# Special Topics in Mathematical Sciences

## ▼ Fall 2024

## 21-366 Topics in Applied Mathematics - Random Graphs and Algorithms

**Instructor:**Prof. Alan Frieze**Level:**Undergraduate**Description:**In this course we study the typical properties of a large graph chosen from a variety of distributions. Much of the course discusses the properties of the Erdos-Renyi graph G(n,m), a random graph with vertex set [n] chosen uniformly at random from all m-edge graphs. The key notion is that of a threshold. There is usually a sharp transition m* so that if the number of edges m<<m* then G(n,m) is very unlikely to have a given property P, whereas if m>> m* then G(n,m) is very likely to have with property P. Later we will discuss random regular graphs and a simple model of "real-world social'' networks. Topics: Basic Models; Evolution; Vertex Degrees; Connectivity; Small Subgraphs; Spanning Subgraphs: Matchings, Hamilton Cycles, Embeddings; Extreme Characteristics: Diameter, Independence Number, Chromatic Number; Digraphs; Fixed Degree Sequence; Preferential Attachment Graphs; Edge Weighted Graphs: Minimum Spanning Tree, Shortest paths.**Prerequisites:**TBA

## 21-801 Advanced Topics in Discrete Mathematics - Topological Methods in Combinatorics

**Instructor:**Prof. Florian Frick**Level:**Graduate**Description:**Recent years have seen rapid progress in the development and application of topological methods in combinatorics. Key problem areas in the field, such as chromatic numbers of hypergraphs, matching theory, piercing and covering problems, and several discrete-geometric questions are often best viewed through the topological lens. This topological viewpoint also provides new insights into algorithmic hardness results. This course will introduce the relevant topological techniques and explore their combinatorial applications. These methods are also useful in a broader mathematical context and have been fruitfully employed across mathematical disciplines and beyond in fields as diverse as data science, economics, and machine learning.**Prerequisites:**- For undergraduate students: (21-355 or 21-235 or 21-455) and (21-373 or 21-237) and permission of the instructor
- For graduate students: None

## 21-830 Advanced Topics in Partial Differential Equations

**Instructor:**Prof. Matthew Rosenzweig**Level:**Graduate**Description:**This course is an introduction to the mathematics of interacting particle systems at and away from equilibrium, with a focus on their macroscopic description by solutions of nonlinear partial differential equations. Topics covered will include mean-field limits, propagation & generation of chaos, and trend to equilibrium. Emphasis will be on continuum, as opposed to lattice, models coming from physics, with the Langevin system as the prototypical example, though connections to other fields (e.g. machine learning) will be explored.**Prerequisites:**Students should have a good understanding of real and functional analysis, some familiarity with ordinary and partial differential equations. But the most important prerequisite is mathematical maturity and a willingness to fill in gaps in one’s knowledge on the fly. In addition, undergraduate students will need (21-355 or 21-235 or 21-455) and (21-373 or 21-237) plus permission of the instructor.

## 21-849 Special Topics - High-dimensional Probability

**Instructor:**Prof. Konstantin Tikhomirov**Level:**Graduate**Description:**The course provides a rigorous introduction to high-dimensional probability and statistics. The students will learn probabilistic aspects of modern data science which are important both for starting research in the field and for applications in other branches of science and technology. In particular, the following topics will be considered:- Bernstein/Chernoff/Azuma concentration inequalities, concentration on the sphere and on the discrete cube;
- suprema of Gaussian and empirical processes, with applications in mathematical statistics;
- spectral theory of random matrices and its applications to covariance matrix estimation, Principal Component Analysis, Compressed Sensing, community detection in social networks.

**Prerequisites:**- For undergraduate students: (21-355 or 21-235 or 21-455) and (21-373 or 21-237) and 21-721 and permission of the instructor
- For graduate students: 21-721 is highly recommended

