equations. Classical topics include first order equations and characteristics, the Cauchy-Kowalevski theorem, Laplace's equation, heat equation, wave equation, fundamental solutions, weak solutions, Sobolev spaces, maximum principles. The second term focuses on special topics such as second order elliptic theory.

Instructor(s): J. Luehrmann.

**AS.110.633. Harmonic Analysis. **

Fourier multipliers, oscillatory integrals, restriction theorems, Fourier integral operators, pseudodifferential operators, eigenfunctions. Undergrads need instructor's permission.

Instructor(s): C. Sogge

Area: Quantitative and Mathematical Sciences.

**AS.110.635. Microlocal Analysis. **

Microlocal analysis is the geometric study of singularities of solutions of partial differential equations. The course will begin by introducing the geometric theory of (Schwartz) distributions: Fourier transform and Sobolev spaces, pseudo-differential operators, wave front set of a distribution, elliptic operators, Lagrangean distributions, oscillatory integrals, method of stationary phase, Fourier integral operators. The second semester will develop the theory and apply it to special topics such as asymptotics of eigenvalues/eigenfunctions of the Laplace operator on a Riemann manifold, linear and non-linear wave equation asymptotics of quantum systems, Bochner-Riesz means, maximal theorems.

Instructor(s): H. Lindblad.

**AS.110.637. Functional Analysis. **

Instructor(s): F. Lu.

**AS.110.643. Algebraic Geometry. **

Affine varieties and commutative algebra. Hilbert’s theorems about polynomials in several variables with their connections to geometry. General varieties and projective geometry. Dimension theory and smooth varieties. Sheaf theory and cohomology. Applications of sheaves to geometry; e.g., the Riemann-Roch theorem. Other topics may include Jacobian varieties, resolution of singularities, geometry on surfaces, connections with complex analytic geometry and topology, schemes.

Instructor(s): X. Zheng.

**AS.110.644. Algebraic Geometry. **

Affine varieties and commutative algebra. Hilbert's theorems about polynomials in several variables with their connections to geometry. General varieties and projective geometry. Dimension theory and smooth varieties. Sheaf theory and cohomology. Applications of sheaves to geometry; e.g., the Riemann-Roch Theorem. Other topics may include Jacobian varieties, resolution of singularities, geometry on surfaces, schemes, connections with complex analytic geometry and topology.

Instructor(s): V. Shokurov

Area: Quantitative and Mathematical Sciences.

**AS.110.645. Riemannian Geometry. **

Differential manifolds, vector fields, flows, Frobenius’ theorem. Differential forms, deRham’s theorem, vector bundles, connections, curvature, Chern classes, Cartan structure equations. Riemannian manifolds, Bianchi identities, geodesics, exponential maps. Geometry of submanifolds, hypersurfaces in Euclidean space. Other topics as time permits, e.g., harmonic forms and Hodge theorem, Jacobi equation, variation of arc length and area, Chern-Gauss-Bonnet theorems.

Instructor(s): C. Mese.

**AS.110.675. High-Dimensional Approximation, Probability, and Statistical Learning. **

The course covers fundamental mathematical ideas for certain approximation and statistical learning problems in high dimensions. We start with basic approximation theory in low-dimensions, in particular linear and nonlinear approximation by Fourier and wavelets in classical smoothness spaces, and discuss applications in imaging, inverse problems and PDE’s. We then introduce notions of complexity of function spaces, which will be important in statistical learning. We then move to basic problems in statistical learning, such as regression and density estimation. The interplay between randomness and approximation theory is introduced, as well as fundamental tools such as concentration inequalities, basic random matrix theory, and various estimators are constructed in detail, in particular multi scale estimators. At all times we consider the geometric aspects and interpretations, and will discuss concentration of measure phenomena, embedding of metric spaces, optimal transportation distances, and their applications to problems in machine learning such as manifold learning and dictionary learning for signal processing.

Instructor(s): M. Maggioni

Area: Quantitative and Mathematical Sciences.

**AS.110.707. Functional Analysis. **

Instructor(s): B. Dodson.

**AS.110.711. Topics in Topos Theory. **

Reading course to discuss Topics in Topos Theory

Instructor(s): E. Riehl.

**AS.110.712. Topics in Mathematical Physics. **

Instructor(s): H. Lindblad.

**AS.110.722. Topics in Homotopy Theory. **

The course will focus on recent developments in homotopy theory, such as Galois theory for E_n (n \geq 2) ring-spectra, and on connections with number theory; in particular, work of Bhatt, Hesselholt, Lurie, Scholze and others on topological Hochschild homology and its applications to geometry over the p-adic complex numbers.

