PIMS First Year Interest Groups

Applications to join a 2026-2027 FYIG are now open - Apply to Join an FYIG

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 2026-27 academic year. PIMS postdocs leading an FYIG may be eligible for a small addition to their PDF travel supplement.

Applications to join the FYIG projects listed below are now open. Interested graduate students are invited to read the project descriptions below and sign-up if they would like to attend. The deadline for student applications to join an FYIG is October 9th.

Apply to Join AN FYIG Project

Applications are now closed for new First Year Interest Groups and will re-open next summer for the following academic year. The deadline for new FYIG project submission is typically mid September.

2026-27 FYIG Topics

The following projects were selected for the 2026-2027 PIMS-FYIG program.

1. Vertex Operator Algebras: From Algebra to Quantization (Maneesha Ampagouni, University of Saskatchewan)

2. The Vector Fields Problem on Spheres (Yang Hu, University of Regina)

3. A first look at commutative algebra (Chi Trung Chai, University of Manitoba)

4. From Particles to Plasmas: An Introduction to the Vlasov-Poisson Equation (Antoine Gagnebin, Simon Fraser University)

5. Network flows and applications (Olha Silina, University of British Columbia)

6. Proportionality and Fairness in Social Choice (Abu Mohammad Hammad Ali, University of Regina)

7. Singularities in positive characteristic (Peter McDonald, Simon Fraser University)

    

1 - Vertex Operator Algebras: From Algebra to Quantization

Maneesha Ampagouni, University of Saskatchewan

Vertex operator algebras (VOAs) provide an algebraic framework for studying structures arising in representation theory and mathematical physics. This reading group will introduce students to the foundations of vertex operator algebras. We shall study examples such as Heisenberg VOA which provides an algebraic description of the two dimensional conformal field theory corresponding to a free boson and an affine VOA. We shall also study their representation theory.  

In the latter part of the year, we shall study how the structures of vertex algebras can be generalized through quantization. The aim is not to develop the full theory, but to give students a first view of an active research direction.  

By the end of the program, students should have an understanding of the basic theory and examples of vertex operator algebras, enough familiarity with their representation theory to begin reading introductory research literature.  

Familiarity with linear algebra and basic abstract algebra is helpful. No prior knowledge of vertex operator algebras, representation theory, or conformal field theory is assumed. Relevant concepts from Lie theory will be introduced as needed.

Reading List

1. Lepowsky, James; Li, Haisheng. Introduction to Vertex Operator Algebras and Their Representations. Progr. Math. 227, Birkhäuser, Boston, 2004. 
    

Additional Material: 

1. De Sole, Alberto; Gardini, Matteo; Kac, Victor G. On the structure of quantum vertex algebras. J. Math. Phys. 61 (2020), no. 1, 011701, 29 pp. 
2. Bakalov, Bojko; Kac, Victor G. Field algebras. Int. Math. Res. Not. 2003, no. 3, 123--159. 
    

2 - The Vector Fields Problem on Spheres

Yang Hu, University of Regina

The hairy ball theorem says that one cannot nicely comb the hairs on the surface of a regular coconut – a 2-dimensional sphere. But can the same job be done on a coconut whose surface is a 2027-dimensional sphere? If so, in how many ways? In this FYIG project, we will embark together on a journey to uncover the answer by 2027. 

This problem-driven project, centered on the question of finding the maximum number of linearly independent vector fields on spheres, will introduce us to a variety of fundamental constructions in algebraic topology, including vector bundles, characteristic classes, and K-theory. We will then venture into more advanced computational techniques, including operations in cohomology theories, the Atiyah–Hirzebruch spectral sequence, and Atiyah duality. This will form the technical heart of the project and pave the way toward understanding Adams’ celebrated solution to the vector fields problem on spheres. 

This FYIG project will not only demonstrate how tools from algebraic topology and homotopy theory can be brought together to resolve a classical problem in geometry, but will also equip participants with a collection of ideas and techniques that can open many doors to future research in topology and related areas.

Reading List

1. (Vector bundles and characteristic classes.) J. Milnor and J. Stasheff, Characteristic classes, Princeton University Press, 1974. 
2. (Topological K-theory and stable homotopy theory.) H. Miller, Course notes: Vector fields on spheres, etc., available at https://chromotopy.org/latex/misc/haynes-notes/haynes-notes.pdf. 
3.  (Solution to the vector fields problem on spheres.) J. F. Adams, Vector fields on spheres, Annals of Math. 75, no. 3 (1962), 603–632."

