## About the Directed Reading Program

### General Information

The **Directed Reading Program** is a program for undergrads and grad students/postdocs. Undergrads are paired with a mentor whose interests align with theirs and they study a topic of their choice for one semester. See the table below for a list of mentors and topics.

Many projects are based around the self-paced reading of a particular book or article with guidance by the mentor. If you'd like to do a topic different from one in the table, then feel free to contact the DRP comittee with your project idea. All projects must be approved by the DRP committee.

Directed Reading Program Fall 2019 Participants (not pictured: Luke Steverango, Emine Yildirim, Molly Liu and Ben Syms-Wilson).

### Requirements

The DRP student is required to have:

- An
**hour-long weekly meeting**with his or her mentor to discuss progress towards the goal of the project, - Students will be expected to do
**four hours of independent work**between each meeting, - At the end of the semester (before finals), each DRP student will give a 15-20 minute presentation on their work (WITH PIZZA!)

### Benefits of Program (for Undergrads)

Undergraduates participating in the Directed Reading Program:

- Learn to work independently by studying math you are interested in
- Develop relationships with graduate student mentors and faculty
- Practice your presentation and communication skills by giving a presentation in a friendly environment

### Benefits of Program (for Grad Students)

Graduates participating in the Directed Reading Program:

- Gain experience mentoring a student with less experience
- Learn more about a subject of your choice
- Great to put on your CV!

### Topics

Mentor | Interests | Topic Name | Description |
---|---|---|---|

Daniel Cloutier | Algebraic Geometry, Combinatorics, Random Matrices | Generating Functions | Following Herbert Wilf's excellently-named Generatingfunctionology (yes, all one word), we will learn about generating functions. At first glance, these are just taking a certain sequence of numbers ( the Fibonacci numbers, the binomial numbers, or any other sequence that counts something) and putting them into a power series, which seems like a counterintuitive idea. However, these turn out to be incredibly useful, and a beautiful insight as to how we can use techniques involving the infinite to solve finite problems. Recommended: First-year Calculus and Linear Algebra |

Daniel Cloutier | Algebraic Geometry, Combinatorics, Random Matrices | Algorithms and Algebraic Geometry | Algebraic Geometry is a beautiful and deep subject, with many ways of approaching it. One technique, developed in Ideals, Varieties, and Algorithms, involves attacking the subject in a very computational way, attempting to find algorithms to solve polynomial equations, which lets us think of algebraic geometry as a kind of 'non-linear algebra'. Along the way, it develops a fascinating correspondence between Algebra and Geometry, using each of them to solve problems coming up in the other domain. Recommended: Rings and Fields |

Henry Kavle | Dynamical systems (particularly hamiltonian/geometric mechanics), calculus of variations, differential geometry | Dynamical Systems (of the continuous variety) | Dynamical systems comprises many diverse areas of research; one of the main branches concerns the qualitative theory of ODEs. By thinking of ODEs as geometric flows, we can often understand the long term tendencies of a complicated system even if we cannot solve it by hand. Using these techniques, we will look at issues of stability and instability, qualitative changes as we adjust the parameters of a system (bifurcations), and possibly even a little bit of chaos.Recommended: Vector/Multivariable Calculus, Linear Algebra, ODEs. |

Henry Kavle | Dynamical systems (particularly hamiltonian/geometric mechanics), calculus of variations, differential geometry | Calculus of Variations | The calculus of variations is the delicate study of extremizing functionals (essentially "functions of functions"). In calculus, we solve extremization problems by setting derivatives equal to zero. But when dealing with functionals, we can no longer just differentiate with respect to a variable - we have to figure out how to take derivatives with respect to whole functions! A famous early application of these techniques was Bernoulli's solution of the brachistochrone problem, in which he proved that a cycloid is the curve that minimizes the time it takes a bead to slide under the influence of gravity between two points in the plane. Other developments (and possible directions for a DRP project) include the study of minimal surfaces, lagrangian mechanics, or geodesics in riemannian geometry. Recommended: Vector/Multivariable Calculus, Linear Algebra, ODEs; Real Analysis and/or PDEs are a bonus |

Stefanie Knebel | Evolutionary Game Theory, Computational Neuroscience, Stochastic Control, Robotics | Stochastic Control | How do the brain and nervous system work together to create movement? How good is the body at doing this? Since the brain is so complex, we require statisical methods in order to tackle this problem, hence the use of Stochastic Control. The main interest of the readings will be to introduce stochastic control. We can read from one or more of the books listed below:
Stochastic Processes, Estimation and Control - Speyer, Chung
Fundamentals of Computational Neuroscience - Trappenberg
Adaptive Optimal Control (Section 3) - Bitmead, Gevers, Wertz
Those who are just interested in an introduction to the brain are also welcome to apply. Recommended: Basic Statistics, Linear Algebra, Calculus |

