# Courses

Find our course schedules, syllabi and exams, and course notes below.

For the most up-to-date courses, visit CUNY's Dynamic Course Schedule.

### General Introductory Courses

Unless otherwise stated, the following courses are 45 hours plus conferences, 4.5 cr. [70000 level courses meet twice a week, 1 hour and 15 minutes per session.]

MATH 70100-70200 Functions of a Real Variable

MATH 70300-70400 Functions of a Complex Variable

MATH 70500-70600 Algebra

MATH 70700-70800 Topology

MATH 70900-71000 Differential Geometry

MATH 71100-71200 Logic

### Special Topics Courses

A variety of courses will be offered in special areas, number of credits and hours to be announced when scheduled. All independent research courses carry variable credits. Appropriate prerequisites will be indicated for each course when it is given. [80000 level courses meet either once a week, 2 hours per session (3 cr.), or twice a week, 1 hour and 15 minutes per session (4.5 cr.)]

Research for the doctoral thesis requires permission of a faculty supervisor and approval by the Executive Officer.

MATH 89901 Independent Research in Analysis

MATH 89902 Independent Research in Algebra and Number Theory

MATH 89903 Independent Research in Geometry and Topology

MATH 89904 Independent Research in Logic

MATH 89905 Independent Research in Applied Math

MATH 90000 Dissertation Supervision (1 credit)

Hours and credits to be announced when given.

Courses in this number series are intended to serve as an introduction to mathematical research and will be focused on problems at a level of difficulty suitable for qualified first-year graduate students. Permission of the instructor is required.

## Courses

*All classes are scheduled for in person only at the Graduate Center. See CUNYFirst for updated room assignments. Short descriptions of the course are shown if provided by the instructor.***MATH 70100: Functions of a Real Variable **

T & Th, 9:30 a.m. - 11:00 a.m., Room 6417

Prof. T. Kucherenko

4.5 cr.

Course Description: The course is the first installment of the year-long course in Real Analysis. We will cover the following topics: the real number system (the least upper bound property, binary and ternary expansions, countable and uncountable sets ); elementary topology (compactness, connectedness, metric spaces, complete metric spaces, metrizable spaces, the topology of n-dimensional Euclidean space); normed spaces (sequence spaces, function spaces, various notions of convergence); and differentiation.**MATH 70500: Algebra **

M & W, 9:30 a.m. - 11:00 a.m., Room 6417

Prof. O. Kharlampovich

4.5 cr.

Outline:

Topics

1) Group Theory: Quotient Groups and Homomorphisms, Group Actions, Direct and Semi direct Products and Abelian Groups, Introduction to Nilpotent, Solvable and Free groups.

(2) Ring Theory: Introduction to Rings, Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains, Rings of Polynomials (Chapters 7, 8, 9).

(3) Introduction to Modules

Text: *Abstract Algebra*, D. Dummit and R. Foote (Third Edition)

Recommended: *Algebra*, Hungerford**MATH 70700: Topology **

T & Th

T: 11:45 a.m. - 1:15 p.m Room 6494

Th: 11:45 a.m. - 1:15 p.m., Room 6417

Prof. J. Behrstock

4.5 cr.

Outline: This is a first semester graduate course in Topology. We will start with a review of point set topology,

and discuss basic properties of topological spaces, including compactness, connectedness and separation properties. Most of the course will then be spent introducing techniques ways in which algebraic invariants arise in topology, and applications thereof. This latter part will cover the fundamental group of a space, the theory of covering space, and end with a brief introduction to homology groups. Along the way we will explore many examples.

Text: Allen Hatcher, Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0.

Additional (optional) texts for point-set topology: John Hocking and Gail Young, Topology and/or Munkres, Topology

Topics: Metric spaces, topological spaces, continuity, Hausdorff condition, compactness, connectedness, product spaces, quotient spaces. Fundamental group, covering spaces. Homology.**MATH 70900: Differential Geometry **

W & F, 2:00 p.m. - 3:30 p.m., Room 6421

Prof. S. Preston

4.5 cr.

