## PIMS First Year Interest Groups

## Program Overview

The PIMS First Year Interest Groups (FYIG) Program aims to bring together early career researchers to study active research topics in the mathematical sciences. Each First Year Interest Group will be led by a PIMS PDF, and center on an accessible subject for beginning graduate students.

The PDF will lead a small reading group (up to 4 students) of early year (1st and 2nd year MSc or PhD) graduate students on books/papers that inspired them, and, of course, are accessible to early graduate students. The groups will meet virtually for an hour once every two weeks through the end of the 2022-23 academic year.

If you are interested, please fill out the application by **September 30th, 2022**. We will announce FYIG groups by October 5th, 2022.

## 2022-23 FYIG Topics

### Tropical Meteorology and Cloud Modeling

##### Kumar Roy, University of Victoria

The desired size of my group will be of 2 students. And I would also like to mention that there are already some students (2 at least) of my supervisor Dr. Boualem Khouider whose research in this area and they will be most likely be interested. Further, I will be happy to extend the group to up to 4 students if there is interest from other students in Mathematics and Statistics departments or other schools and departments such as Earth and Ocean Sciences.

Preliminary book list:

- Models for Tropical Climate Dynamics: Waves, Clouds, and Precipitation. Springer, Berlin (2019), Author: Boualem Khouider
- Current trends in the Representation of Physical Processes in Weather and Climate Models. Springer (2019), Editors: DA Randall, J Srinivasan, Ravi Nanjundiah, Partha Mukhopadhyay.

### Variational Methods and Partial Differential Equations

##### Cintia Pacchiano, University of Calgary

The desired size of my group is 4 students.

Preliminary book list:

- Giusti, E.: Direct methods in the calculus of variations. World Scientific Publishing Co., Inc., River Edge, 2003
- Marcellini, P.: Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions. Arch. Rat. Mech. Anal. 105, 267–284 (1989)
- Giaquinta, M., Giusti, E.: On the regularity of the minima of variational integrals. Acta Math. 148, 31–46 (1982)
- Mingione, G., : Regularity of minima: an invitation to the Dark side of the Calculus of Variations. Appl. Math. 51 (2006), 355-425
- Colombo, M., Mingione, G. Regularity for Double Phase Variational Problems. Arch Rational Mech Anal 215, 443–496 (2015). https://doi.org/10.1007/s00205-014-0785-2

### Modular Forms in Number Theory

##### Joshua Males, University of Manitoba

Imagine that you come across a sequence of numbers in the wild in your work that you want to understand more about. One of the most natural things to do is to assemble them into a generating function, which often turn out to be very special functions known as modular forms.

Modular forms are highly symmetric functions that live on the upper-half plane, and whose symmetries force extraordinary properties. Making use of techniques in modular forms we can then extract a lot of information about the original sequence of numbers that we started with.

In this course, we’ll learn the basic theory of modular forms (and their generalisations), and then focus on the Circle Method that originated with Hardy and Ramanujan 100 years ago. This is a technique that allows one to give both exact and asymptotic formulae for the coefficients of modular forms; we’ll see examples from combinatorics (e.g. partitions) and topology (e.g. Vafa-Witten invariants), and hopefully get our hands dirty computing new examples

Specific focus on asymptotic estimates and exact formulae for arithmetic coefficients.

A group of size 4 would probably be preferable, so that the students can work on reading and tackling problems together through the programme, though I am open to other sizes of group.

Preliminary book list:

- The 1-2-3 of Modular Forms, Bruinier, van der Geer, Harder, Zagier
- The Web of Modularity, Ono
- The Theory of Partitions, Andrews
- Harmonic Maass Forms, Bringmann, Folsom, Ono, Rolen