Math Circle student research honors

Math Circle students Gili Rusak, Matthew Babbitt, and Zubin Mukerjee have won honors this year for their original math research projects.   They will be presenting their research in separate sessions at the 20th annual Hudson Undergraduate Math Research Conference, which will be held at Williams College on Saturday April 6.

Gili's applied mathematics project, An Analysis of Teenage Twitter Communities, which draws on graph theory, probability, and sociology, won top honors at  Capital Regional Science and Engineering Fair at RPI  last week.  This means that Gili will represent our region at the Intel  International Science and Engineering Fair (ISEF) to be held in Phoenix, Arizona.  This is the world's largest international science fair, where over 1,500 students from over 70 countries around the world gather to present their work.  Over $3 million in prizes will be awarded.  Gili is a sophomore at Shaker High School who has been taking advanced college classes at Siena College.  You can read more about Gili's awesome work as a mathematical community builder here.

Matt Babbitt's graph theory research project, Counting number of edges, thickness, and chromatic number of k-visibility graphs, won semifinalist honors in this year's Intel Science Talent Search.  Matt's research benefits from advice from his MIT Research Science Institute mentor, Jesse Geneson, and Dr. Tanya Khovanova, the head mathematics mentor at RSI.  Matt, a homeschooled senior from Fort Edward  who has taken advanced math classes at Union College, has been named a Jack Kent Cooke Scholar and plans to attend MIT next year.

Zubin Mukerjee's number theory research project, Random Involutions and the Number of Prime Factors, is based on his joint work with a fellow student, Uthsav Chitra, at PROMYS last summer.  Their research mentor was Dr. Kristen Wickelgren, a research fellow at Harvard.   The project won semifinalist honors in the Siemens Competition in Math Science and Technology.  Zubin, a senior at Guilderland High School, who has been taking advanced math classes at SUNY Albany, has also won a number of honors in history and music.  You can read more about Zubin here.

Gili Rusak, mathematical community builder

Gili Rusak launches students on investigations of Archimedean solids.

Gili Rusak, a tenth grader at Shaker High School who also takes advanced math classes at Siena College, has been building deep, rich, and inclusive mathematical communities all around the Capital District and even beyond.  For the past two years, she has been helping Doyle Middle School teacher Nancy Smith with coaching Doyle's MATHCOUNTS team in Troy.

Last winter, she participated in the first annual SUMiT, a "fully collaborative, math intensive event" organized by Girls' Angle and the Undergraduate Society for Women in Mathematics at MIT and returned home efferverscent with enthusiasm about the wonderful experiences she had had as a participant in that event.   She came back inspired with a missionary zeal to create a similar math event here in the Capital District, to bring that same mathematical joyful collaboration to students in the Capital District.

After months of thoughtful planning and brainstorming, Gili designed, organized, and led a completely marvelous math treasure hunt inspired by the SUMiT model.  Gili's local event took place at Union College's Kenney Community Center last summer.  Watching Gili and the two AAMC veterans she had recruited to help, Cecilia Holodak and Elizabeth Parizh, orchestrate this event was the single most epic math experience of my entire career as a math outreach volunteer!  (And I have had many awesome ones, so that is saying a lot!)  The photo below shows Gili and Elizabeth with some of their happy treasure  hunters and you can learn much more about that treasure hunt in the writeup and photos on Gili's blog here.

Gili leading a Math Treasure Hunt she designed and organized for younger girls at the Kenney Community Center  at Union College late last summer .

After hearing about Gili's very successful local treasure hunt, Ken Fan at Girls' Angle invited Gili to help him lead a much larger treasure hunt at Microsoft New England Research & Development Center as part of a social event ("Games Night") at the Math Prize for Girls at MIT last fall.  It was a *huge* hit engaging scores of girls from all over the United States and Canada.

Math Prize for Girls @MIT participants enjoy the extremely fun yet challenging math treasure hunt Kan Fan and Gili ran at a "Games Night" social event at Microsoft New England Research and Development (NERD) Center.

Gili's account of that night is here.  Ken describes one fun part of their treasure hunt, Mental Madness, here.  In another event, called "Robo-Ape", Gili and Ken asked the girls to compose algorithms to instruct a robotic ape about how to eat a banana.  Gili then read their algorithms aloud while Ken played the role of the robotic ape, executing their algorithmic instructions quite literally to great amusement.  (You can see a video clip of RoboApe here.)