Richard Leyland | Algebraic Geometry, Number Theory, Representation Theory | Orthogonal Groups of Quadratic Forms | The way we measure distance in $\mathbb{R}^2$ is by the function $d(\mathbf{x},\mathbf{y}) = \sqrt{(x_1-y_1)^2+(x_2-y_2)^2}$. The expression $z_1^2+z_2^2$ is called a quadratic form since all terms have degree 2. But what if we used a different quadratic form like $z_1^2-z_2^2$ or $3z_1^2 -z_1z_2+z_2^2$? How does this generalize to higher dimensions? We will investigate how distance changes when we change the quadratic form and what matrices preserve distance. This topic can serve as a nice introduction to Lie Theory and is an important topic in Special Relativity. Recommended: Abstract Linear Algebra, Group Theory |

Richard Leyland | Algebraic Geometry, Number Theory, Representation Theory | Space of Cartan Tensors | Tensors generalize matrices to higher dimensions. In differential geometry there are vectors and matrices attached to surfaces that tell us information about the rates of change of functions on the surface. The curvature tensor can tell us about how curved a surface is. We can also dive into how the curvature tensor arises from representation theory of tangent vectors. Recommended: Differential Geometry, Vector Calculus. Perhaps Representation Theory as well. |

Luke Steverango | Game Theory, Social Decision Theory, Group Theory, Cryptography, Algebraic Geometry, Combinatorics, Graph Theory | Voting/Social Decision Theory | Is there a fair voting system? How does the specific voting system influence the results? What does a fair voting system look like? In Voting/Social Decision Theory, we grapple with these and other questions, attempting to see the way we attempt to make “fair” decisions, and whether its even possible to have a voting system that leaves everyone satisfied. In this directed reading program we will cover basic types of voting systems devised, consider properties that we would like to have in a voting system, and eventually make our way to Arrow’s Theorem. If time permits, we may cover some reading into the mathematical modelling of gerrymandering and other related topics. Recommended: Interest in the Application of Math, A Course with Mathematical Proofs (Optional). |

Pei-Lun Tseng | Free Probability, Random Matrices | Free Probability | Free probability theory is a research field which in a non-commutative context parallels aspects of classical probability where the notion of independence are replaced by "freely independence". There exist two different approaches to free probability theory at a very basic level; one is analytic and the other one is combinatorics. There are many books, notes, or papers that could be a possible option to read; it depends on the DRP student's background knowledge and their preference (analytical or combinatorial approach.) Recommended: Basic probability, linear algebra, basic analysis. |

Keshia Yap | Algebra, Number Theory | Conjecture and Proof | We’ll be going through the book Conjecture & Proof by Miklós Laczkovich that discusses questions in various fields of mathematics including number theory, algebra and geometry. The book introduces interesting and important concepts in a way that is easily accessible. We can easily tailor the reading to fit the students interests. Recommended: Discrete math and basic knowledge of sets and functions. |

Emine Yildirim | Combinatorics, Algebra (specifically graph algebras and cluster algebras), Category Theory and Homological Algebra | Combinatorics of cluster algebras and graph representations | Many complicated mathematical constructions can be simplified using simple combinatorial ideas. There is a beautiful combinatorics associated with algebras coming from graphs and triangulations of regular polygons. This combinatorics is heavily used in cluster algebras which appear many areas of mathematics nowadays. In this reading program, we will investigate the combinatorics of cluster algebras and their relations with other topics. Recommended: Linear Algebra. |

Troy Zeier | Geometry, Topology, Surfaces | Combinatorics of cluster algebras and graph representations | We will be reading through Geometry of Surfaces by John Stillwell Recommended: Calculus, Analysis |

### Apply Here

**Undergrads**

Complete the following Google form

Application Deadline: September 20th 2019

**Grads and Postdocs**

If you have an idea for a topic, fill out the Google form . Please fill out one form per topic idea.

### Contact

If you have any questions, email the DRP committee here.

**Founding Members of the DRP Committee:**

- Stefanie Knebel - President knebels@queensu.ca
- Richard Leyland - Senior Vice President: leyland.richard@queensu.ca
- Daniel Cloutier - Vice President: 15dc3@queensu.ca
- Henry Kavle - Communications Coordinator: kavle.h@queensu.ca
- Luke Steverango 13lwps@queensu.ca

**Mentors Fall 2019:**

- Stefanie Knebel
- Richard Leyland
- Daniel Cloutier
- Henry Kavle
- Luke Steverango
- Keshia Yap
- Pei-Lun Tseng
- Troy Zeier
- Emine Yildirim

**Mentees Fall 2019:**

- Rylen Sampson - Computational Neuroscience
- Mukund Mauji - Synaptic Neuroplasticity
- Alicia Liu - Robust Control in Motor Functioning
- Jefferson Lin - Dynamic Neural Fields
- David Hoskin - The Riemann Zeta Function
- Hao Suen - Clifford Algebras
- Ethan Morris - Fractals and the Minkowski Dimension
- Jacob Cohen - Fractals and the Minkowski Dimension
- Eesha Lodhi - Minimal Surfaces
- Curtis Wilson - Generating Functionology
- Raymond Jing - The Combinatorial Nullstellensatz
- Michael Strban - Arrow's Impossibillity Theorem
- Victoria Bolitho - Mathematics of Gerrymandering
- Damara Gagnier - Conjecture and Proof
- Tianyu Yang - Free Probability
- Ben Syms-Wilson - Geometry of Surfaces
- Elifnaz Gulsen - Quiver Representations
- Molly Liu - Cluster Algebras