# Virtual Math Circle Paid Mentorships

## Paid Mentorship (up to \$7,400)

Would you like a \$2,500 stipend to virtually mentor 3–4 high school students in a 3-week summer math research project of your choosing and design, with the opportunity for an additional \$1,200 stipend in case the project topic is augmented and receives approval to be continued beyond its initial summer session? Would you like to double both stipends—for a possible total of \$7,400—by mentoring two different projects?

The Virtual Math Circle Summer Research Camp is a collaboration between the LSU Department of Mathematics and the LSU Gordon A. Cain Center for STEM Literacy. We invite graduate students, PhDs, and post-docs from math-related fields to propose research projects for one or both of the following 3-week summer sessions (and possible continuation beyond that initial session):

- 2023 Session I: June 12 – July 1. Paid mentorship: \$2,500 initial stipend plus potential additional \$1,200 stipend when topic extension is approved.
- 2023 Session II: July 17 – August 5. Paid mentorship: \$2,500 initial stipend plus potential additional \$1,200 stipend when topic extension is approved.

Application deadline: March 15, 2023

## Mentorship Job Description

- Mentors plan a mathematics research project to be conducted with a group of 4 high school students.
- The research project design must consist of a primary phase, to be carried out in an intensive 3-week summer session, along with one or more possible topic extensions, to be carried out during the following semester.
- Most days throughout the 3-week summer session, mentors meet virtually with their group for several hours, providing virtual supervision, guidance, and support as the students conduct specified mathematical research.
- Mentors assist with the preparation of a written report and colloquium talk to be presented by the students on the last day of the summer session.
- Mentors ensure research conduct and final presentations meet substantial criteria for peer review.
- Mentors attend all final presentations (from all research groups) given on the last day of the summer session.
- The mentor workload during the 3-week summer session is expected to be 60 hours in total.
- Subject to approval by Math Circle leadership, if one or more of the students in a research group wish to continue the research beyond summer with the intention of presenting at LSU Discover Day the following April, the project will continue from the end of the summer session until the presentation in April.
- During project continuation, mentors touch base with their students at least once a week and are available to meet virtually for a research discussion at least once a month.
- Mentors assist with the preparation of a written paper or report, a poster, and a poster presentation talk, all to be presented at LSU Discover Day.
- Mentors continue to ensure research conduct and final presentations meet substantial criteria for peer review.
- Mentors attend all Math Circle presentations given at LSU Discover Day.
- The mentor workload during project continuation is expected to be 30 hours in total (spread across 8 months).

### Final summer presentations

- At the end of each 3-week summer session, each student group presents both a written report and a 30–45 minute talk at a virtual colloquium with all camp students, all camp mentors, and several university postdocs and professors in attendance.
- Videos of the talks will be recorded. Each written report and each talk video will be published on the Math Circle website.

### LSU Discover Day presentations (for projects continued beyond summer)

- Each student group participating in the research topic extension prepares a written report or paper, and a poster.
- Each student group presents their poster at LSU Discover Day with all research continuation students, all associated mentors, and several university postdocs and professors in attendance.
- Video of the poster presentation will be recorded. Each written report or paper, each poster, and each presentation video will be published on the Math Circle website.

## Research Project Guidelines

- Each research project must be accessible to high school students.
- Each research project must have a primary phase that can be completed during the 3-week summer session.
- Each research project must include one or more topic extensions that will be available after the primary phase is completed.
- Each research project should lead to open research problems with the possibility of future publication.
- You may submit several projects for consideration by camp organizers, but at most one will be chosen per mentor per summer session. (During a given 3-week camp session, mentors are expected to devote their full attention to a single research group.)
- If you plan to lead two projects (one in each session), the two projects may not be identical (but may be similar insofar as they explore different aspects of a single larger research problem).

## Mentor Qualifications and Eligibility

- Applicants must either:
- hold a PhD in a math-related field; or,
- be currently enrolled as a graduate student working towards a PhD in a math-related field.

- International postdocs and PhDs: you must be eligible for employment in the United States.
- International graduate students: you must be enrolled and in good standing at a university in the United States and provide employment authorization from your university. (For example, Curricular Practical Training is an employment authorization issued by International Services offices to allow F-1 students to participate in off-campus employment deemed an integral part of an established curriculum.)

## How to Apply

Complete the mentorship application form once you have gathered and prepared the following:

- copy of your CV;
- copy of your college transcript(s);
- research project proposal(s).

In case you would like to submit a project for Session I and another project for Session II, you may submit both proposals in a single application.

All applications submitted by the application deadline (see top of this page) will be considered. Late applications will be considered only if student enrollment demand calls for the availability of additional projects.

### Selection process

Project proposals deemed acceptable by camp organizers will be published for students to register. If at least 3 high school students register for a given project, then that project is funded, meaning you will be hired to lead that research group. You earn the summer stipend at the completion of the summer mentorship, and earn the potential extension stipend after the presentation at LSU Discover Day.