Ken Fan from Girls' Angle and Gili in the Robo-Ape event

Attendees at the Math Prize social event included Stephen Wolfram and his 15-year-old daughter Catherine, who was intrigued by the treasure hunt idea that Ken and Gili were leading.  Afterwards, Gili and Catherine stayed in touch and worked together to create yet another local treasure hunt back in Schenectady at Union College's Kenney Center in early November, this one with a Halloween theme.  You can see a little bit of their treasure hunt in this video (starting at 3:27).   Gili described some of their activities in her blog here.

Gili is an outstanding role model, a trail blazer who is creating wonderful road maps that other students can follow as well to create their own mathematical community building events!  She is only a tenth grader, but her work thus far exceeds my wildest dreams of what I would have thought possible.  And she started out in a small satellite middle school math circle led by Zagreb Mukerjee at a table in the Clifton Park library back in when she was a fifth grader.

Where it all began years ago:  a younger Gili (center, back to camera) participating in a small satellite middle school math circle led by Zagreb Mukerjee (standing) at the Clifton Park Library.

 Zagreb is now off in college, but Gili is indeed doing her utmost to "pay it forward" and share the magic of creating vibrant local mathematical communities with younger people in new and innovative ways.  And who knows what wondrous activities Gili will--in her turn--inspire the young students with whom SHE is working to do a few years down the road, when it is THEIR turn to pay it forward!

Sign up for AMC10B or AMC12B at Siena College!

Thanks to Siena College Math Department and especially to Professor Mohammad Javaheri (a silver medalist at the 1995 International Math Olympiad) for sponsoring this exciting math contest for high school students in the Capital District.  The AMC10/12 is the first in a series of challenging "extreme math problem solving" events that ultimately leads to selection to the US team for the International Math Olympiad!  It also opens many other doors as well, including the American Invitational Math Exam, the USA Math Olympiad and USA Junior Math Olympiad.  If you are a girl, it is also the entry point to the Math Prize for Girls at MIT next September!

 We especially encourage participation from students who enjoy math challenges but who have never even heard of this contest before. High school students interested in joining Albany Area Math Circle sessions to help prepare you to have an enjoyable extreme math experience should contact AAMC advisor Mary O'Keeffe at mathcircle@gmail.com for more information about how to prepare. Please use the form below to sign up to take the contest at Siena on February 20.

 

Congratulations, Zubin!

Zubin Mukerjee, at right, with other members of the Upstate New York All-Star Math Team at the national high school math tournament, ARML, in June

Zubin Mukerjee, a veteran leader of Albany Area Math Circle who has also organized and led a satellite math circle of his own for younger students, has blazed yet another a new trail for others to follow.

Zubin, a Guilderland High School senior, who is taking advanced classes in math and economics at SUNY Albany, and his co-author, Uthsav Chitra from Delaware, have won semifinalist honors in a prestigious national research contest for high school students.  Zubin and Uthsav worked on an original research project in number theory, "Random Involutions and the Number of Prime Factors of an Integer,"under the guidance of a mentor at PROMYS in Boston last summer.

Research presents new challenges as well as new rewards compared to the contest problems with which many math circle students are familiar.  When you work on a contest problem, it may be very hard, but you KNOW that you are working on a problem that somebody else has already solved and that there must be a clever elegant solution to the problem.   It is indeed exciting to have the Aha! moment when you find the solution to a contest problem, but such moments pale compared to those you can experience in math research, the thrill of discovering an answer to a problem nobody else has ever found before. Sometimes the results are negatives ones, not exactly the ones you were hoping for originally, but even those disappointing results can provide important clues to promising new lines for exploration.

It takes passion, perseverance, and luck to find original new research results, far moreso than in contests.  When doing original research, there are no guarantees at the outset that the problem will even have a solution at all, let alone that it will yield interesting results worth sharing with others.   Even once the problem is solved, it takes excellent writing skills to write up your research results in a way that will allow others to appreciate the importance and validity of what you have discovered.  Zubin's years of helping to write power round solutions for our math circle teams as well as his prize-winning entries in history day competitions have certainly polished his expository writing skills.

Here are the abstract and executive summary for Zubin and Uthsav's research project.

