**Who:** Nine early college students who are deeply but perhaps informally prepared
for mathematics research.

**What:** Eight weeks of mathematical research on incidence geometry over finite fields. More details.

**When:**
June 11th - Aug 5th 2023 (8 weeks).

**Where:** Bryn Mawr College, alongside the participants in the
MathILy summer program. More details.

**Why:** Because students who are prepared for research should have
the opportunity to do it.

**The application process:** Consists of personal information, information about your mathematical
studies, a multi-question personal statement, and two recommendations (instructions will be sent to recommenders when you submit the aforementioned information), all of which must be received by April 5, 2023. More details.

**Stipend:** $4800, plus on-campus housing and meals while at
Bryn Mawr.

**Note about SARS-CoV-2/COVID-19:** MathILy-EST *will* take place in 2023, in person and with full vaccination required. If a change in pandemic conditions makes this impossible, then MathILy-EST will again take place synchronously online.

**Research Description:**
In 2023 the MathILy-EST topic will be **incidence geometry over finite fields**, under the direction of Dr. Brian Freidin.

Given a polynomial \( f(x,y)\) in 2 variables, the line \( y=mx+b\) is *tangent* to the curve \( f(x,y)=0\) if the polynomial \( f(x,mx+b)\) has a repeated root, and otherwise the line is *transverse* to the curve. If multiple lines are tangent to a curve at the same point, the curve is called *singular*, and might involve a crossing or a cusp.

For instance, the curve shown at right, \( x^4-2x^3+x^2+xy+y^3-y^2=0\), is tangent to the green line at the green point because \( f(x,0)\) has a double root at \( x=1\). (Verify this by setting \( y=0\) and factoring.) The curve is also tangent to the orange line at the orange point because \( f(x,1-x)\) has a double root at \( x=0\). (Verify this by setting \( y=1-x\) and... ) And the curve is singular at the red \( (0,0)\) because it is tangent to every line \( ax+by=0\). (Verify this by...)

If we consider all our equations mod \( p\), basically setting \( p=0\), then we only have to think about points with coordinates between \( 0\) and \( p-1\). Similarly, we only have to think about lines and polynomials with coefficients in that same range. (And there are only finitely many possible lines!) In this setting, the curve\( f(x,y)=0\) is called *transverse-free* if it is tangent to every line that it intersects.

There has been some recent work counting the proportion of curves that are transverse-free, mostly focused on non-singular curves. This summer we will investigate the combinatorial and linear algebraic aspects of transverse-free curves. To include singular points, one goal is to understand the number of lines that avoid a given collection of singular points. The number of such lines, to which a transverse-free curve must be tangent, will vary depending on the number and placement of the singular points.

The curve \( x^4-2x^3+x^2+xy+y^3-y^2=0 \), along with two tangent lines and a singular point

In higher dimensions, given all the hyperplanes in \(n\) dimensional space mod \( p\), how many ways are there to select some points on each hyperplane at which a curve or surface should be tangent, so that no point is selected from more than one hyperplane? And if there are singular points, we need only select tangency points from the hyperplanes that avoid singularities.

Once we have selected some singular points and tangent hyperplanes, we can look for our transverse-free curves and surfaces. We can ask: How many polynomials \( f(x_1,\ldots,x_n)\) of degree \( d\) in \( n\) variables define a hypersurface \( f(x_1,\ldots,x_n)=0\) that is tangent to a fixed hyperplane \( a_1x_1+\cdots+a_nx_n=b\)? Fixing a point on the hyperplane, the condition of being tangent at that point imposes an equation relating the coefficients of \( f\) to each other. But if we want the hypersurface \( f=0\) to be tangent to several hyperplanes, or contain several singular points, how do the relations among the coefficients of \( f\) interact? How many hypersurfaces have all the tangencies and singularities we want to impose?

**Research Mentor:** Dr. Brian Freidin is an Assistant Professor at Auburn University after earning a PhD from Brown University and a postdoc at the University of British Columbia. He did research as an undergraduate at the University of Illinois through the Geometry Lab and the Center for Complex Systems Research.