Course Description: This is the first part of a two-semester course. Topics include smooth manifolds, tangent bundles, vector fields, Lie groups, tensors, differential forms, Stokes theorem, connections, and Riemannian curvature. Some books that cover these topics: 1) Michael Spivak, A Comprehensive Introduction to Differential Geometry, 2) Jeffrey M. Lee, Manifolds and Differential Geometry, 3) John M. Lee, Introduction to smooth manifolds, and 4) course notes from the professor that will be distributed in class.**MATH 71600: Modern Harmonic Analysis **

F, 11:45 a.m. - 1.45 p.m., Room 5382

Prof. A. Mayeli

3 cr.

Course Description: This course is designed for students who have a solid foundation in real analysis. We will cover the following subjects:

1) Hilbert and Banach space operator theory fundamentals: This part will be used as a perquisite for the rest of the class.

2) Applications of operator theory to bridge the gap between mathematics, engineering, and data science. Time-frequency concentration problems and the distribution of eigenvalues of time-frequency limiting operators; frames (e.g. wavelets, Gabor bases), and the associated operators; applications of frames and wavelets in signal processing, such as studying the behavior of signals (e.g., time series and stock market signals), detecting jump points using wavelets, and using wavelets as activation functions in neural networks.

3) An introduction to the Whitney Extension Problem and current breakthroughs in Sobolev spaces, with applications to data interpolation and finding smooth solutions.

4) Introduction to Fuglede Conjecture (in Euclidean spaces and finite Abelian groups), Fourier transform on finite Abelian groups, and connections to geometric combinatorics, group theory (Galois theory) and number theory.

**MATH 82530: Understanding Three Space from Several Perspectives**

T, 11:00 a.m. - 1:00 p.m., Room 6417

Prof. D. Sullivan

3 cr.**MATH 83100: Hyperbolic Dynamics **

W, 11:15 a.m. - 1:15 p.m., Room 6417

Prof. E. Pujals

3 cr.

Course Description: This is an introduction to the mathematical theory of chaotic dynamical systems. Topics to be covered:

- Linear hyperbolic systems. Main examples: toral automorphisms, Smale horseshoe, solenoid, geodesic flows on surfaces
- with negative curvature. Hyperbolic sets. Uniformly hyperbolic systems. Structural stability. Attractors and physical measure. More advanced topics (to be selected).

**MATH 84300: Topics in Statistics and Data Science **

M, 11:45 a.m. - 1:45 p.m., Room 4419

Prof. S. Chatterjee

3 cr.

Course Description: This course introduces the fundamental concepts and mathematical methods used in data science, modern statistics, and machine learning, including the description and theoretical analysis of several current algorithms, their theoretical basis, and associated mathematical frameworks. Many of the algorithms that will be discussed have been successfully used in various areas of real-world products and services. The main topics that we aim to cover (if time permits) are:

- Probability tools, concentration inequalities
- Regression problems and algorithms
- Ranking problems and algorithms
- Clustering and classification problems
- Support vector machines (SVMs)
- Spectral graph theory and applications
- Optimization in machine learning
- Probabilistic models: key concepts and examples

**MATH 87000: Topics in Algebraic Number Theory **

Th, 2:00 p.m. - 4:00 p.m., Room 6417

Prof. V. Kolyvagin

3 cr.

Course Description: The accent in the course will be on arithmetic of elliptic curves snd curves of genus 1 over number fields. Main topics will include: congruent numbers, elliptic curves, Mordell-Weil theorem. L-function of an elliptic curve, modular parametrization, The Birch-Swinnerton-Dyer conjecture-rank part. Problem of existence of a rational point on a curve of genus 1, the case of a local field, the case of a global field, Hasse principle. Main homogeneous spaces over an elliptic curve. The Shafarevich-Tate group of an elliptic curve. The full Birch-Swinnerton-Dyer conjecture. It is supposed that the students have algebra course. There is no particular textbook to be used, but useful literature will be recommended.**MATH 87200: Group Theory**

T, 2:00 p.m. - 4:00 p.m., Room 6496

Prof. V. Shpilrain

3 cr.

