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 centre on an accessible subject for beginning graduate students.

New PIMS FYIGs are proposed by PIMS PDFs and opened for applications at the beginning of each academic year. Each PDF leads a small reading group (up to 4 students) of 1st and 2nd year MSc/PhD graduate students through books/papers that inspired them, and which are accessible to early graduate students. The groups meet virtually, for an hour, once every two weeks through the end of the 2025-26 academic year.

Application for this year's FYIG are listed below. If you are a first or second year graduate student who would like to participate in one of these groups, please click the button below and register your interest. The projects listed have been proposed but have not yet been finalized. Project selection depends on various criteria, including enrolment. The application deadline is October 10th, and applicants will be notified at that time of the final results.

Apply to Join a FYIG

2025-26 FYIG Topics

Potential FYIG projects for the 2025-2026 program are listed below. If you are interested in joining one of these groups, please click the button above. The deadline for student applications to join an FYIG will be October 10th and the final selection of FYIGs will be made at this time (click for more details).

1. Quantum Walks on Graphs (Hermie Monterde, University of Regina)

2. Explorations in Approximation Theory (John Jairo Lopez Santander, University of Manitoba)

3. An introduction to Generative Diffusion Models (Tianxia Jia, University of Victoria)

4. Cohomological Methods in the Geometric Theory of Differential Equations (Konstantin Druzhkov, University of Saskatchewan)

5. Forecasting and Mathematical Modeling for Renewable Energy: An analysis of renewable mini-grid projects for rural electrification (Trisha Lawrence, University of Calgary)

6. Primes and the Theory of the Riemann zeta function (Nicol Leong, University of Lethbridge)

7. Introduction to Toric Varieties (Emanuela Marangone, University of Manitoba)

1 - Quantum walks on graphs

Hermie Monterde, University of Regina

Quantum walks describe the propagation of quantum states in a graph. This is particularly important in the construction of quantum computers, where the accurate transmission of quantum information (in the form of quantum states) is a key task. This motivation has attracted a great deal of interest in the study of quantum walks in the last few decades.

This reading group will explore elementary techniques (algebraic and combinatorial) used in the study of quantum walks. Students with a background in undergraduate Linear Algebra and Graph Theory should find the topics accessible and engaging. We plan to explore key topics such as perfect state transfer, periodicity and strong cospectrality. For this, we recommend reading Chapters 1, 4 and 7 of Coutinho and Godsil [1] and Chapters 1, 2, 3 and 5 of Monterde [2].

Reading List

1. G. Coutinho and C. Godsil. Graph Spectra and Continuous Quantum Walks, 2021.
2. H. Monterde. Quantum state transfer between twins in graphs. MSc. Thesis, University of Manitoba, 2021.

2 - Explorations in Approximation Theory

John Jairo Lopez Santander, University of Manitoba

Approximation theory is a central area of mathematics that studies how complicated functions can be represented and analyzed through simpler objects such as polynomials and rational functions. The subject naturally combines analysis, numerical methods, and applications, making it an ideal entry point into research for first-year graduate students. We will use Trefethen’s book Approximation Theory and Approximation Practice (2013), supplemented with selected research papers, to introduce both classical results and modern perspectives that connect rigorous theory with practical computation.

The group will begin with readings and problem-solving from the main reference, complemented by computational experiments in MATLAB or Python to deepen their understanding. In the meetings, we will discuss the main results of each section and reviewing progress on the exercises, comparing different approaches and working collaboratively whenever solutions or topics remain unclear. After covering the foundations, we will turn to research papers such as Olver et al. (2020) on fast algorithms with orthogonal polynomials and Weniger (1989) on nonlinear sequence transformations, showing how approximation theory translates into efficient numerical methods.

By the end of the project, students will have developed a solid understanding of key ideas in approximation theory, along with hands-on computational experience through writing codes in MATLAB or Python to implement algorithms and test theoretical results. They will also produce solutions to a range of problems that connect theory and practice, and gain exposure to current research directions through selected papers. The collaborative structure and the combination of theory, computation will help them build intuition, technical skills, and confidence as they prepare for future research.

Reading list

1. Trefethen, Lloyd N. "Approximation Theory and Approximation Practice; 2013." Philadelphia, SIAM: zbMath ID 1264.

2. Olver, Sheehan, Richard Mikaël Slevinsky, and Alex Townsend. "Fast algorithms using orthogonal polynomials." Acta Numerica 29 (2020): 573-699.

3. Weniger, Ernst Joachim. "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series." Computer Physics Reports 10, no. 5-6 (1989): 189-371.

3 - An introduction to Generative Diffusion Models

Tianxia Jia, University of Victoria

Diffusion models are a new class of generative models that have transformed machine learning research, with wide ranging applications such as image super-resolution, high-quality content generation, and data synthesis across many scientific domains. For example, in atmospheric science, diffusion models are emerging as a powerful tool for downscaling, i.e. generating high-resolution climate fields from coarse-scale simulations. This offers new opportunities for studying extreme weather and improving regional climate projections.