**Graduate Research Apprentice menTor at MathILy-EST (GReAT-EST):** Lily Wang is a PhD student studying theoretical computer science at the University of Michigan. Her research interests are in combinatorics and graph theory. During her undergrad, she worked on research projects involving graph drawing, polytopes, and algorithm design. When she's not working, Lily enjoys hiking, art, and taking train rides.

**Goals and Expectations:**
During the summer, each of the participants will be expected to...

- Meet each weekday (and some weekends) to collaborate with other participants on research projects.
- Record progress, ideas, experiments, conjectures, and proofs each week for submission to the MathILy-EST director.
- Meet regularly (at least twice per week) with the MathILy-EST director, both individually and as groups, to assess progress and receive advice on how to proceed.
- Attend the weekly Ever-EST Seminar to learn how to traverse the mathematical community's peaks and valleys.
- Attend the Daily Gather, an hour-long interactive mathematics lecture or activity that takes place each weekday afternoon.

Blackboards, whiteboards, glass... we have it all.

**Prerequisites:**
Applicants must be undergraduates in good standing. Preference will be given to first-year
college students, with second-year and entering college students also considered for
participation. NSF funding requires that participants must be US citizens or permanent residents.

**Transitioning into Research:**
MathILy-EST participants will have little-to-no formal experience with
mathematics research. To ease participants into the work they will be doing, the program
will start with the research mentor providing an overview of the project that includes
pointers to literature and concrete examples that are associated with questions to
investigate. Participants will be assigned sets of readings that they then must present
to one another, as well as hands-on explorations so that they can familiarize themselves
with relevant examples and generate data as a start to working on open problems.

The weekly record of progress that participants are expected to keep will provide practice in writing formally and building in time for detailed feedback and revisions of writing. In addition, it will provide fodder for paper drafts once research has progressed to the point of publication, and regular presentations on research progress will provide participants with feedback on their research methods and presentation skills.

Professional development for MathILy-EST participants will include...

- Instruction in LaTeX, Sage,
*Mathematica*, and any specialized software relevant to the research project. - How to find and access relevant books and journal articles, both through the library and MathSciNet
- What a CV is, and how to draft one
- How to give a presentation and a conference talk
- How to construct a research paper
- What sort of careers one can go into with a degree in mathematics
- How to apply for and choose graduate schools

**MathILy and MathILy-EST:**
The MathILy-EST REU will overlap with the MathILy summer program. The
two programs share space---MathILy-EST participants will stay in the same dorms as MathILy students
and instructors, and have work space in the same building as the MathILy classrooms---which
will provide plenty of opportunities for social interactions, from casual mathematical
conversations in the public areas of the dorm room to afternoon frisbee,
board, and card games.

**After the Program:**
At the end of MathILy-EST, the research mentor will help participants put together plans
for presenting their research at local and national conferences, and will check in
regularly during the following semesters to make sure they're keeping to a reasonable
timeline for preparing publications and conference presentations. In addition, MathILy-EST
participants will be invited to the {MathILy, MathILy-Er, MathILy-EST} Yearly Gather at the Joint Mathematics
Meetings, which will provide a chance to catch up with their fellow participants and to
network with participants from other years and programs.

**Why Bryn Mawr is an awesome
place to be in the summer:**

The dorms are really nice, and have air conditioning. (So do the classrooms.)
You will have a single room.

The campus is beautiful...really
beautiful. And very safe!

The food is excellent. (Still, if
you suddenly need pizza or snacks, there are several pizza places and a
grocery store near campus.)

Laundry is free.

You're right near Haverford, Bryn Mawr, Swarthmore, UPenn, Drexel, Temple, and
Villanova, all excellent institutions of higher education.

Campus is also right near a hospital, shopping center,
etc.

Philadelphia, which has tons to offer---including art and science and medicine and natural history museums, a zoo, shopping malls, a farmers market, the Italian
Market, the Declaration
of Independence (yes, the real
thing), the Liberty
Bell, and about a zillion other things---is a short train ride away.

**Practical stuff:** We'll give you all the details after you've applied and if you are admitted.

**If after reading this whole page you still have questions,**
please do contact us at .

*MathILy, MathILy-Er, and MathILy-EST are projects of the nonprofit organization Mathematical Staircase, Inc..*