3 - A first look at commutative algebra

Chi Trung Chai, University of Manitoba

The book Twenty-four hours of local cohomology, written by both of my advisors together with others, was an inspiration for me as a graduate student. It is a modern commutative algebra book that is appropriate for graduates, especially new ones. Depending on the background of the participants, we can lower the level to reading Commutative algebra by Atiyah and MacDonald. 

The plan is to explore the first 8 chapters of the book. They include basic materials an algebra student should know, together with many exercises and advanced materials that we may learn depending on the progress and background of the group. Topics include basic ring theory, Krull dimension, depth, and examples. 

I expect students undertaking this project shall have a better and broader idea on what commutative algebra is, and can decide whether they want to pursue further directions.

Reading List

1. Twenty-four hours of local cohomology (by Iyengar, Leuschke, Leykin, Miller, Miller, Singh, and Walther) 
2. Introduction to commutative algebra (by Atiyah and MacDonald)"

4 - From Particles to Plasmas: An Introduction to the Vlasov-Poisson Equation

Antoine Gagnebin, Simon Fraser University

A plasma is a gas of charged particles. When there are many of them, we do not follow each particle. Instead, we describe the gas by a density function in position and velocity. The Vlasov-Poisson equation is one way to model a plasma, under the assumption that collisions between particles can be neglected. It describes the evolution of the density function under the electric field that the particles create themselves. 

We will start with Golse's lecture notes, which explain how to derive the Vlasov-Poisson equation from a system of N particles. This only requires ODEs and basic measure theory. Then we will read the paper of Lions and Perthame, which proves that solutions of the three-dimensional Vlasov-Poisson system exist for all times. This is a classic example of how a priori estimates give global existence for a nonlinear PDE. Finally, if time permits, we will end with linear Landau damping, which describes the behaviour of solutions when time goes to infinity. We will follow the short treatment in Grenier, Nguyen and Rodnianski, which uses the Fourier-Laplace transform and some complex analysis. 

By the end of the project, students will know the main tools of the field, be able to read current papers on Vlasov equations or related kinetic equations, and have a general picture of active research topics.

Reading List

1. F. Golse, Mean Field Kinetic Equations. Lecture notes, École Polytechnique, 2013 
2. P.-L. Lions, B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system. Invent. Math. 105 (1991), 415–430. 
3. E. Grenier, T. T. Nguyen, I. Rodnianski, Landau damping for analytic and Gevrey data. Math. Res. Lett. 28 (2021), 1679–1702."

5 - Network flows and applications

Olha Silina, University of British Columbia

Network flow is a rich area of graph theory with a wide range of real-world applications. For example, suppose water flows from some source (e.g., a treatment plan) to consumers by traveling through different pipes — how much water can such a network sustain? This problem is classically modeled as an edge-capacitated directed graph, where the amount of flow entering any non-terminal vertex equals the amount of flow leaving it. Using tools from graph theory and optimization, one can answer not only the numerical question of “how much flow”, but also determine the structure of such flows. 

In this reading group, we consider the theory of network flows and its connection to various areas of graph theory. This includes maximum flows, the max-flow min-cut theorem, circulations, and nowhere zero flows. We will see connections between flows and graph colorings, perfect matchings, and cycle packing. Furthermore, to illustrate how widespread and active this area is, the group will peer-review a recent breakthrough result of OpenAI proving a long-standing Cycle Double Cover Conjecture. With enough interest and time permitting, we will attempt to generalize this result to weighted setting or delve into some of the numerous open problems regarding existence of flows. 

Familiarity with graph theory and / or optimization is encouraged for this group, though the texts of the reading list are quite self-contained.

Reading List

1. R. Diestel, Graph Theory, 5th edition, Springer, 2017, Chapter 6. 
2. F. Jaeger, ""Flows and generalized coloring theorems in graphs,"" Journal of Combinatorial Theory, Series B, Vol. 26, Issue 2, 1979, pp. 205-216. 
3. A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency, Springer, 2003, Chapters 10, 11, 12. 
4. J. Geelen, ""OpenAI's proof of the Cycle Double Cover Theorem,"" https://arxiv.org/pdf/2607.15399 
5. S. Oum, ""A proof of the cycle double cover conjecture by OpenAI: an exposition,"" https://arxiv.org/pdf/2607.16356"

6 - Proportionality and Fairness in Social Choice

Abu Mohammad Hammad Ali, University of Regina

Modern computational social choice has increasingly expanded beyond simple utility maximization/cost minimization, in order to address fairness and representation in complex collective decision-making. We propose a first year interest reading group group focused on Proportionality and Fairness in Social Choice, with particular emphasis on two closely connected frontiers: static multiwinner elections and repeated single-winner elections.