Instructor(s): E. Riehl

Area: Quantitative and Mathematical Sciences.

**AS.110.724. Topics in Arithmetic Geometry. **

Topics around the subject of Arithmetic Geometry will be covered in this course.

Instructor(s): B. Smithling

Area: Quantitative and Mathematical Sciences.

**AS.110.725. Topics in Analysis: Nonlinear Dispersive Equations. **

**AS.110.726. Topics in Analysis. **

The topics covered will involve the theory of calculus of Functors applied to Geometric problems like Embedding theory. Other related areas will be covered depending on the interest of the audience.

Instructor(s): Y. Sire

Area: Quantitative and Mathematical Sciences.

**AS.110.727. Topics in Algebraic Topology. **

Instructor(s): N. Kitchloo.

**AS.110.728. Topics in Algebraic Topology. **

Instructor(s): N. Kitchloo.

**AS.110.731. Topics in Geometric Analysis. **

Instructor(s): Y. Wang.

**AS.110.733. Topics In Alg Num Theory. **

Instructor(s): D. Savitt.

**AS.110.735. Topics In Hodge Theory. **

Instructor(s): C. Mese; R. Brown; S. Zucker.

**AS.110.737. Topics Algebraic Geometry. **

Instructor(s): V. Shokurov.

**AS.110.738. Topics Algebraic Geometry. **

Introduction to toric varieties. This class is a general introduction to toric varieties. Toric varieties are special kinds of algebraic varieties which can be described by lattices and convex sets. They provide a rich source of concrete examples in complex geometry or mathematical physics. If time permits, we discuss in the end the stability of toric embeddings. Students should know basic notions of algebraic geometry (schemes, sheaves, linear systems), as covered in AS.110.643.

Instructor(s): C. Mese; R. Brown; V. Shokurov

Area: Quantitative and Mathematical Sciences.

**AS.110.742. Topics In Partial Differential Equations. **

In this course we will be discussing some dispersive evolution equations, primarily the nonlinear Schrodinger equation. Topics will include well - posedness theory, conservation laws, and scattering. The course will be accessible to students who have not taken graduate partial differential equations or functional analysis.

Instructor(s): J. Spruck.

**AS.110.749. Topics in Differential Geometry. **

In this class, we will study Aaron Naber and Jeff Cheeger's recent result on proving codimension four conjecture. We plan to talk about some early results of the structure on manifolds with lower Ricci bound by Cheeger and Colding. We will prove quantitative splitting theorem, volume convergence theorem, and the result that almost volume cone implies almost metric cone. Then we will discuss regularity of Einstein manifolds and the codimension four conjecture.

Instructor(s): J. Bernstein

Area: Quantitative and Mathematical Sciences.

**AS.110.755. Topics in Fluid Dynamics. **

Graduate students only.

Instructor(s): J. Bernstein

Area: Quantitative and Mathematical Sciences.

**AS.110.756. Topics in Algebra. **

This will be a course in commutative algebra. Topics may include: Noetherian rings and modules, the Nullstellensatz, Hilbert basis theorem, localization, integrality, Noether normalization, primary decomposition, DVRs, Dedekind domains, dimension theory, smoothness and regularity, and homological methods.

Instructor(s): C. Consani

Area: Quantitative and Mathematical Sciences.

**AS.110.761. Topics in Topology. **

Instructor(s): N. Kitchloo

Area: Quantitative and Mathematical Sciences.

**AS.110.764. Topics in Riemannian Geometry. **

Topics courses are restricted to graduate students only.

Instructor(s): J. Bernstein.

**AS.110.790. Seminar in Complex Geometry. **

Presentations of current research papers by faculty, graduate students and invited guest speakers. For graduate students only.

Instructor(s): B. Shiffman; H. Xu

Area: Quantitative and Mathematical Sciences.

**AS.110.791. Seminar in Analysis and Partial Differential Equations. **

Presentations of current research papers by faculty, graduate students and invited guest speakers. For graduate students only.

Instructor(s): J. Bernstein

Area: Quantitative and Mathematical Sciences.

**AS.110.793. Seminar in Topology. **

For graduate students only. Presentations of current research papers by faculty, graduate students and invited guest speakers.

Instructor(s): N. Kitchloo

Area: Quantitative and Mathematical Sciences.

**AS.110.794. Seminar in Category Theory. **

Presentations of current research papers by faculty, graduate students and invited guest speakers. For graduate students only.

Instructor(s): E. Riehl

Area: Quantitative and Mathematical Sciences.

**AS.110.795. Data Science Seminar. **

Presentations of current research papers by faculty, graduate students and invited guest speakers. For graduate students only.

