Email: firstname [underscore] lastname [at] berkeley [dot] edu
Home Institution: UC Berkeley
Mentor: Lev Borisov
I am a fourth year pure mathematics undergraduate student at UC Berkeley. As far as mathematical interests go, I am a bit of a generalist. That is to say, I have yet to take a math course that I didn't enjoy. With that being said, I have a particular affinity for algebraic geometry and mathematical logic. In my free time, I am an amateur pianist and guitarist.
Any complex surface which has the same Hodge diamond (equivalently the same Betti numbers) as the complex projective plane but is not biholomorphic with the complex projective plane is called a fake projective plane (FPP). There are exactly 100 such surfaces, all of which can be realized as a complex 2-manifold via a quotient of the complex 2-ball B². These surfaces come in pairs, with fifty pairs in total. It is known that every FPP can also be realized as a complex algebraic surface. However, out of the fifty pairs, only twelve so far have been explicitly realized as an algebraic surface via giving an explicit set of defining polynomial equations.
The goal of this project is to find explicit defining polynomial equations for three FPPs which have not yet been explicitly realized as algebraic surfaces. In particular, these three FPPs are labeled as (C18,p=3,\{2\},D_3), (C18,p=3,\{2\},(dD)_3), and (C18,p=3,\{2\},(d^2D)_3) in the Cartwright-Steger classification. The broad outline of how we plan to do this is as follows: we start with the FPP labeled (C18,p=3,\{2I\}) in the classification, which has known defining equations. We will then attempt to lift these equations up to a common cover of these four FPPs, and then bring them back down to the three FPPs with unknown equations. This project will involve a lot of heavy computation, much of which will be carried out in Mathematica.
I would like to thank Professor Lev Borisov for mentoring me through this project. I would also like to thank the DIMACS program for providing me this outstanding opportunity to participate in this research over the summer. Lastly, I would like to thank the NSF for supporting the REU through grant CCF-2447342, and the Rutgers Math Department for supporting my project.
Week 2 Presentation Slides (PDF)
Week 8 Presentation Slides (PDF)
Final Report (PDF)
I arrived at DIMACS on Tuesday 5/26. I met with my mentor on Wednesday and Friday to discuss details of the project. We went over background information, discussed basics of FPPs, and discussed what the project would entail. Over the weekend, I began to read through chapter 10 of Artin's Algebra, to become familiar with complex representations and character theory of finite groups, as it is important to our project. In addition, I read through the paper Finding equations of the fake projective plane (C18,p=3,\{2I\}) by Lev Borisov and Bojue Wang. I also downloaded the mathematica files from the paper and began to run them on my own computer, as the computational data from these will be needed for our project this summer.
This week, I finished reading through chapter 10 of Artin. In addition, I gave a 5 minute presentation on my project on Tuesday morning to the rest of the DIMACS participants (PDF of slides can be found in the Files section above). Per Professor Borisov's GAP calculations from last week (which I doubled checked on my own computer for a quick sanity check), the automorphism group of the common cover between the three unknown FPPs and the one known FPP is C3xC3xS3, and each cover corresponds to an order 18 subgroup. Thus, the next step for us was to calculate and begin to analyze the character table for C3xC3xS3 (Artin chapter 10 is already becoming quite important). I worked with Professor Borisov on Friday to get a script set up in Mathematica to help us do this. I have also been attemtping to run the Mathematica files from the paper by Borisov and Wang, but have had some difficulty (possibly limitations in computing power), so finishing running those files is currently a big short-term goal.
I encountered some difficulty running the previously mentioned Mathematica scripts throughout this week, as they proved to be a bit finicky. After a discussion with my mentor, we decided the best thing to do would probably be to write our own scripts to run instead, as a lot of the steps from the scripts we were trying to re-use were not strictly necessary for our project. Thus, we began to work on our own scripts. The character table of C3xC3xS3 suggested there should be sections f1 and f2 in H0(18FPPHat,K) that transform via the characters (1,1,sgn) and (omega,1,sgn) respectively. We then wrote g1=f1^2, g2=f2^3, and g3=f1f2, as these products are invariant under the covering involution from the 2 to 1 cover from 18 to 9FPPHat (thus the gi descend to seection in H0(9FPPHat,2K)). The gi also satisfy g1g2=g3^2. We then wrote the gi as a linear combination of unknown coefficients over a previously calculated basis of H0(9FPPHat,2K). We subjected these to the relation g1g2=g3^2 and plugged them into previously calculated points in order to obtain a system of polynomial equations in the unknown coefficients. Our hope was to be able to solve them, and then subsequently recover f1 and f2 to give us concrete sections in H0(18FPPHat,K). Unfortunately, all of our attempts to have Mathematica numerically solve this system faield. It may be the case that we simply lack enough computing power to solve this system. For now, we are looking to see if we can find a trick to solve it with our available computing power, or for another angle to approach this problem. On an unrelated note, I took a trip to New York on Saturday the 13th, and witnessed many festivities going on in the city after the Knicks won their first NBA finals in 53 years. This was quite fun.