Course description. The course will focus on algorithms in combinatorial group theory and their complexity, thus providing a bridge between group theory and complexity theory, the core area of theoretical computer science. We will address complexity of classical algorithms like Nielsen's and Whitehead's algorithms in free groups (complexity of the latter algorithm is still unknown), as well as of more modern ones including algorithms for solving the word problem for braid groups, Thompson's group, hyperbolic groups, etc.

In addition to the "traditional" worst-case complexity of group-theoretic algorithms, we will study their average-case complexity (i.e., the expected runtime) and generic-case complexity (i.e., complexity on random inputs). This will require introducing a probability measure on infinite groups and thereby will connect group theory to probability theory, particularly through random walks on groups.

The course is self-contained.**MATH 87400: Topics in Number Theory **

M, 2:00 p.m. - 4:00 p.m., Room 5417

Prof. A. Gamburd

3 cr.**MATH 89901.03: Research Seminar **

M, 4:15 p.m. - 6:15 p.m., Room 6417

Team Taught

1 cr.

Course description: All graduate students without a formal advisor* are required to attend, but even those with an advisor are welcome! The schedule is here. The 1 credit Research Seminar is geared towards introducing graduate students to different areas of mathematics and possible advisors. Each week a different member of the Graduate Center Mathematics Department faculty will discuss a topic that is accessible to all graduate students. Faculty members are strongly discouraged from attending. The formal presentation is 4:15pm - 5pm with informal discussion until 6:15pm.

*Students formally establish an advisor by providing the Math Office with the Oral Syllabus form signed by student and advisor.

*All classes are scheduled for in person only at the Graduate Center. See CUNYFirst for updated room assignments.*

**MATH 70200: Functions of a Real Variable []**

M & F, 11:45 a.m.- 1.15 p.m., Room 6417

Prof. L-P. Arguin

4.5 cr.

We will cover the following subjects: measure theory and the Lebesgue integral, product measures and Fubini's theorem, the convergence theorems, differentiation and the Radon-Nikodym theorem, L^p-spaces, Hilbert and Banach spaces, Fourier series, elements of functional analysis. Most subjects required for the qualifying exams will be covered. Connections with probability theory, complex analysis and functional analysis will be highlighted.

**MATH 70400: Functions of a Complex Variable []**

M & W, 9:30 a.m. - 11:00 a.m., Room 6417

Prof. D. Saric

4.5 cr.

This is second course in Complex Analysis. We will continue the study of complex variables where the first course finished. The plan is to cover most topics from Saeed Zakeri’s book A course in Complex Analysis chapters 8-11. The rest of the course will cover special topics of interest to students.

We will cover Weierstrass’s Theorem, Jensen’s formula, entire functions of finite order, Mittag-Leffler’s theorem, Elliptic functions, Rational approximation, Analytic continuations, Bloch’s theorem, Picard’s theorem, conformal maps of finitely connected domains. In the rest of the course we will give an elementary introduction to quasiconformal maps, Riemann surfaces, hyperbolic geometry and Teichmuller spaces.

**MATH 70800: Topology []**

T & Th, 11:45 a.m. - 1.15 p.m., Room 5417

Prof. J. Maher

4.5 cr.

This is the second part of the graduate Topology sequence. We will learn techniques and applications of algebraic topology to study topological spaces. Topics covered will include simplicial and singular homology, homotopy invariance, exact sequences and excision, cellular homology, homology with coefficients, axioms for homology, cohomology of spaces, cup products, Poincare duality and products. We will cover additional topics if time permits. In addition to rigorous proofs, we will do lots of examples and computations. The text book will be Algebraic Topology by Hatcher, and we will cover chapters 2 and 3, plus additional topics as time permits.

**MATH 70910: Research Seminar []**

M, 4:15 p.m. - 6:15 p.m., Room 6417

Team Taught

1 cr.

All graduate students without a formal advisor* are required to attend, but even those with an advisor are welcome!

View the current seminar schedule

The 1 credit Research Seminar is geared towards introducing graduate students to different areas of mathematics and possible advisors. Each week a different member of the Graduate Center Mathematics Department faculty will discuss a topic that is accessible to all graduate students. Faculty members are strongly discouraged from attending.