## ▼ Spring 2024

## 21-366 Topics in Applied Mathematics

**Instructor:**Prof. David Kinderlehrer**Level:**Undergraduate**Description and Prerequisite knowledge**

## 21-375 Topics in Algebra: Mathematical Paradoxes

**Instructor:**Prof. Clinton Conley**Level:**Undergraduate**Description:**The notion of equidecomposability dates back to the ancient Greeks' attempts to formalize the intuitive notions of area and volume. The idea is that if an object A can be chopped into finitely many pieces which can be re-assembled into another object B, then A and B should have the same "size." Shockingly, Banach and Tarski (building upon work of Hausdorff) showed that this intuition fails dramatically in three-dimensions: for instance, any two balls are equidecomposable, regardless of their diameters. A flurry of mathematics at the interface of group theory, geometry, measure theory, and set theory followed in an attempt to understand this paradoxical result. This course will formalize and prove such paradoxicality results in an algebraic framework, isolate and analyze the salient notion of amenability, and (time permitting) discuss current research directions in this area.**Prerequisite knowledge:**Group theory (21-373 or 21-237) is a prerequisite. Some background in real analysis and/or set theory is recommended but not required, as we will develop the relevant concepts during the course.

## 21-410 Research Topics in Mathematical Sciences

**Instructors:**Prof. Theresa Anderson and Prof. Evan O'Dorney**Level:**Undergraduate**Description:**The course will be offered with a focus on a variety of number theoretic areas of active research with the potential for a professional journal publication. This course will be run in the style of a Research Experience for Undergraduates (REU) to explore the creative world of mathematics research via a concrete, yet flexible project.

This semester's offering will showcase 3 areas of number theory: Diophantine approximation, arithmetic statistics, and the intersection of measure theory and number theory. Co-taught by Professors Theresa Anderson and Evan O'Dorney, this will be offered MWF (time tba) for 9 units. Entrance to the class is by application only (information below), and the nine selected students will be divided into three groups (information below). The main work in the course will be to conduct serious independent research with your group members and professor(s), leading to a final presentation and typed rough draft manuscript. All applicants will be notified of their status in advance of the registration deadline. Interested students must be undergraduates at CMU and fill out the Google form located here: https://forms.gle/eEqKdFjoRQpYixtg8

The group on Diophantine approximation will be led by Dr. O'Dorney and will plan to investigate questions related to approximating reals by rationals (see arxiv.org/abs/2205.12829). The group on arithmetic statistics will be co-led by Dr. Anderson and Dr. O'Dorney, focusing on an interplay of algebraic and analytic tools to improve counts related to polynomials or number fields (see arxiv.org/abs/2107.02914). The group on measure theory and number theory will be led by Dr. Anderson and will attempt to extend the construction of a measure (see arxiv.org/abs/2009.03875) in a variety of ways. Accepted students’ preferences will be taken into account, but final decisions for group placement will be made by the professors. Note that though a published work is not expected, the three previous REU experiences Dr. Anderson has run have all resulted in published or submitted works. See arxiv.org/abs/1809.01075 or arxiv.org/abs/2108.04155 for some ideas.**Prerequisite knowledge:**21-341 (or 21-238), 21-355 (or 21-235), 21-356 (or 21-236), 21-373 (or 21-237) and 21-441.

## 21-820 Advanced Topics in Analysis - Regularity for Minimal Surfaces (Section A)

**Instructor:**Prof. Robin Neumayer**Level:**Graduate**Description:**The study of minimal surfaces — finding the surface with least area among those bounded by a given curve, as a model for soap films — is among the most classical problems in the calculus of variations. Early developments in minimal surface theory led to Douglas being awarded the very first Fields medal 1936 and it remains an active field of research today. One might hope to model soap films by smooth surfaces, but this formulation of the problem turns out to be ill-posed — indeed, in high dimensions minimal surfaces can exhibit singularities! Instead, the minimal surface problem is formulated in a geometric measure theoretic framework, and from here, regularity becomes a central question. In this course, after building up the basic theory of sets of finite perimeter, we will develop the classical regularity theory and analysis of singularities for area minimizing minimal surfaces, and then survey some contemporary results about the structure of singularities.**Prerequisite knowledge:**The course will be largely self-contained and measure theory the only prerequisite.

## 21-820 Advanced Topics in Analysis - Convex Functions (Section B)

**Instructor:**Prof. Giovanni Leoni**Level:**Graduate**Description:**Topics will include: Convex functions defined on Euclidean spaces: Regularity. Subdifferentiablity and monotone multifunctions. Alexandrov’s theorem. Conjugate functions. Convex envelopes. Biconjugate functions. Applications. Convex functions defined on infinite-dimensional spaces: Regularity. Subdifferenttiability. Duality Theory. Applications.**Prerequisite knowledge:**For undergraduates: 21-356 or MS Analysis I or permission from the instructor.

