Spring 2026
M 392C - Open Problems in Topology (Miller)
We will go through the motivation, background, and current status of many open problems in low-dimensional topology. We will discuss interesting (potentially easier) subquestions, what strategies seem natural, and why analogous arguments fail. Primary sources will be the K2 and K3 problem lists, which cover topics related to knot theory, surfaces, 3-manifolds, 4-manifolds, and general dimension (with low-dimensional motivation).
M 392C - Representation Theory (Mason-Brown)
This course will serve as an introduction to the (infinite-dimensional) representation theory of reductive Lie groups. One exciting aspect of this subject is that it meaningfully interacts with an extraordinary spectrum of mathematics, including harmonic analysis, number theory, algebraic geometry, operator algebras, and noncommutative algebra. In this course, we will develop the analytic and algebraic aspects of the theory in parallel, culminating in a proof of the Langlands classification. The prerequisites for this course include abstract algebra and basic representation theory, though exposure to manifolds and Lie groups would be very helpful. Grades will be determined by final projects exploring related topics.
M 393C - Foundational Techniques in Machine Learning (Kileel)
This course will be a mathematically rigorous introduction to topics from linear algebra, high-dimensional probability, optimization and statistics, which are foundational tools for data science, or the science of making predictions from structured data. A secondary aim of the course is for students to gain experience in exploring data science problems through computer programming.
M 393C - Kinetic Plasma Models (Gamba)
Info coming soon.
M 393C - Mathematics in Deep Learning (Tsai)
Info coming soon.
M 393C - Mathematical Fluid Dynamics: Existence, Uniqueness, Stability (Vishik)
Info coming soon.
M 393C - Partial Differential Equations II (Wynter)
M 394C - Stochastic Analysis (Sirbu)
Info coming soon.
Fall 2025
M 392C - Riemann Surfaces (Perutz)
Riemann surfaces, complex manifolds of dimension one, occupy a central place in differential and algebraic geometry and in complex function theory. This is a setting where one can prove deep theorems, using methods of analysis of linear PDE, of sheaf theory, and of algebraic geometry, that extend to higher dimensions, but only at the cost of far more foundational effort and less transparency; hence this is an excellent setting to build intuition. It is also an area distinguished by beautiful examples. One such is Klein’s quartic curve, a compact Riemann surface described by a simple algebraic equation, whose symmetry group, the simple group of order 168, is the largest of any genus 3 Riemann surface; this Riemann surface can alternatively be understood as a modular curve, the home for modular forms of level 7, and can be pictured as a certain tiling of the hyperbolic disc.
This course will start with the basic theory, and with many concrete examples and constructions (algebraic curves, branched covers, elliptic curves as tori and as cubic curves, modular forms, etc.). It will also cover some of the main theorems: existence of non-trivial meromorphic functions on compact Riemann surfaces, the Riemann-Roch theorem, projective embeddings of compact Riemann surfaces; the Hodge decomposition and Serre duality; uniformization. These results can all be proven by the same, potential-theoretic approach, involving the Laplace and Poisson equations, giving a taste of the methods of geometric analysis. The sheaf-theoretic approach, and more advanced topics, will be covered as time allows. The main text will be Riemann Surfaces by S.K. Donaldson, though I will also draw on more algebraic sources.
Regular student presentations and problem classes will be built in.
Prerequisites: complex analysis (as in the prelim) and basic topology up to the fundamental group are essential. Familiarity with the Algebraic/Differential Topology prelims (e.g. homology, differential forms) will be very useful. Some exposure to functional analysis, as in the Methods of Applied Math prelim sequence, is also helpful.
M 393C - Renorm Group Methods in Mathematical Physics (Chen)
M 393C - Partial Differential Equations I (Patrizi)
M 393C - Predictive Machine Learning (Bajaj)
The Fall course this year is on the design and performance analysis of optimally controlled, aka reinforcement leaned statistical machine learning algorithms, trained, verified and validated on filtered, noisy observation data distributions collected from various multi-scale dynamical systems. The principal performance metrics will be on online and energy efficient training, verification and validation protocols that achieve principled and stable learning for maximal generalizability . The emphasis will be on possibly corrupted data and/or the lack of full information for the learned stochastic decision making dynamic algorithmic process. Special emphasis will also be given to the underlying mathematical and statistical physics principles of free-energy and stochastic Hamiltonian flow dyamics . Students shall thus be exposed to the latest stochastic machine learning modeling approaches for optimized decision-making, multi-player games involving stochastic dynamical systems and optimal stochastic control. These latter topics are foundational to the training of multiple neural networks (agents) both cooperatively and in adversarial scenarios to optimize the learning process of all the agents.