Abstract:  For hundreds of years, mathematicians have tried to find good approximations for the function d(n), which counts the number of prime factors of an integer n. In this paper, we examine using random involutions to approximate d(n) by comparing the number of fixed points of a random involution on F₂^2g(n) to the number of fixed points of a specific involution, τ(n). We find and prove that the expected number of fixed points of a random involution converges, so that d(n) cannot be approximated using this method; moreover, we use this to show that the involution τ(n) is not random, as it has more fixed points than a random involution.

Executive summary:  The natural numbers are perhaps the most familiar to humans. They are the counting numbers: 1, 2, 3, etc. A divisor of a number is something that divides evenly into that number. For example, 3 and 14 are divisors of 42, but 42 is not a divisor of 3 or 14. A prime number is a natural number whose only positive divisors are 1 and itself. The first few primes are 2, 3, 5, 7 ... there are also infinitely many of these. There is a well-known function that returns the number of prime divisors of a number n, given that number. We denote this function d(n). Our goal in this project is to further research on modeling d(n).

Our mentor proposed a possible method of modeling d(n) by looking at special functions called involutions that act on the surface of modular curves. In particular, we studied the involution τ(n), which is related to d(n), by comparing it to random involutions. We were able to conclude,through a series of proofs and derivations as well as some graphical analysis using Mathematica, that d(n) cannot be modeled by τ(n) and that, as a result,  τ(n) is not random. In other words, there is something special about τ(n) that makes it so we can’t model d(n).

The consequences of this result are not yet fully clear. Nevertheless, this result can lead the way to studying other types of involutions, some of which may be able to model d(n). An accurate model for d(n) would be incredible, as it would make finding the prime factorization of large numbers much easier; this would have many applications in cryptography and computer science. Much research remains to be done on involutions though; perhaps one day, a closed-form expression for d(n) will be found through random involutions.

Zubin and Uthsav's research mentor was Dr. Kirsten Wickelgren, an American Institute of Mathematics fellow at Harvard University.  Here is a link to a copy of the background document including the problems she suggested they investigate as well as definitions of some key concepts and a helpful list of the supplementary references with which she initially launched them on their way.  If you are interested in understanding more about their work, you may want to take a look at those references yourself.  Students who have not yet studied much number theory will also find the Art of Problem Solving's textbook on introductory number theory very helpful.  [Added later:  Zubin also passed along another recommendation of a classic number theory book, Hardy & Wright's Introduction to the Theory of Numbers, endorsed by PROMYS Director Glenn Stevens as "clear and concise."  Zubin also notes that Hardy & Wright cover many topics in number theory in their book, some relevant to their project and some not.]

You will also note that Zubin and Uthsav used Wolfram Mathematica computer software to help create graphs to give them insights into their problem analysis.  Thanks to Wolfram's sponsorship of contests such as American Regions Math League (ARML), Harvard-MIT Math Tournament (HMMT), and Princeton University Math Contest (PUMaC), Zubin and all our veteran math circle students who have participated in one or more of those contests have received free student licenses to use this very powerful software.  Those licenses will remain valid as long as they are students, including college and graduate school years ahead.

Math Circle travel contest dates this year

Math contests are a fun (but entirely OPTIONAL) part of our Albany Area Math Circle experience.

To help our members plan ahead, I will describe the entire calendar of contests in which members can participate below.  I have broken them down into two categories: travel contests and locally administered contests.  This post describes the travel contests we plan for this coming year.  My next post will describe the locally administered contests we plan.


Harvard-MIT November Tournament (HMNT):  
Saturday November 10 at Harvard University in Cambridge, MA   
This is an ideal travel contest for our younger veterans who have done at least NYSML in the past, but who have not yet done much proof-writing.   Note that participating in this contest rules out participating in PUMaC the following weekend.  It also rules out participating in the (much harder) February 2013 HMMT.  If you are seriously considering this contest but not yet sure of your schedule, please email Mr. Babbitt ASAP to let him know of your tentative (or definite) interest.
HMNT November Cost per student:  $10 registration fee plus modest costs for lunch.  AAMC can put parents in touch with one another to arrange carpools so that parents can share gas and tolls.  Math contests at Harvard and MIT have been done as a daytrip by AAMC mathletes who were comfortable getting up early and sleeping in the car.   If your parents agree, it may also be possible for students to arrange to stay overnight in a Harvard or MIT dorm with a college student host or hostess.   (You would need to bring a sleeping bag, and you will probably be sleeping on the floor.  Note that dorms can be noisy on Friday nights!)  Parent drivers may want to work out arrangements to share hotel rooms (unless they are driving as a day trip.)   Although we will put parents in touch with each other to facilitate whatever arrangements seem most suitable to them, it is entirely up to the parents involved to work out mutually satisfactory travel arrangements.