## 21-882 Advanced Topics in Financial Mathematics: Advanced Optimal Transport

**Instructor:**Prof. Johannes Wiesel**Level:**Graduate**Description:**Building on Dejan Slepcev’s course “Optimal transport (OT) and applications” in the fall semester, this class will take a closer look at the state of the art of research in optimal transport in mathematical finance, probability theory, statistics, machine learning and computer science. In particular we will focus on optimal transport of stochastic processes (also called causal or adapted OT), martingale OT, weak OT, transport inequalities, convergence of empirical measures in Wasserstein distance, estimation of OT maps, Wasserstein barycenters and entropic OT.**Prerequisite knowledge:**TBA

## ▼ Fall 2023

## 21-322 Topics in Formal Mathematics: Proof Assistants and Elementary Differential Topology

**Instructor:**Prof. Patrick Massot**Level:**Undergraduate**Description:**Formal mathematics refers to mathematics that is carried out so precisely that it can be fully processed and checked by computer, using software known as a "proof assistant." This course will first teach you how to use the Lean proof assistant, which will consolidate your understanding of how mathematical proofs work, and also help you understand how proofs should be written on traditional media such as paper or a blackboard. The second part of the course will focus specifically on reviewing multivariate calculus and providing an introduction to differential topology in Euclidean spaces, and students will be required to carry out formalization projects in these areas. Differential topology uses multivariate calculus to study properties of shapes that are global and do not change when shapes are deformed. Typical applications include the hairy ball theorem and Smale's sphere eversion theorem.**Prerequisite knowledge:**Multivariable Calculus (21-259/21-266/21-268/21-269) and Real Analysis (21-235/21-355)

## 21-387 Monte Carlo Methods and Applications (cross-listed with 15-327)

**Instructors:**Prof. Gautam Iyer and Prof. Keenan Crane**Level:**Undergraduate**Description:**The Monte Carlo method uses random sampling to solve computational problems that would otherwise be intractable, and enables computers to model complex systems in nature that are otherwise too difficult to simulate. This course provides a first introduction to Monte Carlo methods from complementary theoretical and applied points of view, and will include implementation of practical algorithms. Topics include random number generation, sampling, Markov chains, Monte Carlo integration, stochastic processes, and applications in computational science. Students need a basic background in probability, multivariable calculus, and some coding experience in any language.**Prerequisite knowledge:**Multivariable Calculus (21-259/21-266/21-268/21-269) and Probability (15-259/21-325/36-218/36-225)

## 21-800 Advanced Topics in Logic: Dynamics of Polish Groups

**Instructor:**Prof. Aristotelis Panagiotopoulos**Level:**Graduate**Description:**This is a topics course in topological groups and dynamics. We will be focusing on techniques and phenomena that go beyond the realm of locally compact groups, but which are highly relevant in the context of more general Polish groups such as: automorphism groups of countable structures; diffeomorphism groups of manifolds; the unitary group of the separable infinite dimensional Hilbert space; etc. In the absence of classical techniques which rely on local compactness, we will develop various alternative frameworks for analyzing the structure of 'large' topological groups, such as: Fraïssé theoretic methods; Baire category methods; structured completions and compactifications.

In the first part of the course we will develop these techniques and we will provide a wide range of applications and examples from topology, analysis, and logic. We will also discuss various dynamical phenomena which can only occur when the Polish groups are 'large'. Phenomena such as: the lack of complete left-invariant metrics; complete absence of any non-trivial unitary representations; automatic continuity results; and (time permitting) extreme amenability phenomena. In the second part of the course we will discuss the role that Polish groups play in 'descriptive dynamics' and how to use 'wild dynamics' in order to prove negative anti-classification results for various classification problems in mathematics. For example, we will cover Hjorth's turbulence theory and use it to show that the collection of all irreducible representations of the free group in two generators cannot be classified using isomorphism types of countable structures as invariants.**Prerequisite knowledge:**Algebraic Structures (21-237/21-373) and Real Analysis (21-235/21-355)

## 21-820 Advanced Topics in Analysis: Optimal Transport and Applications

**Instructor:**Prof. Dejan Slepčev**Level:**Graduate**Description:**The course will develop foundations of optimal transportation and investigate the properties of the resulting spaces. Dynamical description of optimal transport and variants, such as entropy regularized transport and unbalanced will also be presented. We will cover a number of applications, in particular gradient flows in spaces of probability measures. The course will include numerically computing optimal transportation maps, plans, and distances and statistical properties of approximating optimal transportation in random setting. The material will be covered rigorously and rigorous arguments are expected on the homework. However the course will be structures so that there will be options for some more applied assignments and projects.**Prerequisite knowledge:**Knowledge of measure and integration (21-720), differential equations (21-632), some advanced real analysis (basics of Sobolev spaces), functional/convex analysis (Fenchel-Rockafellar duality), probability (Prokhorov compactness theorem), calculus of variations (direct method, lower semicontinuity), and differential geometry.