## How to Prepare Your Research Project Proposal

Using LaTeX when possible (otherwise, you may use Microsoft Word), write a clear proposal on a topic that high school students could reasonably explore over a 3-week session. Include the following:

- which summer session you are available to lead this project;
- your name, position, and institutional affiliation;
- a project title;
- a mathematical subject area category (graph theory, probability theory, combinatorics, topology, etc.);
- specifically state whether high school calculus is a prerequisite, or will be taught as needed, or isn't expected to occur in the project;
- clearly state all assumed background knowledge (e.g., high school algebra, geometry, probability, calculus, coding) so students know whether the project might align with their interests and skill level;
- provide an abstract (1–2 paragraph descriptive summary of the research proposal, at most 250 words, avoiding math jargon and notation as much as possible);
- explicitly state that you will be available, for interested students, for a research topic extension into the following academic year culminating in a poster presentation at LSU Discover Day;
- briefly describe one or more possible topic extensions that will be available;
- include a project outline/timeline (1–2 pages) showing a topical plan along with goals/milestones for each week of the summer camp;
- list of citations (material that will be used to conduct the research).

You will need to submit a PDF as well as the LaTeX (or Word) source file from which the PDF is generated.

For topic ideas, you may find it helpful to peruse our archive of past projects. Note that your research proposal must also include a project outline/timeline for each week of the summer camp. For an example of this, expand the sample proposal below.

### Expand to view a sample research proposal

#### LSU Math Circle Research Proposal

- Session
- 2021 Session II (Jul 18–Aug 6, 2021)
- Mentor
- Isaac Michael, postdoc, Louisiana State University
- Project title
- Probability and Coin Flipping
- Topic area
- Probability Theory
- High school calculus
- Not a prerequisite; some calculus will be introduced and taught as needed
- Skills and Background
- This research project requires little background knowledge other than basic algebra and arithmetic. All required skills needed to approach the problem will be taught by the instructor. This project serves as an excellent introduction to
*probability theory*,*counting methods*,*sequences*,*series*, and*mathematical proofs*. During the experimental stage, we will implement computational tools such as*Excel*,*Python*, and/or*R Studio*. During the development of the final presentation, we will use tools such as*PowerPoint*,*Beamer*,*ShareLaTeX*, and/or*TeXShop*. - Abstract
- A
*fair coin*has equal probability of obtaining heads or tails. Say you want to make an equal-probability decision between two outcomes, such as which team in a sports game gets to go first. You can accomplish this using the fair coin by assigning heads to one outcome and tails to the other. But what if you have to make an equal-probability decision between*three*outcomes? Is it possible to use a fair coin in some way to accomplish this?We will develop an algorithm to use a fair coin to select from three mutually disjoint events with equal probability. We then hope to extend this result to the case of $n$ pairwise disjoint events, developing an algorithm so that, using a (two-sided) fair coin, each event has probability $1/n$.

If this can be achieved, we seek to generalize further by considering the case where the coin is no longer fair; that is, for some $p\in(0,1)$, we have $$P(\text{heads})=p,\qquad P(\text{tails})=1-p.$$

Under this modification, is it still possible to choose from among $n$ pairwise disjoint events so that each would occur with equal probability?

- Possible extension
- Once the iteration algorithm is established, let $N$ denote the random variable associated to the number of iterations needed to produce the desired outcome of equal probabilities. Can the expected value $E(N)$, variance $V(N)$, and standard deviation $STD(N)$ of $N$ be computed?
- Outline/timeline
- This is a general (tentative) outline of how the research project will progress. We will adjust the schedule as needed throughout the 3-week period.
**Week 1 (Background & Examples).**The first week will be more instruction-based. We will begin with a brief overview of probability theory using references [1, 2]. The students will be introduced to basic notions such as*probability functions*,*sampling with/without replacement*,*mutual independence*,*conditional probability*, and*Bayes’ Rule*.We will introduce the basic notions of a rigorous mathematical proof, including

*proof by exhaustion*,*proof by induction*, and*proof by contradiction*. The students will be given several examples and assignments to help strengthen their skills in proof writing.**Week 2 (Experimentation & Main Problem).**The second week will be based heavily on experimentation. The goal during this stage is to determine an algorithm (‘game’) between three people, which, using a coin, yields a 1/3 probability of winning. All experiments will be implemented via*Excel*,*Python*, and/or*R Studio*.Once a method has been found with strong empirical evidence of success, the next stage is to prove the assertion rigorously. We will first attempt to prove the base case $n=3$ and then generalize to the case $n≥3$. We will then extend, if possible, to the case where the coin is no longer ‘fair’. That is, for some $p\in(0,1)$, $$P(\text{heads})=p,\qquad P(\text{tails})=1-p.$$

**Week 3 (Finalizing Results & Presentation).**The first half of week three will be spent finalizing the empirical results of the experiments and the mathematical proofs.The second half of the final week will be spent creating the final presentation. We will decide whether to use

*PowerPoint*or the more sophisticated*Beamer*presentation using*LaTeX*. This will depend on the comfort levels of the students with*LaTeX*typesetting.

- References References are given below.
- G. Casella, R. Berger, Statistical Inference, 2nd ed., Duxbury Advanced Series, 2001.
- J. Rice, Mathematical Statistics and Data Analysis, 3rd ed., Duxbury Advanced Series, 2006.
- J. Stewart, Calculus: Early Transcendentals, 8th ed., Cengage Learning, 2016.

## Questions?

Contact Isaac Michael <imichael@lsu.edu>.