Princeton University Math Contest (PUMaC):  
Saturday November 17 at Princeton University in Princeton, NJ 
This is a very challenging contest which includes a collaborative team round.   Only experienced veteran students should consider this contest.  It is considerably harder than the AIME, so students who have not yet taken the AIME should not consider it.  But even the AIME is not sufficient preparation, because the team round requires mathematical expository skills.  The team round is proof-based and essay style.  Students need to be prepared to work collaboratively (and remotely) on a google document from Nov 10 through November 16.  In addition, there will be mandatory on-line practice in October for all team members on our AoPS forum.  Non-team members are also welcome to join the on-line practice forum.  Our Math Circle veteran leaders, Matt Babbitt and Zubin Mukerjee, will provide further information about what is expected of team members on the AoPS forum.  If you are seriously considering this contest but not yet sure of your schedule, please email Mr. Babbitt ASAP to let him know of your tentative (or definite) interest.
Cost per student:  $12.50 registration fee (which includes lunch).  AAMC can put parents in touch with one another to arrange carpools so that parents can share gas and tolls.  Parent drivers will probably want to work out arrangements to share hotel rooms.  Due to the distance involved as well as the need to finalize the power round with a final team meeting in Princeton on Friday evening, this event is NOT doable as a day trip.  Although we will put parents in touch with each other to facilitate whatever arrangements seem most suitable to them, it is entirely up to the parents involved to work out mutually satisfactory travel arrangements.

Harvard-MIT Math Tournament (HMMT)
Saturday February 16, 2013 at MIT in Cambridge, MA
This contest has always been extremely hard, and we expect that trend to continue to unprecedented epic levels of difficulty this year.  Only those with extreme tolerance for mathematical endurance should consider this event.  However, mathematical strength is not enough to be selected for this team, since collaboration with other team members will be very important.  It is very important that all members of our HMMT team know one another's strengths and weaknesses so that the "whole can be more than the sum of the parts."  It is also important that all team members encourage one another when the going gets tough, because--believe me!--it will get tough!

Our advanced veteran students who wholeheartedly participate in fall math circle activities including our PUMaC on-line practices on the AoPS forum and regularly attend (most of) our Friday meetings with an extremely positive and supportive spirit that encourages others not to give up when the going seems hopeless will be given first preference in team selection.   All other factors being equal, seniors will be given preference over younger students.   For younger students, participation in previous years' HMMT on-line events or travel events will be a plus factor.  All other factors being equal, younger students who have done well in prior year on-line HMMT events will be given preference over those who attended the travel event and did equally well.

Cost per student for Feb HMMT:  $10 registration fee plus modest costs for lunch.  AAMC can put parents in touch with one another to arrange carpools so that parents can share gas and tolls.  The event can be done as a daytrip if your mathletes are comfortable getting up early and sleeping in the car.   If your parents agree, it may also be possible for students to arrange to stay overnight in a Harvard or MIT dorm with a college student host or hostess.   (You would need to bring a sleeping bag, and you will probably be sleeping on the floor.  Note that dorms can be noisy on Friday nights!)  Parent drivers may want to work out arrangements to share hotel rooms (unless they are driving as a day trip.)   Although we will put parents in touch with each other to facilitate whatever arrangements seem most suitable to them, it is entirely up to the parents involved to work out mutually satisfactory travel arrangements.

New York State Math League (NYSML)
Saturday April 13, 2013 at Biram Hills  High School in Westchester County
NYSML is an ideal "first" travel competition for our students.  It is the statewide math championship for high school students.  It is held in a different place each year.  We (Albany Area Math Circle) were the host league in 2010, Suffolk County in eastern Long Island hosted in 2011, Southern Tier was the host league in 2012.  The 2013 host will be the Westchester-Putnam Math League.    All high school students who attend meetings regularly and demonstrate mathematical and behavioral maturity as well as enthusiasm and persistence are welcome to join our NYSML teams.  Unlike the other travel tournaments, we expect to be able to take as many 15-person teams as we would like to NYSML.  (In 2010, we had three teams with almost 45 students.  If a team is a bit short of 15 students, we can use alternates from other teams and vice versa at NYSML.)