## 21-849 Special Topics: Introduction to Arithmetic Statistics

**Instructor:**Prof. Tess Anderson**Level:**Graduate**Description:**Do you like to count? Arithmetic statistics is an exciting area of mathematical research where one asks questions about how random mathematical objects behave. Instead of slowly learning techniques from hundreds of years ago, we will instead jump right in to fresh developments, learning/reviewing what we need as we go. A delightful interplay of number theory, combinatorics and analysis take center stage, working in ways you likely haven't seen before. Evaluation will be based on participation and a final presentation of a recent paper in the area. Just bring your curiosity.**Prerequisite knowledge:**Field Theory (21-374), Number Theory (21-441), Real Analysis (21-235/21-355), and Probability (15-259/21-325/36-218/36-225)

## ▼ Spring 2023

## 21-366 Topics in Applied Mathematics: Mathematical Biology

**Instructor:**Prof. David Kinderlehrer**Level:**Undergraduate**Description:**Collaborating, in these unsettled times and our chaotic environment, we can enjoy some respite looking to the future, seeking to resolve fundamental problems in the life sciences. Biology, and more generally life sciences, is a vast diverse ﬁeld of study. Thus, the range of possible mathematical applications is both vast and diverse. Or, at least, becoming familiar with the mathematical ideas and methods that might pertain to them will cover a wide spectrum of mathematics and as well as inspire the development of new mathematics. It is also a burgeoning, and extremely exciting, ﬁeld of research. As novices, what should be our approach? This is a developing course here at CMU and, as we progress, I solicit your ideas about its content and format. Biological systems are comprised of many interacting units participating at many physical and time scales. Their precise details are generally unknown. Our task is to describe the parts and functions over which we wish to have predictive capability. In our modeling we choose what we think to be appropriate mathematics. For this we need to establish both the capabilities and the limitations of the mathematical description.

The plan is to cover the material in Theoretical Modeling Tools, the ﬁrst seven chapters of the book, laced with considerable supplementary material. When discussing a particular topic, you might consult several of the reference books. We begin with infectious diseases. We shall discuss some classical successes that include these highlights:- basic genetics: what is the connection between Mendelian genetics and evolution?
- mRNA: the basis of the covid vaccine
- the Hodgkin-Huxley equations
- the remarkable Turing instability, with recent surprises
- the fundamental Luria Delbruck Experiment

## 21-410 Research Topics in Mathematical Sciences

**Instructor:**Prof. Wesley Pegden**Level:**Undergraduate**Description:**Research projects in Discrete Mathematics. In this course, students will work alone or in small groups on research problems, aiming to prove original results.

## 21-800 Advanced Topics in Logic

**Instructor:**Prof. James Cummings**Level:**Graduate**Description:**Cardinal Arithmetic. One of the oldest problems in set theory asks (in modern language) which behaviours of the continuum function kappa :-> 2^kappa are consistent with ZFC set theory. This course will cover two main aspects of this problem: 1) Constraints on the continuum function which are provable in ZFC, 2) Forcing constructions which show that our ZFC constraints are optimal. Some of the forcing constructions use strong hypotheses, and we will discuss why this is necessary. Prerequisites: Familiarity with graduate set theory at the level of 21-602. As the course proceeds, some material about forcing at the level of 21-702 will be needed.

## 21-849 Special Topics

**Instructor:**Prof. Tomasz Tkocz**Level:**Graduate**Description:**Analytic and Probabilistic Methods in Convex Geometry. This course will be a snapshot of several classical and modern topics in convex geometry and high dimensional probability (mainly nonasymptotic), where analytic methods play a prominent role. The topics will include concentration of measure and isoperimetry, functional inequalities, log-concavity, Gaussian space, etc., as well as applications of the tools developed in other fields, primarily convexity and metric geometry. This course will be self-contained, but basic solid knowledge in linear algebra, measure theory and probability (mostly at the undergraduate-level) will be assumed.