An initial listing of lecture topics and reference material are given in the syllabus below. This is subject to some modification, given the background and speed at which we cover ground. Homework exercises shall be given almost bi-weekly. Assignment solutions that are turned in late shall suffer a 10% per day reduction in credit and a 100% reduction once solutions are posted. There will be a mid-term exam in class. The exam content will be similar to the homework exercises. A list of topics will also be assigned as take-home final projects to train the best of scientific machine-learned decision-making (agents). The projects will involve modern ML programming, an oral presentation, and a written report submitted at the end of the semester.
This project shall be graded and be in lieu of a final exam.
The course is open to graduate students in all disciplines. Those in the 5-year master's program students, and in the CS, CSEM, ECE, MATH, STAT, PHYS, CHEM, and BIO, are welcome. You’ll need an undergraduate level background in the intertwined topics of algorithms, data structures, numerical methods, numerical optimization, functional analysis, algebra, geometry, topology, statistics, stochastic processes . You will need programming experience (e.g., Python ) at a CS undergraduate senior level.
M 393C - Tensor Methods (Kileel)
Description:
In recent years, tensors have emerged as a powerful tool for modeling complex relationships across multiple dimensions in applied and computational mathematics. This course will constitute an introduction to higher-order tensor methods at the graduate topics level. Topics may include: low-rank tensor decompositions; numerical algorithms for computing tensor decompositions; nonconvex and manifold constrained optimization techniques; relevant randomized numerical linear algebra; uniqueness guarantees based on algebraic geometry; numerical analysis properties of tensors; and applications to data analysis, scientific computing, and machine learning. Course assessment will occur through homework assignments and a final project. Our primary textbook will be “Tensor Decompositions for Data Science” by Grey Ballard and Tamara Kolda, although this may be supplemented by other sources such as “Tensors: Geometry and Applications” by J.M. Landsberg and/or “An Introduction to Optimization on Smooth Manifolds” by Nicolas Boumal.
M 394C - Stochastic Processes (Sirbu)
Course description
The course centers on the study of Ito-diffusion processes and their applications. After an introduction to stochastic calculus and stochastic integration with respect to Brownian motion, it develops the theory of stochastic differential equations and their connection to classical analysis. It also exposes the students to optimal stochastic control of diffusion processes, the Hamilton-Jacobi-Bellman equation, singular stochastic control and linear filtering. Some applications, mainly from mathematical finance, inventory theory, decision analysis and insurance will be presented.
Topics
1. Review of fundamental concepts in probability
2. Martingales and filtrations
3. Brownian Motion
4. Stochastic Integration
5. Stochastic Calculus
6. Stochastic Differential Equations
7. Feynman-Kac formula and connection with linear PDE
8. Introduction to Optimal Stochastic Control
9. Introduction to Singular Stochastic Control
10. Filtering
11. Applications
Background
1. Knowledge of Probability and fundamental concepts of Stochastic Processes.
2. Measure Theory and Real Analysis are highly recommended.
3. The level of the class will be mostly like the one in references 1-3 below
Suggested readings
1. “An introduction to Stochastic Differential Equations”, L. C. Evans.*
2. “Stochastic Differential Equations”, B. Oksendal (6th edition)
3. “Brownian Motion and Stochastic Calculus”, I. Karatzas and S. Shreve
4. “The theory of Stochastic Processes, I”, I. Gihman and A. Skorokhod
5. “The theory of Stochastic Processes, II, I. Gihman and A. Skorokhod
6. “Controlled Markov Processes and Viscosity Solutions”, W.H. Fleming and M. Soner
*: The class will mostly follow this text.
Some organizational issues about the course:
The grade will be based on homework assignments (approximately 5 assignments during the semester).