While static multiwinner rules formalize proportional representation through axioms such as Justified Representation (JR), Extended Justified Representation (EJR), and Proportional Justified Representation (PJR) under approval ballots, real-world systems often make sequential choices over time -- such as daily scheduling, repeated public goods selection, or committee rotations. Bridging these domains allows us to explore the interaction between optimal social choice and proportionality and fairness.

The reading group will dedicate the first phase to foundational results in static committee selection, reviewing rules such as Proportional Approval Voting (PAV), the Phragmén family, and Method of Equal Shares, among others. In the second phase, the focus will shift to temporal elections and repeated single-winner mechanisms, analyzing how persistent minority groups avoid perpetual marginalization across rounds through history-dependent scoring, virtual budget balancing, and round-by-round quota tracking. By treating sequential decisions as the temporal analogue of multiwinner representation, participants will examine where classical axioms succeed, where they fail, and where approximation algorithms can bridge the computational tractability gap.

This group is intended for graduate students in computer science, mathematics, and related fields interested in algorithmic game theory and theoretical computer science. Our eventual goal is to explore open problems in the domain of fairness in online repeated elections, including proportionality notions that are suited for the online setting, as well as the interaction between welfare maximization, proportionality, and bounds on the cost of proportionality, namely the ratio between the welfare of globally optimal outcomes and that of the best proportionately fair outcome.

Reading List

1. Aziz, H., Brill, M., Conitzer, V., Elkind, E., Freeman, R., & Walsh, T. (2017). Justified representation in approval-based committee voting. Social Choice and Welfare, 48(2), 461–485. 
2. Sánchez-Fernández, L., Elkind, E., Lackner, M., Fernández, N., Fisteus, J. A., Basanta-Val, P., & Skowron, P. (2017). Proportional Justified Representation. In Proceedings of the 31st AAAI Conference on Artificial Intelligence (AAAI-17), 31(1), 670–676. 
3. Kalayci, Yusuf Hakan, Jiasen Liu, and David Kempe. ""Full proportional justified representation."" arXiv preprint arXiv:2501.12015 (2025).  
4. Peters, Dominik, Grzegorz Pierczyński, and Piotr Skowron. ""Proportional participatory budgeting with additive utilities."" Advances in Neural Information Processing Systems 34 (2021): 12726-12737.  
5. Lackner, M. (2020). Perpetual Voting: Fairness in Long-Term Collective Decision Making. In Proceedings of the 34th AAAI Conference on Artificial Intelligence (AAAI-20), 34(02), 2110–2117. 
6. Lackner, M., & Maly, J. (2021 / 2023). Perpetual Voting: The Axiomatic Lens. In Proceedings of the 30th International Joint Conference on Artificial Intelligence (IJCAI-21). 
7. Bulteau, L., Hosseini, H., Lackner, M., & Talmon, N. (2021). Justified Representation in Approval-Based Multi-Issue and Sequential Voting. 
8. Chandak, Nikhil, Shashwat Goel, and Dominik Peters. ""Proportional aggregation of preferences for sequential decision making."" Journal of Artificial Intelligence Research 85 (2026).  
9. Phillips, Bradley, et al. ""Strengthening proportionality in temporal voting."" arXiv preprint arXiv:2505.22513 (2025).  
10. Elkind, E., Obraztsova, S., & Teh, N. (2024). Temporal Fairness in Multiwinner Voting. In Proceedings of the 38th AAAI Conference on Artificial Intelligence (AAAI-24), 38(20), 22633–22640. 
11. Teh, Nicholas. ""The price of proportional representation in temporal voting."" arXiv preprint arXiv:2605.11157 (2026). 

7 - Singularities in positive characteristic

Peter McDonald, Simon Fraser University

Rings and varieties of characteristic p>0 come equipped with a special morphism - the Frobenius. For a ring, this is the map sending an element to its pth power. Thanks to a 1969 theorem of Kunz, we know that a ring is regular (and the corresponding variety is smooth) if and only if the Frobenius is a flat map. By relaxing this condition, we get many interesting classes of singular rings and varieties, often corresponding to singularities of rings and varieties over the complex numbers. Classification of these singularities has been a major project in commutative algebra and algebraic geometry over the past 40 years, and the past decade has seen many exciting developments following André's proof of the direct summand conjecture in 2018. 

As a starting point, we will use the recent book of Karl Schwede and Karen Smith ""Singularities defined by the Frobenius map"" as well as the book of Linquan Ma and Thomas Polstra ""F-singularities: a commutative algebra approach"". We will start by learning about Kunz's theorem before introducing a few different classes of singularities in positive characteristic as well as the notion of test ideals. From there, we will decide as a group where we'd like to head. One option is to learn more of the positive characteristic theory, such as tight closure, F-signature, and Hilbert-Kunz multiplicity. Another is to explore connections between singularities in positive characteristic and singularities defined over the complex numbers. Yet a third to discuss connections with the theory of big Cohen-Macaulay modules and the homological conjectures. 