Instructor(s): M. Maggioni

Area: Quantitative and Mathematical Sciences.

**AS.110.798. Seminar in Number Theory. **

Presentations of current research papers by faculty, graduate students and invited guest speakers. For graduate students only.

Instructor(s): B. Smithling.

**AS.110.799. Seminar in Algebraic Geometry. **

For graduate students only. Presentations of current research papers by faculty, graduate students and invited guest speakers.

Instructor(s): X. Zheng.

**AS.110.800. Independent Study-Graduates. **

Instructor(s): J. Murphy

Area: Quantitative and Mathematical Sciences.

**AS.110.801. Thesis Research. **

Instructor(s): Staff.

### Cross Listed Courses

#### Applied Mathematics & Statistics

**EN.553.738. High-Dimensional Approximation, Probability, and Statistical Learning. 3.0 Credits.**

The course covers fundamental mathematical ideas for certain approximation and statistical learning problems in high dimensions. We start with basic approximation theory in low-dimensions, in particular linear and nonlinear approximation by Fourier and wavelets in classical smoothness spaces, and discuss applications in imaging, inverse problems and PDE’s. We then introduce notions of complexity of function spaces, which will be important in statistical learning. We then move to basic problems in statistical learning, such as regression and density estimation. The interplay between randomness and approximation theory is introduced, as well as fundamental tools such as concentration inequalities, basic random matrix theory, and various estimators are constructed in detail, in particular multi scale estimators. At all times we consider the geometric aspects and interpretations, and will discuss concentration of measure phenomena, embedding of metric spaces, optimal transportation distances, and their applications to problems in machine learning such as manifold learning and dictionary learning for signal processing.

Instructor(s): M. Maggioni.

For current faculty and contact information go to http://www.mathematics.jhu.edu/people/

### Faculty

#### Chair

David Savitt

Number theory, Galois representations.

#### Professors

Caterina Consani

Arithmetic geometry, number theory, and non-commutative geometry.

Nitu Kitchloo

Symplectic geometry, topology of Kac-Moody groups, classical algebraic topology

Hans Lindblad

Harmonic analysis, PDE, fluid dynamics, relativity.

Chikako Mese

Geometric analysis.

Mauro Maggioni

Bloomberg Distinguished Professor; Harmonic Analysis and Signal Processing, Approximation Theory, Statistical and Machine Learning, Stochastic Processes, Spectral Graph Theory, Numerical Linear Algebra.

Bernard Shiffman

Several complex variables, complex geometry.

Vyacheslav V. Shokurov

Algebraic geometry.

Yannick Sire

Harmonic analysis, real geometry, complex geometry.

Christopher Sogge

J.J. Sylvester Professor; Fourier analysis, partial differential equations.

Joel Spruck

J.J. Sylvester Professor; Partial differential equations, geometric analysis.

W. Stephen Wilson

Algebraic topology, homotopy theory.

Steven Zucker

Hodge theory, algebraic geometry.

#### Associate Professors

Jacob Bernstein

Minimal surface theory, mean curvature flow.

Ben Dodson

Partial differential equations, harmonic analysis.

#### Assistant Professors

Fei Lu

Malliavin Calculus and stochastic partial differential equations, data-driven model reduction and data assimilation.

Emily Riehl

Homotopy theory.

Brian Smithling

Arithmetic, algebraic geometry.

Yi Wang

Geometric analysis, nonlinear partial differential equations

#### Associate Teaching Professor

Richard Brown

Director of Undergraduate Studies; Dynamical systems, low-dimensional topology.

#### Emeriti

J. Michael Boardman

Differential topology, algebraic topology.

Jack Morava

Algebraic topology, mathematical physics.

Takashi Ono

Algebra, number theory, algebraic groups.

#### J.J.Sylvester Assistant Professor

Christian Gavrus

Partial differential equations, harmonic analysis

Jingjun Han

Algebraic geometry

Jonas Luehrmann

Partial differential equations, mathematical physics.

Joel Specter

Number theory

Liming Sun

Geometric analysis, partial differential equations

Hang Xu

Complex geometry.

Valentin Zakharevich

Algebraic topology, quantum field theory

Xudong Zheng

Algebraic geometry.

#### Associate Research Scientist/Lecturer

Jian Kong

IT Senior Lecturer; Algebraic geometry.

#### Assistant Research Professor

Sui Tang

Applied harmonic analysis, mathematical signal processing.

Stefano Vigogna

Machine Learning, Harmonic Analysis.

#### Joint Appointments

Gregory Eyink

Professor (Applied Mathematics); Mathematical physics, fluid mechanics, turbulence, dynamical systems.