# Special Topics in Mathematical Sciences

## ▼ 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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