This First Year Interest Group will introduce both the mathematical foundations and scientific applications of diffusion models. On the theoretical side, we will study key concepts such as stochastic differential equations (SDEs), stationary distributions, and connections to partial differential equations (PDEs) that underlie diffusion processes. On the applied side, we will explore how these ideas enable generative modeling in machine learning, and we will examine applications to atmospheric science, including statistical downscaling.

At the end of the project, students are expected to have a working understanding of the mathematics behind diffusion models, some hands on experience experimenting with open source implementations, and a sense of how these tools might connect to their own research interests in mathematics, atmospheric science, or related fields.

Reading List

1. DDPM: Denoising Diffusion Probabilistic Models (Ho et al. 2020)

2. Improved Denoising Diffusion Probabilistic Models (Nichol & Dhariwal 2021)

3. DDIM: Denoising Diffusion Implicit Models (Song et al. 2020b)

4. SR3: Image Super-Resolution via Iterative Refinement (Saharia et al. 2021)

5. Stable Diffusion: High-Resolution Image Synthesis with Latent Diffusion Models (Rombach et al. 2022)

6. SRDiff: Single Image Super-Resolution with Diffusion Probabilistic Models (Li et al. 2021)

7. Probabilistic weather forecasting with machine learning (Price et al. 2024)

8. CorrDiff: Residual Corrective Diffusion Modeling for Km-scale Atmospheric Downscaling (Mardani et al. 2025)

9. Fast, scale-adaptive and uncertainty-aware downscaling of Earth system model fields with generative machine learning (Hess et al. 2025)

4 - Cohomological Methods in the Geometric Theory of Differential Equations

Konstantin Druzhkov, Universtity of Saskatchewan

Many geometric structures that arise in nonlinear differential equations have a cohomological origin. These include, in particular, symmetries, conservation laws, variational principles, presymplectic structures, and even Poisson brackets, when viewed from an appropriate perspective.

The aim of this course is to present some classical ideas and applications of the geometric theory of differential equations while delving into its modern cohomological foundations. This approach will allow us to address subtle conceptual questions: why conservation laws are best understood as geometric rather than physical objects; what fundamental gaps remain in the current understanding of Poisson brackets in the theory of PDEs; and in what essential sense gauge systems differ from other systems, with this distinction manifesting in their intrinsic geometry.

Reading List

1. A. M. Vinogradov, I. S. Krasil'schik (eds.), Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, American Mathematical Society, Vol. 182, 1999.

2. J. Krasil’shchik, A. Verbovetsky, Geometry of jet spaces and integrable systems, Journal of Geometry and Physics 61 (2011) 1633–1674.

3. Jet Nestruev, Smooth Manifolds and Observables. Graduate Texts in Mathematics, Vol. 220. Springer, Cham, 2020.

4. P. J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed., Springer-Verlag, 1993.

5. G. W. Bluman, A. F. Cheviakov, and S. C. Anco, Applications of symmetry methods to partial differential equations. Applied Mathematical Sciences, Vol. 168, 2010.

5 - Forecasting and Mathematical Modeling for Renewable Energy: An analysis of renewable mini-grid projects for rural electrification

Trisha Lawrence, University of Calgary

The global population with access to electricity has steadily increased—from 84% to 92%—but approximately 700 million people still live without reliable access to electricity. As the world strives to meet sustainable energy goals, such as those outlined in the United Nations Sustainable Development Goal 7 (SDG 7)—which advocates for affordable, reliable, sustainable, and modern energy for all by 2030 [1]—the challenge of rural electrification remains significant.

In parallel, the Mining Association of Canada supports the federal government’s initiatives to reduce greenhouse gas (GHG) emissions. In 2023, Canada reported emissions of 694 megatons of carbon dioxide equivalent (Mt CO eq), representing a reduction of 65 Mt (8.5%) compared to 2005 and a modest decrease of 6.0 Mt (0.9%) from 2022. Despite this progress, Canada still contributes approximately 1.4% of global emissions, ranking as the 12th largest emitter [2].

This interest group focuses on the paper “An Analysis of Renewable Mini-Grid Projects for Rural Electrification” [3], which evaluates 104 renewable energy mini-grid projects implemented globally. The study presents a compelling application of statistical methods to a global development challenge, making it highly relevant to both undergraduate and graduate-level statistics courses.

The project will conduct both analysis and meta-analysis of the compiled dataset from 104 mini-grid projects [4] with the following objectives:

• Use data visualization tools (e.g., pie charts, histograms, and box plots) to produce meaningful comparisons and summaries related to project success and cost per capita.

• Interpret statistical models to identify key factors associated with mini-grid success or failure.

• Examine whether region and technology type are significant predictors of project success, including any interaction effects.