The formal presentation is 4:15pm - 5pm with informal discussion until 6:15pm.

*Students formally establish an advisor by providing the Math Office with the Oral Syllabus form signed by student and advisor.

**MATH 71200:** **Logic I []**

M & W, 6:30 p.m. - 8:00 p.m., Room 6417

Prof. R. Miller

4.5 cr.

Math 712 is the spring-semester half of the yearlong logic sequence Logic I. It presents Kurt Gödel's Completeness and Incompleteness Theorems in full detail, and then spends three to four weeks introducing the students to computability theory and another three to four weeks introducing them to set theory. (Broadly speaking, the four main subdisciplines within logic are computability theory, model theory, proof theory, and set theory.) For the first part, we will follow Chapters 2 and 3 of Enderton's text *A Mathematical Introduction to Logic*. The following two segments will refer intermittently to the texts *Recursively Enumerable Sets and Degrees*, by Soare, and *Set Theory: an Introduction to Independence Proofs*, by Kunen; but students should not need to purchase these two texts. In concert with Math 711, this course provides full preparation for the Math Program's qualifying exam in logic.

Naturally, the normal prerequisite for Math 712 is Math 711, the first half of the sequence. The content of 711 is mainly model theory. However, not all of this content will be necessary for 712. Some background in model theory will be assumed, but many students may have adequate preparation from a decent undergraduate logic course, or may be able to pick it up on their own before 712 begins. Of course, the ideal plan is to take 711 in the fall and then 712 the following spring (and then, most likely, the logic qualifying exam). However, students who missed 711 should not consider themselves ineligible for 712, especially given that the 711-712 sequence will not be offered again until the 2023-24 academic year.

**MATH 82000: Comparison Geometry []**

Th, 9:30 a.m. - 11:00 a.m., Room 6417

Prof. R. Bettiol

3 cr.

In this course, we will develop several modern techniques in Geometric Analysis and Differential Geometry, focusing on the interplay between curvature, topology, and global shape on smooth manifolds. The unifying theme is to understand the geometric (and sometimes topological) effects of curvature bounds using many different types of tools, from PDEs to the triangle inequality, from convexity of distance functions to representation theory and optimization. Various open research problems will be discussed.

Main results to be covered include the Bochner technique, the theorems of Rauch and Toponogov for sectional curvature, of Bishop-Gromov for Ricci curvature, and their applications such as the Diameter Sphere Theorem, the Soul Theorem, and Gromov's bounds on generators of the fundamental group.

I will prepare lecture notes throughout the semester, but some general references on this subject are:

- Comparison Theorems in Riemannian Geometry, by J. Cheeger and D. Ebin
- Notes on Comparison Theorems in Riemannian Geometry, by J. Eschenburg
- Toponogov’s theorem and applications, by W. Meyer
- Riemannian Geometry (3rd ed), by P. Petersen

**MATH 82600**: **Topology and geometry in low dimensions []**

T, 11:00 a.m. - 1:00 p.m., Room 6417

Prof. D. Sullivan

3 cr.

The spatial continuum was discretized by Poincare' (1900) giving birth to combinatorial topology

which developed into algebraic topology and combinatorial manifold geometry during the last century.

Now these ideas can be usefully combined with computer algorithms related to fluid dynamics.

**MATH 83200: Research topics in dynamical systems []**

W, 11:15 a.m. - 1:15 p.m., Room 6417

Prof. E. Pujals

3 cr.

This course will cover different (self-contained) areas of dynamical systems introducing specific research topics. They include:

- Dynamics of rational maps
- Dynamics of translation surfaces
- Renormalization of Polygon exchange maps
- The use of transfer operators to understand ergodic properties.

Each topic is self-contained, and each of them will include 3 to 4 lectures.

**MATH 83900: Symbolic Dynamical Systems []**

W, 2:00 p.m. - 4:00 p.m., Room 6417

Prof. C. Wolf

3 cr.