M 392C - Representation Theory (Ben-Zvi)
I will present an unorthodox introduction to the representation theory of Lie groups and Lie algebras, focussing entirely on the group of two by two matrices with determinant one. Thus we hope to cover a breadth of topics in representation theory and harmonic analysis, that is usually sacrificed for the depth of treating general Lie groups. Besides its intrinsic beauty, this subject is ubiquitous in number theory, topology and physics. We will start with finite dimensional representations of SL_2(C) (or equivalently SU_2) and their relation to the geometry of the sphere and spherical harmonics. We will then move on to the unitary representations of SL_2(R), and their relation to hyperbolic geometry, special functions and modular forms. We will (ideally) conclude with representations of SL_2(Q_p) and their relation to the beautiful geometry of the tree, a p-adic analog of hyperbolic space. The prerequisites include abstract algebra, though exposure to Lie groups, manifolds and/or representations of finite groups would be very helpful. The grade will be determined by final projects exploring related topics.
M 392C - Gauge Theory (Montague)
M 393C - Math Foundations of Generative AI (Ward)
Spring 2025
M 392C - Algebraic Topology (Conway)
M 392C - Closed Smooth 4-Manifolds (Piccirillo)
One can readily write down myriad interesting examples of simply connected smooth 4-manifolds with boundary, but aside from a few basic constructions, building closed (i.e. compact, no boundary) simply connected 4-manifolds is much harder. In this class we will discuss methods for building interesting simply connected closed smooth manifolds, with an emphasis on building and distinguishing smooth manifolds that have the same topological type. The algebraic topology prelim course is a prerequisite. The course will briefly review handles, Dehn surgery and intersection form, but prior familiarity with these things will be helpful.
M 392C - Riemannian Geometry (Danciger)
This course will be an introduction to Riemannian geometry, with a focus on Riemannian symmetric spaces and their isometry groups. Prerequisites: Differential Topology and some familiarity with the basic theory of semi-simple Lie groups (e.g. Fall semester’s Lie groups course).
M 393C - PDEs II (Blochas)
In this class, we will study the stability of travelling waves solutions to PDEs. Travelling waves are particular solutions to Partial Differential Equations. We will discuss different families of techniques used to prove stability results. Part of this class will be based on the book "Spectral and Dynamical Stability of Nonlinear Waves" by Kapitula and Promislow.
M 394C - Stochastic Analysis - Interacting Particle Systems (Zitkovic)
Fall 2024
M 390C - Algebraic Number Theory (Ciperiani)
This will be an introductory course. We will study the ring of integers of a number field: prove the finiteness of the class group, prove Dirichlet's unit theorem, and analyze the decomposition of prime ideals when lifted to a bigger field. We will continue with a brief discussion of local fields and analytic methods in number theory. Finally we will conclude with an introduction to class field theory without proofs.
M 390C - Arithmetic Groups (Allcock)
examples: SL(n,Z), Sp(2n,Z), orthogonal groups over Z
hyperbolic space, more general symmetric spaces, and applications in algebraic geometry.
finiteness of covolume; K(G,1) spaces.
residual finiteness, existence of finite-index torsion-free subgroups.
elements of algebraic groups (characteristic 0 only, quite concrete, the aim being to be able to give the full definition of arithmetic groups).
p-adic groups and the adelic perspective on arithmetic groups. Maximal subgroups of SL2R that are commensurable with SL2Z. The affine Bruhat-Tits building on which an arithmetic group acts, with applications.
Survey of strong approximation.
Prerequisites: familiarity with semisimple Lie groups, measure theory, group theory, finite algebraic extensions of Q, and the p-adic numbers.
Text: no official required text. But we will refer to Arithmetic Groups by Witte-Morris, and Algebraic Groups and Number Theory by Platonov and Rapinchuk, and other sources. Note that the Witte-Morris book is available for free online, and in an inexpensive print version.
M 391C - Analytic Group Theory (Bowen)
Course Description: The goal of this course is for students to learn the background material needed to research modern analytic group theory. This includes: growth of groups (especially Gromov’s Polynomial Growth Theorem), amenability, Gromov hyperbolic groups, graphs of groups, Margulis’ super-rigidity, Normal Subgroup and Arithmeticity Theorems for higher rank lattices, random walks on groups, property (T), invariant random subgroups (IRS’s) and sofic groups.
Prerequisites: students are expected to be familiar with measure theory, L^p spaces, smooth manifolds and fundamental groups. Otherwise prerequisites will be kept to a minimum.
