# PIMS First Year Interest Groups

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.

New PIMS FYIG are proposed by PIMS PDFs and opened for applications at the beginning of each academic year, typically in October. 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 2024-25 academic year.

The call for new First Year Interest Group projects has now closed, please see below for a list of the accepted proposals. We are now accepting applications to join these FYIGs from graduate students around the PIMS network. If you would like to join one of the projects, please use the button below to complete the application form before **October 1**.

## 2024-25 FYIG Topics

The following FYIG topics are now open and accepting applications. *Please note this list may not be complete until the project submission deadline. Please see above for the deadline and check back for additional projects at that time.* The deadline for student applications to join an FYIG will be **October 1**.

### An introduction to lattice-based cryptography

*Abbas Maarefparvar, University of Lethbridge*

#### Project Description

The advent of quantum computing challenges the security of the 'RSA' and other classical cryptosystems, which motivated the cryptographic community to turn its focus toward developing algorithms that can resist quantum computers. The result has become a new direction of cryptography, called Post-Quantum cryptography (PQ). As one of the most modern and hottest topics in data security, PQ has been deeply investigated and has made significant progress in the last decade.

The foundations of PQ cryptosystems are based on various "hard mathematical problems", including lattice-based, code-based, multivariate polynomials-based, and elliptic curve isogeny-based cryptography. Among these different classes, lattice-based cryptography is one of the most promising primitives for designing quantum-resistance algorithms which provides a striking trade-off between security and practicality. The lattice-based primitives, the hard lattice-based problems that are believed to be resistant to quantum computers, are some interesting problems in the lattice theory that can be stated in terms of some specific ideals of the ring of integers of number fields. This course aims to get elementary familiarity with lattice-based cryptography focusing on algebraic number theory. In particular, we will see some applicable aspects of mathematics in cryptography.

#### Selected Reading List

- Peikert, Chris. "A decade of lattice cryptography." Foundations and trends in theoretical computer science 10.4 (2016): 283-424.
- Jeffrey Hoffstein, Jill Pipher; Joseph H. Silverman. ''An Introduction to Mathematical Cryptography''. Springer-Verlag New York, 2016.

## Shift-cyclicity in Hardy spaces

*Jeet Sampat, University of Manitoba*

#### Project Description

Suppose X is a Banach space of holomorphic functions in one or more variables. A function f is said to be shift-cyclic if the collection of polynomial multiples of f forms a dense subspace of X. The problem of determining these shift-cyclic functions in general has connections to many deep problems in mathematics such as the invariant subspace problem, the dilation completeness problem, and even the Riemann hypothesis (although, loosely). The goal of this reading group is to hone in on the Hardy spaces over the unit polydisk, and discuss the properties of their shift-cyclic functions. We include a wide range of topics and cover the basics of function theory in one as well as several complex variables. Along the way, we employ tools such as Fourier series, Poisson integrals, subharmonic functions, etc. which have wide applicability in many different areas of mathematics.

I conducted a learning seminar on this topic last year, and was able to write up notes worth 20-30 hours of lectures. These notes can be found on my personal web-page.

In the first half, we should be able to cover most of the basic material from my notes. This allows us to branch into specialized topics for the second half. For instance, we could arrange for weekly contributed lectures from the participants, or discuss a paper that one of the participants found while doing their own research on the topic, or conduct discussions on several open problems that are motivated throughout my notes. In fact, my most recent preprint is joint work with a graduate student who attended my seminar, and it is based on such an open problem! Regardless of the research potential of this project, we will not compromise on learning the basics of the theory to ensure that every participant gets something of value out of this program.

#### Selected Reading List

- My own notes
- Chapters 1-3 and 7 from P. Duren's book "Theory of H^p spaces".
- Chapters 1-4 from W. Rudin's book "Function theory in polydiscs".
- Specific topics from N. Nikolski's book "Hardy spaces".

## Polynomial Methods in Combinatorics

*Himanshu Gupta, University of Regina*

#### Project Description

In recent years, several longstanding open problems in Combinatorics have been solved using novel algebraic techniques. This reading group will focus on exploring these techniques, collectively known as Polynomial methods in Combinatorics. Students with a background in undergraduate Linear Algebra and Algebra should find the topics accessible and engaging.

We plan to explore key topics such as combinatorial nullstellensatz, finite field Kakeya problem, polynomial methods in error-correcting codes, and joints problem. For this, we recommend reading Chapters 1-4 of Guth [1], and Chapters 16-17 of Jukna [2]. These polynomial techniques are both elegant and versatile, and we hope participants will find them not only interesting but also useful.

#### Selected Reading List

- L. Guth, Polynomial Methods in Combinatorics, Vol. 64, American Mathematical Society, 2016.
- S. Jukna, Extremal Combinatorics: with Applications in Computer Science, Vol. 571, Second Edition, Berlin: Springer, 2011.

## Sieve methods, twin primes, and beyond

*Emily Quesada-Herrera, University of Lethbridge*

#### Project Description

The unsolved twin prime conjecture states that there are infinitely many prime numbers p such that p+2 is also a prime number. With what frequency can we expect twin primes to appear among all integers? Mathematicians think we know the answer, even though no one is able to prove it.

For the last century, sieve methods have been useful to attack additive problems involving prime numbers such as the twin prime conjecture, obtaining interesting partial results, and are still an important part of recent research in analytic number theory. At the same time, many aspects of sieve methods are elementary and accessible: we can think of them as starting from, and improving upon, the classical sieve of Eratosthenes, which we learn in school to find primes.

The book by Pollack and the expository article by Soundararajan inspired me during my studies. With (selected sections of) our reading list, we will first learn some heuristics to understand why we believe what we believe about prime numbers (and twin primes in particular). We will then learn about sieve methods and some of their applications.

#### Suggested Reading List

- P. Pollack, Not Always Buried Deep, American Mathematical Society; New ed. edition (October 14, 2009).
- K. Soundararajan, Small Gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc. 44 (1), 2007, 1–18.
- A. Cojocaru and M. R. Murty, An Introduction to Sieve Methods and Their Applications, Cambridge University Press; 1st edition, January 30, 2006.
- L. Thompson and S. Carrillo Santana, Analytic number theory. Part II: Sieve methods. [Lecture notes: will be provided to participating students]

The FYIG projects listed above are now accepting applications, please click the button below to apply to join one of them

## Previous FYIG Iterations

### 2023-24 FYIG Topics

The following FYIG projects are no longer active and are provided for reference only.

## Jane Shaw MacDonald (SFU):

*Modelling Ecological Population Dynamics with Reaction-Diffusion Equations*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 limited and 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.

Some other supporting include:

- 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.

## Himanshu Gupta (UR)

*Linear Algebra Methods in Combinatorics*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.

## Gregory Knapp (UC):

*Diophantine Approximation*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).