biotech Research
Although the BCA Math Team has been typically geared towards math competitions, we have begun a new initiative to inform team members and other highschoolers about the world of mathematics research. This page will eventually serve a similar role to our "Contest Prep" page, detailing opportunities and information about math research programs and competitions.

2023 Research Information Session
The BCA Math Team ran its first studentled informational session on Math Research for Highschoolers on Friday, August 25th, 2023. The event was four hours with six sessions. In session 1, Krish Ramkumar introduced math research basics, approaches, programs, and competitions (3:003:30). In session 2, Rushabh Mehta discusses REU experiences and themes of successful projects (3:304:00). In session 3, Michal Lewkowicz covers becoming a first author and securing research opportunities (4:004:30). In session 4, students collaborate on creating research ideas (4:455:30). In session 5, Justin Zhang shares insights on presenting and publishing research (5:306:00). In session 6, we went over interviews with successful researchers. (6:006:30). The 'resources' tab has a summary, notes, and the recording of each session. A playlist containing all of the recordings can be found here. Images of the event can be found in the 'photos' tab.
Session 1
In Session 1, Krish Ramkumar presented "What is Math Research?", setting the base for the rest of the day. He discusses what is math research, ways to approach math research, math programs available, and ways to compete with math research. All research is essentially an inquiry to gain knowledge, while math research has ideas independent of reality. The scientific method is altered slightly, as mathematical ideas can be proven. One can approach math research in many ways (Directed Reading, Expository Work, Original Research Work). Directed reading involves reading papers with mentors to learn knowledge about research work. Exposition involves more focused reading to find a niche result with write ups being able to submit to journals. Krish also describes the various amounts of math camps and research programs available, with some notable ones being MITPRIMES, RSI, and PROMYS. Lastly, Krish discusses ways to compete with math research, such as competitions as ISEF, or conferences and journals to publish work in.
 What is math research?
 “Inquiry to gain knowledge”
 Common example of research is Scientific Method
 Math research
 Ideas independent of reality
 Conjectures (patterns that hold)
 Often looks for a new result
 Logical deduction
 Applied Math Research
 Mathematical modeling, data analysis, simulations
 Infrequently Revolutionary
 Scientists build off each other
 Results are difficult to achieve
 Months, if not years without results
 Higher entry barriers
 What are ways to approach math research?
 Directed Reading
 Work with mentors to read a particular math book/paper
 Present summaries of what they have learned
 Helps learn knowledge about research work and learn deeply
 However, you do not get to contribute
 Expository Work
 More focused reading
 More niche result
 Presentation/writeup
 Contributes to math knowledge, writeups can be submitted to journals
 However, lots of prerequisites
 Original Research Work
 Come up with a research problem
 Read extensively about subfield
 Develop a strategy to study the question
 Work on problem/Solve/writeup
 Making a problem is one of the hardest parts
 Choose a nontrivial question
 Find patterns
 Read interesting papers
 Directed Reading
 What math programs are out there? Which are specific to research?
 AwesomeMath
 {MathILy, MathILyEr}
 HCSSiM
 PROMYS
 Ross
 SUMaC
 Canada/USA Mathcamp
 Etc (SWiM, Idea Math, SAMS, HSMC, MMSS, RYSP, Prove It!, MOP)
 Builds knowledge
 Know your deadlines!
 PRIMESUSA/MIT PRIMES
 Hard problem set
 Huge hourly commitment per week
 Very advanced background needed
 High intensity, High Enrichment
 Find mentors
 Research Science Institute (RSI)
 Rising Seniors only
 Stony Brook University’s Simons Summer Research Program
 Rising Seniors only
 REUs
 PROMYS (sort of)
 What are ways to compete with math research?
 Compete for
 College Apps
 Research Programs/Opportunities
 Actionable Deadlines
 Prize Money
 Competitions
 ISEF
 Regeneron STS
 Yau Science Award
 NJAS > AJAS
 Sharing Research
 Conferences
 Journals
 JSM, JMM, ICIBM, ISSB, IEEE
Session 2
Rushabh Mehta (UChicago ‘21; ISEF 3rd place; Regeneron STS Scholar) discusses his experiences at the UChicago REU (Research Experience for Undergraduates), themes of successful math projects he has seen at the Long Island Science Fair, and general advice and resources for Math Research. He provides strategies to develop a research topic, as well as giving sources of knowledge in order for one to learn more. One of his key ideas is that writing up work and receiving feedback lends itself to valuable insights. Rushabh believes that "we are all imposters", and that you should never be afraid to discuss ideas with mentors. In Rushabh’s own words, he says the key point of research is to “explain in a way that makes sense to everyone”.
