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.

These courses are certified to be graduate level courses by both the School of Mathematical and Computational Sciences (SMCS) at University of Prince Edward Island and by the Atlantic Association for Research in the Mathematical Sciences (AARMS). Upon successful completion of a course, SMCS and AARMS will award a certificate confirming this, which students can then take back to their home institutions if they plan to receive credit for these courses towards their degree. Students hoping to receive academic credit for summer school courses are strongly encouraged to consult with their home institution about this process before the school begins.

For student attendees, the AARMS Summer School will pay the tuition and the accommodation expenses, but will not cover the cost of travel. Note that the accommodation is provided only when a student is taking a course (the night before the first lecture of a course is covered). If a student arrives early or leaves late, the extra accommodation is not covered. Note that those who will finish their bachelor/master degree in the spring/summer term of 2019 and will enter the graduate school in the fall term of 2019 are considered to be students.

To apply for AARMS Summer School 2019, please complete the online application form before April 15, 2019.

Please note that the evaluation committee of the summer school will review all applications shortly after April 15, 2019 and thereafter notify the applicants whose applications are approved by email. Please do not book your air tickets before receiving the approval email.

### Rough paths theory

**Laure Coutin**

*Institut de Mathématiques de Toulouse, France
*July 2 to July 12, 2019

The aim of this course is to provide an introduction to rough paths theory. Introduced by T. Lyons in 1998, rough paths theory provides a deterministic way to define and solve differential equations driven by rough signals. The sample paths of Gaussian processes is a main field of their application. We will start with ordinary differential equations (for continuous paths with bounded variations). We shall then turn to the Young’s integral. We will introduce the central notion of signature and study rough differentials equations. We will conclude with applications to stochastic differential equations.

Prerequisites: Previous course on ordinary differential equations, and knowledge of fixed point theorems.

Bibliography:

- A.M. Davie Differential equations driven by rough paths: an approach via discrete approximations. Appl. Math. Res. Express AMRX 2008 n. 2 (2008) 1-40
- P. Friz and M. Hairer A course on Rough Paths With an Introduction to Regularity Structures Universitext Springer 2014.
- P. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough paths, Vol 120 of Cambridge Studies in Advanced Mathematics Cambridge University Press, 2018
- P. Hartman Ordinary differential equations, Second Edition, Birkhauser, 1982
- M. Gubinelli Controling Rough paths. J. Funct Anal. 216, n 1, (2004), 86- 140
- T. Lyons and Z. Qian, System Control and Rough Paths, Oxford Science Publication, 2002

### q-series in Analysis and Combinatorics

**Mourad Ismail**

*University of Central Florida, USA
*June 17 to July 12, 2019

We will develop the theory of combinatorial and analytic identities, summation theorems, and related topics through analytic and combinatorial techniques. The combinatorics involves counting subspaces and mapping vector spaces over finite fields, and partitioning theoretic identities of number theory. The Mobius function on partially ordered sets will also be mentioned.

We will pay special attention to identities like the Rogers-Ramanujan identities and their various generalizations in some detail. A central piece of the analytic development is the Askey--Wilson integral and its generalizations.

Over all the course will be a bridge between analysis and discrete mathematics through the use of combinatorial and analytic tools. The treatment we propose is very conceptual and is a major improvement over the earlier approaches.

The classical approach to q-series is available in [2] and [3]. One classic reference on partitions and number theory is [1].

The lectures will be based on the lecture notes [4]. A copy of these notes will be made available to the students in the class.

Prerequisites: Advanced calculus.

References:

- G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Massachusetts, 1976, reissued by Cambridge University Press, Cambridge, 1998.
- G. E. Andrews, R. A. Askey and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.
- G. Gasper and M. Rahman, Basic Hypergeometric Series, second edition, Cambridge University Press, Cambridge, 2004.
- M. E. H. Ismail and D. Stanton, Introduction to Quantum Calculus, Lecture notes.

### Fractals: using Iterated Function Systems (IFS) to construct, explore, and understand fractals

**Franklin Mendivil**

*Acadia University
*June 17 to June 28, 2019

This course will present an introduction to one viewpoint (that of IFS) in the study of “fractal” objects (such as the one pictured to the right). Although there is not a universally accepted definition of a “fractal”, for our purposes it is enough to think about objects which have “similar behaviour” at finer and finer resolutions (smaller and smaller length scales). An IFS is a convenient encoding of this “similar behaviour” between scales and lets us (to some extent) both control this relationship and analyze the structure of the resulting object.

We will discuss both geometric fractals (viewed as subsets of $\mathbb{R}^d$) and fractals which are functions or probability distributions. After discussing the construction and basic properties of fractal sets, we will present various notions of “dimension” and discuss relations between these notions and ways of computing them. However, the precise list of topics will depend greatly on the interests and background of the students. As an example, some applications of IFS fractals in digital imaging could be presented. The aim of the course is to develop intuition about what it means to be self-similar and introduce techniques of analyzing fractal objects.

The tools we will use include metric geometry and topology, probability and measure theory, and some aspects of function spaces. We will certainly take the time to make sure that all students have a chance to understand, filling in any gaps in the background knowledge as we go.

### The mathematics and science of chaos

**James Yorke**

*Maryland, USA
*June 17 to June 28, 2019

This is a course on chaos, aimed at the math or science student to reveal ideas needed to understand science. Chaos is the phenomenon where very small changes result in huge changes in the future state of the system. This is what happens in life and in the lab and in computer experiments in many situations. Chaos theory suggests the flap of a butterfly’s wings might ultimately cause a tornado in Texas -- or elsewhere -- and the flapping might prevent a tornado as well. This so-called “butterfly effect” has profound consequences: even moderately long-term forecasts can be nearly impossible in the presence of chaos. But short-term predictions remain possible.

Along with discussions of the major topics, including discrete dynamical systems, chaos, fractals, nonlinear differential equations and bifurcations, I will also illustrate relevant concepts from the physical, chemical and biological sciences. And some time will spend on some of the most important topics in current science.

For the mathematically inclined, each chapter of the course text ends with a Challenge, guiding students through an advanced topic in the form of an extended exercise.

There are computer experiments throughout the text that present opportunities to explore dynamics through computer simulations, designed for use with any software package. A software package “Dynamics” will be provided.

Prerequisites: Calculus, Differential Equations, and Linear Algebra

Required Textbook: Chaos: An Introduction to Dynamical Systems by K. Alligood, T. Sauer, J. Yorke

### Contact information

Dr. Shafiqul Islam (Director)

Email: sislam@upei.ca

School of Mathematical and Computational Sciences

University of Prince Edward Island

Charlottetown, Prince Edward Island

Canada C1A 4P3

### Apply to attend the AARMS Summer School

Application deadline: April 15, 2019