References:
Bekka, Bachir; de la Harpe, Pierre; Valette, Alain Kazhdan's property (T). New Mathematical Monographs, 11. Cambridge
Capraro, Valerio (NL-MATH); Lupini, Martino (1-CAIT)
Introduction to sofic and hyperlinear groups and Connes' embedding conjecture.
With an appendix by Vladimir Pestov. Lecture notes in Mathematics, 2136. Springer, Cham, 2015. viii+151University Press, Cambridge, 2008. xiv+472
Juschenko, Kate
Amenability of discrete groups by examples.
Mathematical Surveys and Monographs, 266. American Mathematical Society, Providence, RI, [2022], ©2022. xi+165 pp.
Woess, Wolfgang Random walks on infinite graphs and groups. Cambridge Tracts in Mathematics, 138. Cambridge University Press, Cambridge, 2000. xii+334 pp.
Zimmer, Robert J. Ergodic theory and semisimple groups. Monographs in Mathematics, 81. Birkhäuser Verlag, Basel, 1984. x+209 pp. ISBN: 3-7643-3184-4
M 392C - Algebraic Geometry (Damiolini)
One could define algebraic geometry as the discipline that studies varieties, that is spaces of solutions of polynomial equations. This course is a first introduction to the subject: as such we will discuss what we mean by varieties (and their more general cousins, schemes) and what are the tools that we can use to describe these geometric objects. The course will be built so that, towards the end of the class, we will be able to understand the Riemann-Roch theorem and some of its applications.
The prerequisites for this course are basic algebra (rings and modules) and point-set topology. We will use results from commutative algebras and some categorical language; both will be introduced/recalled during the course. If you are unsure if you satisfy these prerequisites, drop me an email!
Book: In the first part of the course, I plan to follow the book "Algebraic Geometry I: Schemes" by Görtz and Wedhorn; however, purchasing the book is not required. I will also use Gathmann's notes (https://agag-gathmann.math.rptu.de/de/alggeom.php) and Vakil's "The rising Sea" book (https://math.stanford.edu/~vakil/216blog/FOAGfeb2124public.pdf).
M 392C - Lie Groups (Perutz)
Continuous symmetry groups are ubiquitous in mathematics. Linear-algebraic examples include the general and special linear groups, and the groups of linear isometries of inner products, Hermitian products and symplectic pairings. Then there are the groups of isometries of Euclidean and hyperbolic spaces, which appear physically as the Galilean and Lorentz groups of classical mechanics and special relativity; and the spinor groups first encountered in the study of electron spin.
How should one treat continuous symmetries mathematically? An effective method is to treat the symmetries as a Lie group, which is simultaneously a group and a smooth manifold, whose multiplication map is smooth. The theory of Lie groups grows naturally out of linear algebra. The fact that commuting diagonalizable matrices have simultaneous eigenspaces leads to a structural understanding of unitary groups, and eventually of all compact Lie groups. Lie’s insight was that the algebraic structure of the tangent space at the identity element, as a Lie algebra, determines almost everything about the Lie group. While a fully algebraic approach is therefore conceivable, the theory is richer – and closer to its applications - when one admits insights from differential geometry and occasionally from algebraic topology.
This course will center on the classic theory of compact connected Lie groups, notably the principles that such groups can be classified by combinatorial “root data”; that their irreducible representations are classified by their highest weights; and that there is a uniform formula for the characters of these representations. This is an example-driven subject, and I will encourage you to work through examples. By the end of the course, you should have a concrete understanding of groups such as SU(n).
Prerequisites: The Algebraic Topology and Differential Topology prelims are more than sufficient. Algebra: Sound understanding of abstract linear algebra, basic group theory and topology are indispensable. Representation theory of finite groups is helpful but not assumed. Topology: I will assume you know the rudiments of differential topology, e.g. fluency with vector fields on manifolds, and of algebraic topology, e.g. fundamental groups in relation to covering spaces. Knowledge of (co)homology is helpful but not vital.
Main texts: J. F. Adams, Lectures on Lie Groups; T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups; R. Carter, G. B. Segal and I. MacDonald, Lectures on Lie Groups and Lie Algebras (especially Segal’s part).
