## The 2016 AARMS Summer School

**Theme:** Applications of Category Theory, Combinatorics and Number Theory

**Time and location:** July 11 – Aug. 5, 2016, Dalhousie University

**School Directors:** Dr. Dorette Pronk, Dalhousie and Dr. Geoffrey Cruttwell, Mount Allison

The summer school is intended for graduate students and promising undergraduate students from all parts of the world. Each participant is expected to register for at least two of the four courses. Each course consists of five ninety-minute lecture sessions each week. These are graduate courses approved by Dalhousie and we will facilitate transfer credit to the extent possible.

### Courses

**Higher Category Theory and Categorical Logic**

**Instructors**: Drs. Michael Shulman, University of San Diego, and Peter Lumsdaine, Stockholm University

**Description: **This course will introduce students to the relationship between category theory and logic, and its emerging generalization to higher category theory. On the one hand, category theory provides a flexible and powerful semantics for logic, uniting for instance forcing models of set theory with domain semantics for programming languages. On the other hand, logic provides an “internal language” for categories, that can greatly simplify the proofs of general theorems. We will discuss the classical version of this correspondence that applies to ordinary categories, including elementary toposes. Then we will introduce some basic concepts of higher category theory, and end with a brief introduction to homotopy type theory, a logic that corresponds to certain higher categories.

**Prerequisites**: A basic course in category theory, covering functor categories, limits, adjoint functors and cartesian closed categories, among other things. There will be a workshop reviewing the prerequisites for the category theory courses during the week before the summer school.

**Categories, Quantum Computation and Topology**

**Instructor**: Dr. Jamie Vicary, University of Oxford

**Description: **This course will introduce the theory of monoidal categories, an approach to mathematics which combines algebra and geometry into a single subject, and investigates its applications in quantum computation and topology. An emphasis will be on graphical calculi, which allow us to prove theorems using pictures, rather than traditional mathematical syntax. The course will also have a practical component, using the proof assistant Globular to formalize and investigate the results we encounter. Topics studied will include coherence, linear structures, duality, monoids and comonoids, Frobenius and Hopf algebras, quantum groups, quantum protocols, higher categories, and topological quantum field theory.

**Prerequisites**:There are no formal prerequisites, as we will introduce everything from the beginning. However, some previous experience of category theory would be an advantage, as we will cover the basics quickly.

**Stable polynomials: with applications to graphs, matrices, and probability**

**Instructor**: Dr. David Wagner, University of Waterloo

**Description: **Univariate polynomials with only real roots, while special, occur often enough that their properties can lead to interesting conclusions in diverse areas. Due mainly to the recent work of two young mathematicians, Julius Borcea and Petter Brändén, a very successful multivariate generalization of this method has been developed. The first part of the course sketches the main results of this theory — the most central of these results is the characterization of linear transformations preserving stability. The second part presents various applications of this theory in complex analysis, matrix theory, probability and statistical mechanics, and combinatorics. This includes the very recent work of Marcus, Spielman, and Srivastava resolving two decades-old conjectures.

**Prerequisites**: Some linear algebra (matrices, determinants, etc.), some complex analysis (but just a little bit — nothing very complicated) and some graph theory (just some basic familiarity with the concepts). The course will be accessible to any strong student who has finished two years of a mathematics B.Sc. degree.

**An Introduction to Special Functions and WZ Theory**

**Instructor**: Dr. Armin Straub, University of South Alabama

**Description: **Special functions, that is those functions which are useful but not elementary, include the gamma function, Bessel functions, orthogonal polynomials, hypergeometric functions, elliptic integrals, modular functions and an increasing number of further functions. In this introductory course, we will learn about many of them. Our focus will be on the hypergeometric functions, because a lot of other special functions can be expressed in terms of these and because of their natural place in the theory of linear differential equations. Among other interesting properties, hypergeometric functions satisfy a huge number of transformation laws and identities. WZ theory is an algorithmic toolkit that makes it possible to automatically prove and discover many identities involving special functions. It is very versatile and, for instance, applies to identities involving discrete and/or continuous variables. Throughout the course, we plan to reach out to combinatorics, algebra and number theory for applications and motivation.

**Prerequisites**: A first course in complex analysis, covering analytic continuation and residue calculus.

### Preparatory Lectures

During the week before the summer school, Dr Geoff Cruttwell will offer preparatory lectures in category theory to help you review the material that you will need for the two category theory courses in the school. These lectures will be offered at Dalhousie University. If you are interested in coming a week early to attend these lectures, please indicate this on your registration form.

### To Apply

Please download the application form in one of the formats below, fill it out and return it to gcruttwell@mta.ca by March 15, 2016:

## Contact Information

For further information contact

- Dr. Dorette Pronk
- Department of Mathematics and Statistics
- Dalhousie University
- Halifax, Nova Scotia
- Canada B3H 4R2
- email:pronk@mathstat.dal.ca

- Dr. Geoff Cruttwell
- Department of Mathematics and Computer Science

Mount Allison University

Sackville, NB, Canada

E4L 1E6 - email:gcruttwell@mta.ca

- Department of Mathematics and Computer Science