The course will provide a graduate level introduction to Symbolic Dynamical Systems (SDS). SDS are an important research topic in the modern theory of dynamical systems in part since they provide a powerful tool to model the topological properties of a wide range of dynamical systems. Moreover, SDS have applications in other areas including statistical physics, complexity theory, graph theory, ergodic theory, information theory and computability. The aim of this course is two-fold: First to give a thorough introduction to the classical objects of SDS including sub-shifts of finite type, sofic shifts, coded shifts, factors, entropy and information. Second, the course will elaborate on selected applications of SDS including applications to the thermodynamic formalism, computability theory and complexity theory.

**MATH 87000: Intro in Algebraic Number Theory []**

Th, 2:00 p.m. - 4:00 p.m., Room 6417

Prof. V. Kolyvagin

3 cr.

The main purpose of the course is to study basics of Algebraic Number Theory. In particular, it would provide a ground for further more advanced study.

The center theme will be study of theory of divisibility in rings of algebraic integers ( divisor theory ) - far reaching generalization of the main theorem of arithmetic about uniqueness of decomposition in primes of a rational number.

One of the goals of Algebraic Number Theory is getting applications to solving diophantine equations, it is one of its origins as well.

Historically, Kummer work on Fermat’s equation x(power l)+y(power l)=z(power l) over the cyclotomic field K(l), generated over the field of rational numbers by l-th roots of unity, was very important for development of the theory.

We will study arithmetic of cyclotomic fields and Fermat’s equations over them as a motivation for development of theory of algebraic numbers and as a nice example of how it works.

One year of algebra course is recommended for students to attend.

Book to be used - Borevich, Shafarevich “Number Theory.

**MATH 87800:** **Topics in number theory []**

M, 2:00 p.m. - 4:00 p.m., Room 5417

Prof. A. Gamburd

3cr.

**MATH 88100: Free groups and their automorphisms []**

F, 9:30 a.m. - 11:30 a.m., Room 4419

Prof. I. Kapovich

3 cr.

Free groups are universal algebraic objects of fundamental importance in group theory, topology, logic and other areas of mathematics. The course will discuss the construction and algebraic properties of free groups, with an emphasis on the group Out(Fn) of outer automorphisms of a free group. The study of the group Out(Fn) is informed in large part by the parallels with the study of mapping class groups of hyperbolic surfaces. We will discuss both the similarities and the differences between Out(Fn) and mapping class groups and develop the theory necessary for exploring these connections, particularly the Culler-Vogtmann Outer space, its Thurston-type boundary, and train track maps representing free group automorphisms.

We will use this machinery to obtain algebraic, geometric and dynamical information about Out(Fn) and free group automorphisms.

**MATH 89905.02: ITS Tutorial Series []**

T, 1:30 p.m. - 3:00 p.m., Room 5209

Prof. J. Terilla & others

1 cr.

The Initiative for Theoretical Science (ITS) hosts a symposium series organized around mathematical and math-adjacent theoretical science, such as high energy and quantum physics, statistical physics and chaotic systems, neuroscience and biophysics, machine learning, and quantum information theory.

In advance of the symposia, people have an opportunity to interact with the material in an elementary way at tutorial sessions, usually held on Tuesdays 1:30pm in room 5209. The tutorial topics will be announced as the Fall symposium schedule develops and speakers and visitor dates are confirmed. Students who participate in the ITS tutorials can earn credit by registering for this course. Credit will be awarded P/F based on attendance.

*All classes are scheduled for in person only at the Graduate Center. See CUNYFirst for updated room assignments.*

**MATH 70100: Functions of a Real Variable [41000]**

T &Th, 9:30am - 11:00am, Room TBA

Prof. S. Chatterjee

4.5 cr.

**MATH 70300: Functions of a Complex Variable [40998]**

M & W, 11:45am – 1:45PM, Room TBA

Prof. D. Aulicino

4.5 cr.

**MATH 70700: Topology [40997]**

M & W, 2:00pm – 4:00pm, Room TBA

Prof. J. Terilla

4.5 cr.

**MATH 71100:** **Logic I [40996]**

W & F 10:00am – 11:30am, Room TBA

Prof. P. Rothmaler

4.5 cr.

**MATH 82530**: **3D topology and fluid dynamics [40995]**

T, 11:15am - 12:45pm, Room TBA

Prof. D. Sullivan

3 cr.