Resources and contact information can be found at https://rmehtany.github.io/.
 Think about math 24/7
 How do you read?
 Same conclusions as author
 Material (understand material)
 Motivation (why present?)
 Inference (predict what the author is going to teach next)
 Rabbit Holes
 Avoid them
 Exchange between time / what to learn
 Ask mentor as many questions as you need
 Learning Places
 Real Not Complex
 MIT OCW
 PDF Drive
 Math Stack Exchange
 Project Gutenberg
 Youtube
 Learn the Lingo
 Example of not learning the lingo: "Interuniversal Teichmueller" horror story
 Claimed to prove a,b,c conjecture
 Spent 20 years
 No professor could understand it
 Author laughed at by everyone
 Search
 Convert ideas/keywords in head to people
 Explain like the audience is a 5yearold
 Approach Zero
 Pi Base
 Sci Lag
 DOAJ
 OEIS
 Banff Math Center
 Zentralblatt Math
 Not to produce something new
 Produce a newer/better to understand expectation
 Bash Away
 Demo & Visualize
 Use LLM with caution
 Not good tools to produce demonstrations
 Wolfram Alpha
 Sage Cell
 GPT4
 Practice writing papers
 High school level paper will have >50% background information, <50% results
 Start in the middle of the paper
 Write introduction last
 MOM test
 If your mother can understand the paper, then it should be sufficient.
 Helps write introductions
 Get Feedback ASAP
 Detexify
 Quiver
 Overleaf
 Professors
 Platforms for type of papers
 ArXiv
 Join Communities
 We are all imposters
 Main takeaway: EXPLAIN IN A WAY THAT MAKES SENSE TO EVERYONE!
Session 3
Michal Lewkowicz (Yale ‘24) discussed how he was able to become a first author on publication at IEEE MRS 2019 and strategies to secure research opportunities. He goes over sample cold emails he has sent and discusses strategies to reach out to professors, postdocs, and PHD students. He believes that conferences are critical for learning about the research process and everyone should try to attend a conference. He recommends strategies like connecting with student inclusion chairs, navigating the publication cycle, and volunteering (at conferences).
 CONFERENCE WORK IS NOT NECESSARILY GOOD FOR SCIENCE FAIRS & VICE VERSA
 Science fair is more about practical application and simple explanation
 Conference work is about incremental advances to science
 Reach out to PHD/Grad Students
 Identify your niche
 Go to conferences
 Student inclusion chairs & funds
 Vicious Publication Cycle
 Grants / Funding
 Networking
 Volunteering
 Organizational Work
 Workshops
 Example: Mini Hackathon
Session 4
Session 4 was a working session. Students developed courses of study for the academic year, collaborating and identifying topics of interest for research, exposition, or reading. Students were guided in choosing their topics, with emphasis being placed on challenging one's ability and having a manageable size of background reading. Examples of topics chosen include Zermelo's theorem (game theory) and loss minimization algorithms in data compression.
Michael Klyachman (UChicago '25) provides a list of topics approachable at the expository level.
 Classification of Platonic Solids (3d, then look at nD) as well as 3D Archimedean solids
 Constructible polygons (the road to Galois theory)
 Why alternating groups are simple
 Classification of frieze/wallpaper groups
 Intro to the complexity zoo (L = NL)
 Determinants and Resultants (W.R.T systems)
 Group addition operators (direct product, wreath product, etc.)
