Winter 2024

Quantum Computing

• • Instructor: Adam Marks

  • Quantum computing leverages quantum mechanical effects to perform computation of a rather different style than ordinary (classical) computation. We will learn a mathematical framework to describe quantum computation and prove some key theorems, leading up to Shor's algorithm for factoring integers. We will primarily use tools from linear algebra and basic group theory.

Intro to Continuous Logic

• • Instructor: Jessica Schirle

  • "Classical" or "discrete" model theory provides logicians with powerful tools to build bridges between first order logic and algebraic structures. However, this logic does not do a good job at describing geometric spaces, where mathematicians care much more about the local structure than what happens to individual points in a space. In this course, we'll give an introduction to what's known as "continuous model theory," wherein we change our notion of logic to better capture geometric concepts. Students should be comfortable with elementary real analysis and propositional (and ideally first order) logic.

An Invitation to Applied Category Theory

• • Instructor: Ari Rosenfield

  • Compositional structures, where we're able to "chain processes together," show up all over mathematics and the physical sciences. Compositionality is what category theory is all about! Students can expect to learn both elementary and advanced topics from category theory in the context of their applications to computer science and physics. In particular, expect to come away having learned some basic theory of monoidal categories, enriched categories, and toposes.

Representation Theory

• • Instructor: Deepesh Singhal

  • We will read about group representations, character theory and representations of S_n and their connection to Young diagrams.

Free Probability and Random Matrices

• • Instructor: Jenny Pi

  • Free probability is a non-commutative analog of probability theory which is useful for dealing with spaces of operators and also has deep connections with the asymptotics of random matrices. In this course, we will do a brief introduction into this theory and its connection with the eigenvalue distribution of random matrices via the free central limit theorem.

Symbolic Dynamics and Information Theory

• • Instructor: Karl Zieber

  • Originally developed as a way to approach otherwise intractable problems in dynamics, symbolic dynamics now has broad and sometimes surprising applications, particularly in probability theory and computer science. One famous branch that arose was information theory, which helped design some of the first compression and encryption algorithms and continues to be an important field of research. We will study symbolic dynamics by exploring the fundamentals of sequence space and its operators. We will also learn about shifts of finite type, where we will use the deep connections to infinite graphs. 

Facets of the Prime Number Theorem

• • Instructor: Thurman Ye

  • The density of prime numbers among all natural numbers is remarkably well-estimated by nice functions. Instead of working through one proof of this result in detail, we will spend our meetings examining several different proofs, as well as results intimately related to the PNT. 

Possible topics include: heuristic justifications; computational results; the preluding estimates of Chebyshev and Mertens; estimates following from or equivalent to the PNT (partial summation); the original complex analytic proofs of Hadamard & de la Vallee Poussin; relation to the Riemann hypothesis; proof via the Wiener-Ikehara Tauberian theorem; proofs of Erdos-Selberg and Newman; Dirichlet's theorem on arithmetic progressions (or the Chebotarev density theorem); analogues over global fields or arbitrary rings; probabilistic interpretations of the primes (central limit theorems of Erdos-Kac and Selberg, the Riemann zeta distribution).

Nielsen-Thurston Theory

• • Instructor: Sidhanth Raman

  • What do symmetries of surfaces look like? Here, the group of symmetries we are referring to is called the "mapping class group". Before 1973, we didn't have a good answer to this question! It turns out there is a classification of surface homeomorphisms (called the Nielsen-Thurston Classification Theorem), which illustrates a rich dynamical trichotomy satisfied by the mapping class group. Along the way to understanding the Nielsen-Thurston classification of symmetries, we will meet hyperbolic geometry, complex analysis, algebraic geometry, and covering space theory!

An Introduction to Topology

• • Instructor: Sireesh Vinnakota

  • Students have expressed interest in advanced topics in topology, and thus this course is to be a primer on some of the important point-set topology invariants, building towards the Classification Theorem for Surfaces.