• Assess the influence of energy storage, demand management systems, and the renewable energy proportion on both project costs and success outcomes.

Reading List

1. Design and Analysis of Experiments” (8th Edition) by Douglas C. Montgomery, specifically Chapters 2, 3, and 5 (if time permits).

6 - Primes and the Theory of the Riemann zeta function

Nicol Leong, University of Lethbridge

The Riemann zeta function is intricately linked to the study of prime numbers and plays a crucial role in the area of number theory. In fact, solving the famous Riemann hypothesis would give us an extremely good understanding on the distribution of primes.

While any course in analytic number theory surely includes some material on the zeta function, the theory of the zeta function itself is deep and has numerous implications. The zeta function is not only in itself a very active area of research, but also quite often a starting point for generalisations, for instance to various L-functions, which are useful for elliptic curves, or even in algebraic settings (Langlands program).

We will mainly go through the classic that is Titchmarsh's book. Rather than focusing on technical details, the goal rather is to provide everyone with a big picture understanding on active research areas on different aspects of the zeta function, and their implications and connections to various conjectures, as well as other areas in number theory. The supporting reading list will also supplement our study by keeping us up to date with more modern results. Depending on which areas the group are more interested in (up for discussion), we could explore generalisations to different L-functions and their consequences (supplemented with material from the supporting reading list).

Reading List

1. E. C. Titchmarsh (Revised by D. R. Heath-Brown), The theory of the Riemann zeta-function. Oxford university press, 1986.

Supporting material:

1. H. M. Edwards, Riemann's zeta function. Vol. 58. Academic press, 1974.

2. A. Ivic, The Riemann zeta-function: theory and applications. Courier Corporation, 2012.

3. H. Davenport, Multiplicative number theory. Vol. 74. Springer Science & Business Media, 2013.

4. H. L. Montgomery and R. C. Vaughan, Multiplicative number theory I: Classical theory. No. 97. Cambridge university press, 2007.

5. H. Iwaniec and E. Kowalski. Analytic number theory. Vol. 53. American Mathematical Soc., 2021.

7 - Introduction to Toric Varieties

Emanuela Marangone, University of Manitoba

The study of Toric varieties is an interesting part of algebraic geometry that has strong connections with polyhedra, combinatorics, commutative algebra, symplectic geometry, and topology. Moreover, Toric varieties have unexpected applications in areas such as physics, coding theory, algebraic statistics, and geometric modeling.

This reading group will provide an introduction to the subject, following roughly the first part of "Toric Varieties" by Cox, Little, and Schenck. The precise content will be adapted to students’ backgrounds and interests. We will focus primarily on Chapters 1–6, with the possibility of exploring Chapters 8–9 (and, if time and interest permit, Chapters 10–11).

The group will have an interactive format, with meetings alternating between short presentations of material, problem sessions led by students, and open discussions and questions. The expected outcomes for the participants are to gain a first understanding of toric varieties, explore their connections to combinatorics and commutative algebra, and practice working through concrete examples and exercises.

The group is aimed at early-year graduate students. As a prerequisite, we assume familiarity with the material covered in basic algebra and topology courses. While some prior exposure to algebraic geometry (for example, through an introductory course or an undergraduate algebraic geometry class) would be beneficial, it is not required. Key background concepts will be introduced during meetings or through the readings.

Reading List

1. "Toric Varieties" David A. Cox, John B. Little, Henry K. Schenck

2. Notes “Lectures on Toric Varieties” David A. Cox

Previous FYIG Iterations

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

2024-2025 FYIG Topics

• Abbas Maarefparvar, University of Lethbridge An Introduction to lattice based cryptography

  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

  1. Peikert, Chris. "A decade of lattice cryptography." Foundations and trends in theoretical computer science 10.4 (2016): 283-424.
  2. Jeffrey Hoffstein, Jill Pipher; Joseph H. Silverman. ''An Introduction to Mathematical Cryptography''. Springer-Verlag New York, 2016.
• Jeet Sampat, University of Manitoba Shift-cyclicity in Hardy spaces

  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.

  1. My own notes
  2. Chapters 1-3 and 7 from P. Duren's book "Theory of H^p spaces".
  3. Chapters 1-4 from W. Rudin's book "Function theory in polydiscs".
  4. Specific topics from N. Nikolski's book "Hardy spaces".
• Himanshu Gupta, University of Regina Polynomial Methods in Combinatorics

  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.

  1. L. Guth, Polynomial Methods in Combinatorics, Vol. 64, American Mathematical Society, 2016.
  2. S. Jukna, Extremal Combinatorics: with Applications in Computer Science, Vol. 571, Second Edition, Berlin: Springer, 2011.
• Emily Quesada-Herrera, University of Lethbridge Sieve methods, twin primes, and beyond

  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.

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

2023-24 FYIG Topics

• 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).