## 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.

## 2023-24 FYIG Topics

### Modelling Ecological Population Dynamics with Reaction-Diffusion Equations

##### Jane Shaw MacDonald, Simon Fraser University

In this reading group we focus on the contributions of mathematicians and theoretical ecologists in spatial ecology through the modelling framework of reaction-diffusion equations. Species interact not only with each other but also with their spatial environment, and the topography and limitations of the landscape then impact a species ability to grow. Thus we study how population densities change in both space and time. Themes of our discussions will follow species dynamics and persistence conditions in the case where space is limitedand there is dispersal across space. This will lead us to topics such as persistence, coexistence, and invasion capacity of populations.

The reading group is suitable for up to 4 graduate student participants.

Selected Readings:

The main text for this reading group will be- Cantrell, Robert Stephen, and Chris Cosner. Spatial ecology via reaction-diffusion equations. John Wiley & Sons, 2004.

- Cantrell, Robert Stephen, Chris Cosner, and Shigui Ruan, eds. Spatial ecology. CRCPress, 2010.
- Kierstead, Henry, and L. Slobodkin. “The size of water masses containing plankton blooms.” Journal of Marine Research 12 (1953): 141–147.
- Maciel, Gabriel Andreguetto, and Frithjof Lutscher. “How individual movement response to habitat edges affects population persistence and spatial spread.” The American Naturalist 182.1 (2013): 42-52.
- Maciel, Gabriel Andreguetto, and Frithjof Lutscher. “Allee effects and population spread in patchy landscapes.” Journal of Biological Dynamics 9.1 (2015): 109-123.
- Segel, Lee A., and Julius L. Jackson. “Dissipative structure: an explanation and an ecological example.” Journal of theoretical biology 37.3 (1972): 545-559.
- Potapov, Alex B., and Mark A. Lewis. “Climate and competition: the effect of moving range boundaries on habitat invasibility.” Bulletin of mathematical biology 66.5 (2004): 975-1008.
- Berestycki, Henri, et al. “Can a species keep pace with a shifting climate?.” Bulletin of mathematical biology 71 (2009): 399-429.
- MacDonald, Jane S., and Frithjof Lutscher. “Individual behavior at habitat edges may help populations persist in moving habitats.” Journal of Mathematical Biology 77 (2018): 2049-2077.

### Linear Algebra Methods in Combinatorics

##### Himanshu Gupta, University of Regina

It is widely recognized that both Linear Algebra and Combinatorics find extensive applications in various fields. Due to their significance, they are frequently integrated into university curricula. However, there is a remarkable connection between the two fields. In fact, numerous results in Combinatorics have been proved using elementary linear algebra concepts that would otherwise be difficult to prove. Using elementary concepts such as vector spaces, linear independence, eigenvalues, and eigenvectors, it is possible to establish intriguing links between these two subjects. As a result, it strengthens the impact and understanding of both subjects.

The purpose of this reading course is to learn various techniques and methods from Linear Algebra that can be applied to Combinatorics. We plan to cover [1, Ch. 4 & Ch. 5], [2, Ch. 11], and [4, Ch. 31]. We will also discuss the recent proof of a sensitivity conjecture [3] that relied heavily on Linear Algebra and Graph Theory. This material has inspired researchers across both disciplines including my own mathematical journey, and I hope participants will also find it useful.

To foster collaboration and discussion, the ideal group size is of four students.

References:

- [1] L. Babai and P. Frankl, Linear Algebra Methods in Combinatorics, to appear, 2020.
- [2] C. Godsil and G. Royle, Algebraic Graph Theory, Vol. 207, Springer Science and Business Media, 2001.
- [3] H. Huang, Induced subgraphs of hypercubes and a proof of the sensitivity conjecture, Annals of Mathematics, 190(3), 949-955, 2019.
- [4] J.H. Van Lint and R.M. Wilson, A Course in Combinatorics, Cambridge University Press, 2001.

### Diophantine Approximation

##### Gregory Knapp, University of Calgary

his reading gorup would start by reading the first chapter of Schmidt’s book, “Diophantine Approximation,” on rational approximations to algebraic numbers and then we would pick a direction from there. We could read about continued fractions, games, or landmark results like Liouville’s Theorem and Roth’s Theorem. If we chose to read about games or the named theorems, we would probably continue reading Schmidt’s book. If we read about continued fractions, we would continue briefly in Schmidt’s book and then possibly continue to Khinchin’s book, “Continued Fractions,” to read about some interesting results and techniques in the measure theory of continued fractions.

### Partial Differential Equations under Various Metrics

##### Cintia Pacchiano, University of Calgary

The topic for this years FYIG is: Partial Differential Equations under Various Metrics. For 4 students. The initial selection of text is the following:

- Bjorn, A. and Bjorn, J. “Nonlinear Potential Theory on Metric Spaces” (EMS Tracts in Mathematics, Vol. 17) First Edition.
- Shanmugalingam, N. “Newtonian spaces: An extension of Sobolev spaces to metric measure”. Revista Matematica Iberoamericana Vol 16, No 2, (2000).
- Evans, L. and Gariepy, R. “Measure Theory and Fine Properties of Functions”. CRC Press, (1992).
- Heinonen, J. “Analysis on metric spaces”. Lecture Notes. University of Michigan, (1996).
- Kinnunen, J. and Shanmugalingam, N. “Regularity of quasi-minimizers on metric spaces”. manuscripta mathematica. 105 401-423. (2001).
- Giaquinta, M. and Giusti, E. “Quasi-minima”. Annals l’Institut H. Poincaré: Anal. Nonlineaire 1, 79-107. (1984).