# Special Topics in Mathematical Sciences

## ▼ 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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