## 21-881 Topics in Stochastic Calculus

**Instructor:**Prof. Mykhaylo Shkolnikov**Level:**Graduate**Description:**Particles interacting through their mean field and applications to finance.

## ▼ Fall 2022

## 21-366 Topics in Applied Mathematics: Random Graphs

**Instructor:**Prof. Alan Frieze**Level:**Undergraduate**Description:**In this course we study the typical properties of a large graph chosen from a variety of distributions. Much of the course discusses the properties of the Erdos-Renyi graph G(n,m), a random graph with vertex set [n] chosen uniformly at random from all m-edge graphs. The key notion is that of a threshold. There is usually a sharp transition m* so that if the number of edges m<<m* then G(n,m) is very unlikely to have a given property P, whereas if m>> m* then G(n,m) is very likely to have with property P. Later we will discuss random regular graphs and a simple model of "real-world social'' networks. Topics: Basic Models; Evolution; Vertex Degrees; Connectivity; Small Subgraphs; Spanning Subgraphs: Matchings, Hamilton Cycles, Embeddings; Extreme Characteristics: Diameter, Independence Number, Chromatic Number; Digraphs; Fixed Degree Sequence; Preferential Attachment Graphs; Edge Weighted Graphs: Minimum Spanning Tree, Shortest paths.

## 21-801 Advanced Topics in Discrete Mathematics (Section A)

**Section A:**TBD

**Instructor:**Prof. Wesley Pegden**Level:**Graduate**Description:**TBD

## 21-801 Advanced Topics in Discrete Mathematics (Section B)

**Section B:**Mathematical Games and Puzzles

**Instructors:**Profs. Alan Frieze & Daniel Sleator**Level:**Graduate**Description:**The course studies the mathematics behind finding optimal strategies for playing combinatorial games. In particular we study Tic Tac Toe in high dimensions and its generalization to Maker-Breaker games. We study sums of games and the relations between numbers and games. In addition, we will look at some interesting puzzles.

## 21-820 Advanced Topics in Analysis

**Instructor:**Prof. Irene Fonseca**Level:**Graduate**Description:**In this course we will use modern methods of the Calculus of Variations to study minimization problems for integral functionals depending on vector-valued fields and their gradients. Applications to nonlinear elasticity, singular perturbations, dimension reduction, homogenization, and image denoising in computer vision will be addressed as time will permit.

## ▼ Spring 2022

## 21-366 Topics in Applied Mathematics: Mathematical Biology

**Instructor:**Prof. David Kinderlehrer**Level:**Undergraduate**Description:**Collaborating, in these unsettled times and our chaotic environment, we can enjoy some respite looking to the future, seeking to resolve fundamental problems in the life sciences. Biology, and more generally life sciences, is a vast diverse field of study. Thus, the range of possible mathematical applications is both vast and diverse. Or, at least, becoming familiar with the mathematical ideas and methods that might pertain to them will cover a wide spectrum of mathematics and as well as inspire the development of new mathematics. It is also a burgeoning, and extremely exciting, field of research. As novices, what should be our approach? This is a developing course here at CMU and, as we progress, I solicit your ideas about its content and format. My plan is to cover the Theoretical Modeling Tools, the first seven chapters of the book, laced with supplementary material. We begin with infectious diseases. Especially we shall discuss some classical successes that include basic genetics, the Hodgkin-Huxley equations, the remarkable Turing instability, and the fundamental Luria Delbruck Experiment. Looking through math biology books at a similar level, all cover similar material. From this we can likely conclude that the material for a first course is nearly canonical. The books cover material at a more advanced level and are very interesting as well. Note that many of the books are in paperback; they are generally available through the library as downloads.

## 21-820 Advanced Topics in Analysis: Functions of Bounded Variation

**Instructor:**Prof. Irene Fonseca**Level:**Graduate**Description:**The goal of the course is to give a general introduction to the theory of functions of bounded variation (BV) and of sets of nite perimeter. This theory provides the natural setup for several problems in the Calculus of Variations, in particular those characterized by the onset of (free) discontinuity surfaces. Main properties of BV functions will be studied, and (time permitting) applications to phase transitions and image pr0cessing will be presented.

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