 Look at any of the later chapters of Evan Chen's napkin
 Look at anything from the 2022 UChicago REU. Examples include
 Jessica Cao  ERDOS DISTANCE PROBLEMS
 Ishaan Goel  INTRODUCTION TO MARTINGALES WITH AN APPLICATION IN FINANCE
 David Hu  MARKOV CHAINS IN FINITE STATE SPACES
 Elena Li  THE FERMATEULER THEOREM AND ITS APPLICATION TO PUBLIC KEY CRYPTOGRAPHY
 Jake Zummo  AN INTRODUCTION TO GEOMETRIC GROUP THEORY
 Eli Baur (2021)  THE APPEAL OF NONSTANDARD ANALYSIS
 Rebecca Golovanov (2021)  FINITE FIELDS AND THE MOBIUS FUNCTION
 Finite state machines and how Google's RE2 works
 Look at 3b1b's Summer of Math Exposition for inspiration
Session 5
In Session 5, Justin Zhang (2022 ST Yau Silver medalist; PRIMES ‘22, ‘23) discussed how he presented his research at conferences and published it. Justin goes over his research journey, providing tips applying to research programs and camps. He also discusses his projects and the steps leading up to them. One of Justin's key points is to just get started as your first project will never be perfect. He also emphasizes the importance of conferences, making note of the fact that conference acceptance rates are higher than journal rates, making it a great place to discuss your work and receive feedback.
 Great start for research
 Math Competitions
 Tests your mentality
 Try
 You can’t go wrong
 Just pick a topic
 Justin’s Topics
 Be able to present your ideas
 Topological Data Analysis
 Math Research w/ Dr. Penev
 Physics Research w/ Dr. Dogru
 Be able to explain it to anyone
 Find niche topics
 For Justin, he took the Quantum Mechanics Elective
 ArXiv.org
 Read the papers posted on your topic
 Conferences
 Golden opportunity to learn
 Conference acceptance rate > journal acceptance
 Heart Arrhythmia Project < 10th grade
 Camps
 RSI
 Make MOP or win ISEF
 PRIMES
 PROBLEM SET IS KEY
 How you do on the problem set
 REALLY FOCUS
 Page of Recommended Readings on Website
 Look at problems
 GOOGLE STUFF
 SUMAC
 Do the video
 It helps
Session 6
In Session 6, we went over interviews with successful researchers. This includes Joseph Vulakh (PRIMES ‘22, ‘23; RSI Top 10 Presentation ‘23; ISEF AMS Karl Menger Award ‘23) and Rinni Bhansali (Stanford ‘24; PRIMES ‘19; Regeneron STS Scholar; ISEF 2nd Award ‘19) . During his interview, Joseph talks about how to start research, with anecdotes as to how he himself got into research. Joseph says a key part of learning research is able to understand the material to conduct high level research. He emphasizes the importance of mentorship at all stages of the research process. Rinni emphasizes the importance of getting your hands dirty through the story of trying to read a dense Analytic Number Theory paper. She also mentions the importance of problem solving, recommending the book "The Art and Craft of Problem Solving" by Paul Zeitz.
Joseph Vulakh Interview Notes
Q: Could you share some of your early experiences or challenges when you started conducting research what advice do you have about getting started?
A: Getting caught up to the frontiers of math is hard you need to do a lot of reading. Get excited about learning math without worrying too much about your contribution.Q: What roles did mentors play?
A: In all three phases of research (reading, active research, writeup) mentors are VALUABLE. Many readings will call certain steps 'trival' when they aren't to highschoolers. This is a place where mentors can clear this up. However, if the mentor provides you with a good question, they won't know the answers themselves. Mentors will help guide you to other areas of reading when you get stuck during the active research phase. Finally, for writingup mentors can be useful for checking your work and formatting.Q: What is general advice you would give?
A: Be open minded. Don't feel like you have specialized as you are still only in high school. Every new project is an opportunity to explore a new area.
Rinni Bhansali Interview Notes
Rinni started doing math research seriously after returning from Ross. She started reading an Analytic Number Theory paper and conversed with a professor. She expresses that you can learn a lot just by diving head first into an area. This experience showed her that she wanted to start work in a different area.
She applied to PRIMES with a preference in Applied math research. PRIMES and its application is a lot of effort. Working through every exercise of "The Art and Craft of Problem Solving" by Paul Zeitz helped greatly in gaining the skills. PRIMES offers structure that is very helpful in long term projects.
Rinni says to try to do math in group formats wherever possible. She emphasizes that even having an hour to talk with someone who knows what they are talking about is extremely useful.
Photos
 What is math research?