By the end of the year, students will have a strong foundation in the main classes of singularities defined by the Frobenius. They will also have a solid introduction to one of the additional topics listed above and be aware of the various active research areas in the field.

Reading List

1. Karl Schwede and Karen Smith. Singularities defined by the Frobenius map. 2026. 
2. Linquan Ma and Thomas Polstra. F-Singularities: a commutative algebra approach. 2026. 
    

Supporting material 

1. Bhargav Bhatt. Derived splinters in positive characteristic. 2012. 
2. Bhargav Bhatt, Srikanth Iyengar, and Linquan Ma. Regular rings and perfect(oid) algebras. 2018. 
3. Neil Epstein, Peter McDonald, Rebecca R.G., and Karl Schwede. Closure operations in equal characteristic zero defined via resolution of singularities. 2025. 
4. Craig Huneke. Tight closure and its applications. 1996. 
5. Karen Smith. F-rational rings have rational singularities. 1997 
6. Kevin Tucker. F-signature exists. 2012. 

Previous FYIG Iterations

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

2025-2026 FYIG Topics

• Hermie Monterde (UR): Quantum walks on graphs

  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.

   

• John Jairo Lopez Santander (UM): Explorations in Approximation Theory

  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.

• Tianxia Jia (UV): 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)

  More details

  The following papers require knowledge of stochastic differential equations.

  1. Generative Modeling by Estimating Gradients of the Data Distribution (Song & Ermon 2019)

  2. A Pedagogical Introduction to Score Models: https://ericmjl.github.io/score-models/

  3. Improved Techniques for Training Score-Based Generative Models (Song & Ermon 2020a)

  4. Score-Based Generative Modeling through Stochastic Differential Equations (Song et al. 2020c)

  Survey-style papers that summarize recent developments in diffusion models:

  1. Understanding Diffusion Models: A Unified Perspective (Luo 2022)

  2. Diffusion Models, Image Super-Resolution And Everything: A Survey (Moser et al. 2024)

  3. Diffusion Models: A Comprehensive Survey of Methods and Applications (Yang et al. 2022)

  4. An Introduction to Flow Matching and Diffusion Models(Holderrieth & Erives, 2025)

  References on U-Net:

  1. Attention Is All You Need (Vaswani et al. 2017)

  2. Searching for Activation Functions (Ramachandran et al. 2017)

  3. U-Net: Convolutional Networks for Biomedical Image Segmentation (Ronneberger et al. 2015)

  4. PixelCNN++: Improving the PixelCNN with Discretized Logistic Mixture Likelihood and Other Modifications (Salimans et al. 2017)

  5. Group Normalization (Wu & He 2018)

  6. Non-local Neural Networks (Wang et al. 2017)

• Konstantin Druzhkov (US): Cohomological Methods in the Geometric Theory of Differential Equations

  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.

• Trisha Lawrence (UC): Forecasting and Mathematical Modeling for Renewable Energy: An analysis of renewable mini-grid projects for rural electrification

  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).

  Additional Details

  The project explores theoretical and applied statistical methods, including: Data Analysis, Hypothesis testing, Analysis of Variance, Factorial design experiments, Advanced statistical methods (time permitting)

  Hypotheses Tested in the Paper

  1. H1: Controlling for the region, a higher share of renewable components in generation capacity increases total mini-grid project costs.

  2. H2: The presence of energy storage or demand management technologies significantly influences mini-grid project success.

  3. H3: The ownership model has a significant impact on mini-grid project costs.

  4. H4: Community-based ownership significantly drives the success of mini-grid projects.

  References

  1. United Nations Department of Economic and Social Affairs (UN DESA). The Sustainable Development Goals Report 2025.

  2. Environment and Climate Change Canada. National inventory report [1990–(2023)]: Greenhouse Gas sources and sinks in Canada.

  3. Asligul Serasu Duran and Feyza G. Sahinyazan. An analysis of renewable mini-grid projects for rural electrification. Socio-Economic Planning Sciences, 77:100999, 2021.

  4. Asligul Serasu Duran and Feyza Sahinyazan. Database of reneweable mini-grid projects for rural electrification, mendeley data (2020) v2, 2020.

• Nocol Leong (UL): Primes and the Theory of the Riemann zeta function

2024-2025 FYIG Topics

• Abbas Maarefparvar (UL): 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 (UM): 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 (UR): 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 (UL): 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).