M 392C - Moduli and invariants in symplectic and algebraic geometry (Siebert)
Description: The aim of this class is an introduction to the study of various moduli spaces in algebraic and symplectic geometry. The focus will be on moduli spaces appearing in the construction of invariants and structures in quantum geometry, such as Gromov-Witten and Donaldson-Thomas invariants, or various Floer-theoretic structures such as symplectic cohomology or some simple cases of the Fukaya category. The techniques will both be algebraic-geometric as well as analytic. This is an advanced class assuming some mastery of abstract algebraic geometry, as well as basics of algebraic and differential topology, and symplectic geometry. The format will alternate between traditional lectures and talks by participants on selected aspects.
M 393C - Mathematics in Deep Learning (Tsai)
We will discuss some key mathematical ingredients in the components of a typical deep learning algorithm. The components include (but not limited to): approximation theory of neural networks, including new theories connecting data and initialization and over-parameterization, theory of optimal control, numerical optimization, and optimal transport.
A survey of the prominent deep learning models will be provided.
This course should be regarded as a numerical analysis course.
The course will be conducted with a mixture of regular and student-led lectures, and discussion.
Participants of this course are expected to present/lecture on relevant concepts from suggested reading assignments.
Prerequisites
For undergraduate students who want to enroll in this class, please talk to the instructor. The following qualification is highly recommended:
M341 or 340L with a grade of at least B.
M348, M368K with a grade of at least B.
M365C with a grade of at least B.
M 393C - Partial Differential Equations I (Gualdani)
The topic of the graduate course will be on Nonlinear partial differential Equations, with emphases on elliptic and parabolic equations.
Second order Elliptic equations
- Existence of weak solutions
- Interior and exterior regularity
- Maximum principle
- Harnack’s inequality
- Cacciopoli’s inequality and applications
- Campanato’s spaces
- Schauder’s theory
- Hölder regularity estimates
- Calderon Zygmund decomposition
Second order Parabolic equations
- Existence of weak solutions
- Regularity
- Maximum principle
- Schauder’s estimates
Applications to kinetic equations
- Landau equation
- Landau-Fermi-Dirac equation
Bibliography
L.E. Evans, Partial Differential equations
E. H. Lieb and M. Loss, Analysis
G. M. Lieberman, Second order parabolic differential equations
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural’ceva, Linear and quasilinear equations of parabolic type
D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order
M 393C - Methods in Mathematics Physics
The purpose of this graduate course is to provide an introduction to mathematical aspects of Quantum Mechanics and Quantum Field Theory, with connections and applications to neighboring research areas, which tentatively include kinetic equations and deep learning. No background in physics is required, but some knowledge of Analysis/PDE is useful.
M 394C - Stochastic Processes I (Zariphopoulou)
The course offers a comprehensive study of Ito-diffusion processes, stochastic calculus and stochastic integration. It covers stochastic differential equations and their connection to classical analysis. It also exposes the students to optimal stochastic control of diffusion processes, the Hamilton-Jacobi-Bellman equation (classical and viscosity solutions), singular stochastic control and linear filtering. Applications, mainly from mathematical finance, inventory theory, decision analysis and insurance will be presented. If time permits, the course will offer a brief overview of multi-scale problems in stochastic analysis.
Topics
- Review of fundamental concepts in probability
- Martingales and filtrations
- Brownian Motion
- Stochastic Integration
- Stochastic Calculus
- Stochastic Differential Equations
- Feynman-Kac formula and connection with linear PDE
- Introduction to Optimal Stochastic Control
- Introduction to Singular Stochastic Control
- Filtering
- Applications
- Knowledge of Probability and fundamental concepts of Stochastic
- Measure Theory and Real Analysis are highly
- The level of the class will be mostly similar to the one in references 1-3
- “An introduction to Stochastic Differential Equations”, L. C.
- “Stochastic Differential Equations”, Oksendal (6th edition)
- “Brownian Motion and Stochastic Calculus”, Karatzas and S. Shreve
- “The theory of Stochastic Processes, I”, Gihman and A. Skorokhod
- “The theory of Stochastic Processes, II, Gihman and A. Skorokhod
- “Controlled Markov Processes and Viscosity Solutions”, H. Fleming and M. Soner
Some organizational issues about the course
- Class notes will be distributed via email. Other readings will be suggested and distributed later
- Homework will be assigned every week with strong expectation to be completed (and returned to me) within a
- Two take-home exams will be
- The grade will consist of the take-home exams (2x20%) and the HW assignments (60%).