**MATH 83100: Introduction to Renormalization [40994]**

W, 11:15am - 1:15pm, Room TBA

Prof. E. Pujals

3 cr.

**MATH 85600: Intro. to Partial Differential Equations [xxxxx]**

Th, 11:45am -1:45pm, Room TBA

Prof. V. Martinez

3 cr.

**MATH 86600: Structures on Manifolds [40993]**

F, 2:00pm - 4:00pm, Room TBA

Prof. S Wilson

3 cr.

**MATH 87000: Topics in Algebraic Number Theory [52588]**

Th, 2:00pm - 4:00pm, Room TBA

Prof. V. Kolyvagin

3 cr.

**MATH 87700:** **Intro. to Algebraic Geometry [40999]**

M, 9:30am - 11:30am, Room TBA

Prof. A. Obus

3cr.

**MATH 89901.03: Research Seminar**

M, 4:15pm - 6:15pm, Room TBA

Team Taught

1 cr.

**MATH 89905.02: Independent Research – ITS Tutorial Series**

T, 1:30pm-3:00pm, Room 5209

Prof. J. Terilla

1 cr.

*Classes will be in person or hybrid (online & in person) as indicated. In person dates for Hybrid classes are specified by clicking on the blue highlighted Course description provided by the instructor.*

**Math 70400: Functions of a Complex Variable [62221]**

M & W, 2:00pm-3:30pm, In Person, Rm.6417

Prof. S. Mitra

4.5 cr.

**Math 70600: Algebra II [62223]**

M & W, 12:00pm-1:30pm, In Person, Rm.6417

Prof. O. Kharlampovich

4.5 cr.

**Math 70910: Research Seminar [62224]**

M, 4:15pm-6:15pm, In Person, Rm.6417

Team Taught

1 cr.

**Math 71000: Differential Geometry I [62226]**

W: 9:30am-11:00am, In Person, Rm.6417

F: 2:00pm-3:30pm, In Person, Rm.6417

Prof. L. Fernandez

4.5 cr.

**Math 71200: Logic I [62228]**

T & Th, 4:15pm-5:45pm, In Person, Rm.TBA

Prof. G. Fuchs

4.5 cr.

**Math 82600: Topology and Geometry in Low Dimensions [62353]**

T, 11:00am-1:00pm, Hybrid, Rm.6417

Prof. D. Sullivan

3 cr.

**Math 83200: Introduction to Conservative and Symplectic Dynamics [62229]**

W, 11:15am-1:15pm, In Person, Rm.5417

Prof. E. Pujals

3 cr.

**Math 83600: Probability II [62359]**

Th, 9:30am-11:30am, Hybrid, Rm.6417

Prof. S. Chatterjee

3 cr.

**Math 85600: Intro. to Functional Analysis [62379]**

Fri, 9:30am-11:30am, Hybrid, Rm.5417

Prof. A. Mayeli

3 cr.

**Math 87000: Topics in Algebraic Number Theory [62387]**

Th, 2:00pm-4:00pm, Hybrid, Rm.6417

Prof. V. Kolyvagin

3 cr.

**Math 87100: Knots and 3-Manifolds [62241]**

Th, 11:45am-1:45pm, In Person, Rm.6417

Prof. A. Champanerkar

3 cr.

**Math 87800: Topics in Number Theory [62393]**

M, 2:00pm-4:00pm, Hybrid, Rm.5417

Prof. A., Gamburd

3 cr.

## Syllabi and Exams

## Course Notes

**Prof. Abhijit Champanekar**:**Topology I****Prof. Joel David Hamkins**:**Mathematical Logic****Prof. Jeremy Khan**: Teichmuller Theory and Dynamical Systems**Prof. Olga Kharlampovich**:**Group Theory****Prof. Ilya Kofman: Knot Theory****Prof. Joseph Malkevitch**:**Geometry;****Game Theory (and Fairness); Mathematical Modeling****Prof. Alexey Ovchinnikov**:**Differential Algebra****Prof. Yunping Jiang**:**Dynamical Systems**