Cost per student for NYSML:  Unknown at this time, but probably around $20 per student (including lunch).  It is a potential day trip (for mathletes who don't mind getting up early and sleeping in the car.)   Some parents and mathletes may prefer to stay in a hotel.  As with all travel contests, we will facilitate parents getting in touch with each other to make their own carpool/travel arrangements.   Although we will put parents in touch with each other to facilitate whatever arrangements seem most suitable to them, it is entirely up to the parents involved to work out mutually satisfactory travel arrangements.

American Regions Math League (ARML)
Saturday June 1, 2013 at Penn State (depart Albany morning of 5/30 and practice with team on 5/31 at Penn State, so students need to miss several days of school for this.)
ARML is the national high school championship event.  It has a format similar to NYSML but the problems are harder.  Albany Area Math Circle students can apply to be on the Upstate NY ARML team.   You will want to wait until you have your AMC, and (if applicable) AIME and NYSML scores before you apply to the team.  NYSML experience and performance will also be a big plus for ARML team selection.

Important info about conflict with SAT testing date:  if you are thinking of doing ARML, please plan ahead for the SAT conflict.  June 1 is an SAT administration date.  It is expensive and time-consuming and may not be possible to arrange a makeup date.  It is MUCH better if you plan to take your SAT subject tests in May rather than June, and/or your SAT reasoning tests in January or March rather than June.  The SAT schedule for this coming year is here.

Cost per student for ARML:  Unknown at this time, but in recent years the cost has been around $300 to $325.  That included a chartered bus from Binghamton to Penn State and back as well as two night's housing in Penn State dorms and some, but not all, meals.  Students also needed about $25 to buy additional meals on their own.  Thanks to Cecilia and Gili's moms efforts in fundraising the NYSSPE donated funds that helped to offset about $100 of the costs per student.

In the next post, I will describe the contests that we expect to be offered locally, without the need to travel.

Upstate NY All Stars!

Albany Area Math Circle members of the New York State All Star Math Team thank the New York State Society of Professional Engineers for their support:  
Yujun (Etheal) Chen, Emma Willard School
Sherry He, Emma Willard School
Gili Rusak, Shaker High School
Jien Ogawa, South Colonie High School
Cecilia Holodak, Niskayuna High School
Matt Babbitt, heeg
Zubin Mukerjee, Guilderland High School
Ziqing (Bill) Dong. Farnsworth Middle School

Complete 2012 Upstate NY AllStar Math team roster for 2012 ARML, including students from all over Upstate New York (defined as New York State minus {New York City, Nassau County, Suffolk Count})

Matthew	Babbitt	heeg
Jemmin	Chang	Somers
Yujun	Chen	Emma Willard
Yiqun	Cui	Brighton
Matt	DeCross	Pittsford-Sutherland
Pranav	Devarakonda	Brighton
Matt	Dobbins	Corning West
Ziqing	Dong	Farnsworth
Ryan	Gao	Brighton
Huajun	Gu	Corning West
Jeff	Guo	Penfield
Sherry	He	Emma Willard
Cecilia	Holodak	Niskayuna
Doug	Knowles	Churchville-Chili
Jeremy	Koob	Corning West
Benjamin	Lei	Arlington
Devin	Li	Corning West
Allen	Liu	Penfield
Ben	Lowenstein	Pittsford-Sutherland
Dongze	Lu	Harley-Allendale-Columbia
Evan	Lustick	Canandaigua
Zubin	Mukerjee	Guilderland
Jien	Ogawa	South Colonie
Jordan	Roeder	Canandaigua
Gili	Rusak	Shaker
Perry	Wang	Corning West
Felix	Weilacher	Penfield
Yunmei	Zhang	Harley-Allendale-Columbia
Simon	Zheng	Somers

Guerrilla Math venues?

Schenectady--and more broadly, the Capital District, have many wonderful free outdoor activities.  They are fun family places to go and enjoy our beautiful summers.  And they are great places for you and your families  to do Guerrilla Math!  Just bring some sidewalk chalk or a portable whiteboard and markers, and your ideas for fun math explorations to share with families waiting for the event to begin or during intermissions, afterwards. 

Here are few links I got from a colleague at Union College.  If you can suggest other links, please post in the comments below.

Central Park (Music Haven) Schenectady summer concerts

Jazz on Jay Street

Freedom park summer concerts (Scotia)

Other Schenectady events