## 2020 Diversity in Mathematics: Online Undergraduate Summer School Information and Registration

- Start Date: 08/04/2020
- End Date: 08/14/2020

Vancouver, BC

**Onine Undergraduate Summer School Eligibility Requirements:**

The online summer school is open to *female-identifying, non- binary and two- spirit undergraduate students* studying mathematics or a related discipline such as computer science, physics and statistics at a university in Canada or in the northwest United States, with at least one year of studies remaining in their program. Priority will be given to second and third-year students. Each participant will receive a certificate of participation for attendance.

Please note that the purpose of the program is to introduce the undergraduate participants to a wide variety of professions and careers, in academia and in industry, where advanced mathematics is used every day with spectacular success. It is NOT a purely research-oriented or purely industry-oriented summer school, rather a sampler of different flavours of mathematics-based careers.

A new component of the program will facilitate online engagement and interaction with high school students who are good at math, who may not have had enough opportunities to learn about the impact of advanced mathematics and who can thrive under the mentorship of bright and strong undergraduate women in STEM.

Please use the above information to assess your interest and fit for the program, as well as indicate the weeks you would like to attend.

### 2020 Course Overview:

__Course 1: ____Inverse problems and images__

**Course Dates**: August 4- August 8 (inclusive)

**Course Instructor**: Tracey Balehowsky, University of Helsinki

**Brief Course Description:** Many problems which we would like to solve in the sciences can be abstractly described as follows: given our observations (measurements) of the system we seek to understand, determine the properties of the system which gave rise to the observations. Mathematically, we call this situation an inverse problem, since we know something about solutions (observations) to our model of the system, but we want to determine the coefficients of our model (which encode information about the system’s properties). In this course we will introduce inverse problems arising in imaging, such as computed tomography (CT scan) and magnetic resonance imaging (MRI). We will explore some mathematical techniques used for image construction, denoising, and sampling.

**Prerequisites**: Multivariable calculus, second year linear algebra (in particular comfortability with SVD, bases, eigenvalues and eigenspaces).

**Tech Requirements:** Access to computers with Matlab. I plan to have the student fill in some prewritten code to run.

__Course 2: ____Self-complementary graphs and cyclic hypergraph decompositions__

**Course Dates**: August 10- 14

**Course Instructors****:** Shonda Dueck, University of Winnipeg

**Prerequisites****:** Discrete Mathematics, Graph Theory, basic Group Theory is an asset (mainly Cyclic Groups).

**Course Description**: A hypergraph consists of a set of points called vertices, and a set of subsets of this vertex set called edges. A simple graph is a hypergraph in which every edge has cardinality 2. Hypergraphs are used to model many types of networks, such as transportation systems, the link structure of websites, and data organization networks. One interesting problem in hypergraph theory is that of decomposing a hypergraph into smaller subhypergraphs which all have the same structure. Such decompositions in which the subhypergraphs have desirable properties, such as high symmetry or regularity, are of special interest since they correspond to key structures in combinatorial design theory that have useful applications in cryptography.

We will begin by studying the self-complementary graphs. These are the simple graphs which are isomorphic to their complement, so a self-complementary graph and its complement together decompose the complete graph into two isomorphic subgraphs. We will determine necessary conditions on the order of the self-complementary graphs and look at some nice algebraic techniques for constructing them. The self-complementary graphs are well studied due to their relation to the graph isomorphism problem, which has unknown complexity. Next we will generalize this idea to study the self-complementary hypergraphs, using the same algebraic construction techniques we applied to graphs. We will also construct some self-complementary graphs and hypergraphs which have high symmetry using basic cyclic group theory. Finally, we will study the t-complementary hypergraphs, which are the parts in a cyclic decomposition of the complete hypergraph into t isomorphic hypergraphs. At the heart of this mini course is the study of how permutations of a given finite vertex set V act on the subsets of V, and some basic number theory will come into play.

**Tech Requirements****:** It might be helpful for students to have access to GAP - Groups, Algorithms, Programming - a System for Computational Discrete Algebra. Distributed freely at https://www.gap-system.org/

__Program Delivery:__

__Program Delivery:__

**Courses: **The courses above will be taught by female instructors Each course is taught over one week from 9am PDT. The students will do group work with presentations the last day for each course.

**Panel discussions****: **There will be four panel sessions from academic and industry individuals

**Joint Sessions with High School Students**: There will be two joint "mentorship" sessions each week with high school students.

**Attendance**: Students can attend either or both weeks. Particpants will be issued with a certificate of attendance at the end of the program.*Please note that there is now only space available for week 2 courses.

**Registration f****ees**: Registration for week 2: $25 CAD:

**Delivery**: This summer school will be delivered online through zoom or other similar platforms. Accepted participants need to ensure that they have adequate bandwidth and connectivity to facilitate the online session, as well as running any mathematical programs concurrently.

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### Program Schedules:

Week 1 Schedule: Online Version.

Week 2 Schedule: Online Version.

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