At the suggestion of my mentor, we decided to try a new strategy, as last week's method of trying to solve for the gi seemed like it may be beyond the capabilities of our hardware to solve. We instead began to look for the nonreduced linear cut on FPPHat. The end idea here is that there is a section s on the double cover which is a (-1)-eigenvector for the covering involution, and thus its square is invariant and will be a section on 2K of FPPHat. It is then the case that s^2=equation for nonreduced linear cut (the reasons for this are a bit long and I am not going to type them). My mentor calculated the Hilbert polynomial of the known FPP mod 17 and it was as expected, thus it seemed like things may be well enough behaved over the finite field F17 to work in this space. The benefit of the finite field is as follows: there are only finitely many hyperplanes over F17, so it is possible in principle to simply do an exhaustive search to see which ones of them are nonreduced. Our only problem is that there are ~17^9 hyperplanes, and checking all of them for nonreducedness in Magma would take about a million days. To rectify this, I wrote a program in C that checked each hyperplane for a weaker but necessary condition for the hyperplane to be nonreduced. This weaker condition is still very restrictive, and also can be checked extremely quickly. The program ran in about 18 hours on my laptop, and reduced the possible candidates from ~17^9 down to ~1000. My mentor then checked the surviving 1000 hyperplanes in Magma to see if any were nonreduced, and as expected, there was exactly one such hyperplane. Our next step is to lift this hyperplane modulo 17^k, and then use the resulting 17-adic approximation to recover an explicit equation for this nonreduced cut over an algebraic number field
This week, we first finished lifting the coefficients of the nonreduced cut to an algebraic number field. I then did some more computational work with sections of the form WiWj, which had been computed in my mentor's previous work. They were pairwise products of basis elements Wi and Wj of H0(9FPPHat,K), such that each pair had opposite gradings for the C3xC3 action and thus the product was invariant and descends through the quotient. We then looked at the reducible curve WiWj=0 on FPPHat, and wanted to find the connected components of these (ie, seperate WiWj=0 into "Wi=0" and "Wj=0", although this is not 100% precise because W1 and W2 individually are not sections of H0(FPPHat,K) but rather of H0(9FPPHat,K); to be precise we are looking at the points that come from W1=0 and W2=0 on 9FPPHat)). To do this, we chose a random point p on WiWj=0. Then we used Mathematica to solve WiWj=0 on the FPP in a power series near p (up to degree 30). We then looked at all quadratic monomials in our unknowns and their restriction to this power series. At this point, our original plan was to read off these series coefficients into a matrix, and then use some linear algebra from that point to find the individual equations for the connected components "W1=0" and "W2=0". But in the interest of saving time, we instead opted to generate a bunch of random points on W1W2=0 then use the above matrix to sort them into which lie on "W1=0" and which on "W2=0", which is sufficient for what we need in the next step.
This week, we construced points on the 18-fold cover of FPPHat. We started with the Wi from last week, which form a basis of H0(9FPPHat,K). We first labeled them according to their C3xC3 grading. We know that each section should have two preimages under the double covering, and we also know that if we multiply four of them together such that the gradings add up to zero, the resulting section descends to H0(FPPHat,4K). Thus, we looked for quadratic relations amongst three of the sections which we already knew, and then solved those relations together with the equations for FPPHat and an intersection with a random linear cut. By looking at solutions to these equations that did not lie on any of the three sections that we knew, it must then be the case that those solutions came from the unknown section, allowing us to get arbitrarily many points with arbitrary precision on all 17 of the Wijk. Our next goal is to extend the C2 action; for the action of C3xC3xC2 that gives the 18-fold cover, C2 is a subgroup of S3. 18FPPHat is the common cover of the four FPPs, and C3xC3xS3 is its automorphism group. FPPHat comes from quotienting by the C3xC3xC2 previously mentioned, but the three new FPPs come from quotienting by other subgroups of index 18 inside of C3xC3xS3. Thus, we need to know the full action of this group on the Wijk in order to be able to take invariants down to the three new FPPs.
This week, we were able to solve for the full action of C3xC3xS3 on our basis we calculated for H0(18FPP,K). Since we already know how an involution inside of S3 acts (the C2 from last week), we only needed to figure out how the three-cycle inside of S3 acts. Lucky for us, since these actions are linear (and thus representations), we could simply look at the irreps of S3 to figure out how the 3-cycle acts up to multiplication by some unknown coefficients. We then looked for degree 4 relations amongst the basis of sections, and we were lucky enough to get one that was 1-dimensional, thus we had one relation up to scaling. We know that if we apply the action of S3 with unknown coefficients to this relation, the result must be in the same space of relations. But, since the space is 1-dimensional, the result after applying the action with unknown coefficients is simply a scalar multiple of the original relation! This gave us equations on the Cs that we could solve for. In addition, we had computed the degree four relation with exact coefficients, and Mathematica solved the system for the Cs exactly without any issues. This gave us the full action of C3xC3xS3 on the basis. Next, we took degree two monomials in the basis for H0(18FPP,K), and then averaged them over one of the subgroups of C3xC3xS3 that gives the quotient to one of the unknown FPPs (degree 2 gives us sections that are in 2K, and then averaging them gives us invariant sections under the action that will then descend through the quotient to be sections in H0(FPP,2K)). We were again fortunate in that this space was 10 dimensional - thus if we wanted, we could look for degree 3 relations amongst these averaged polynomials and get our equations! However, at this point, the Cs that we solved for were not in Q(Sqrt[-2],Sqrt[-3]), but rather in some ugly looking field extension of this field. So, before we take our equations, we first want to try making a change of basis so that the Cs are defined over the correct field, which will result in equations that are much nicer.
This week, we successfuly scaled the Wijks in a change of basis so that in the new basis, all the Cs from the action simply became 1. We then began to look at possibilities for further change of bases in order to make the equations as nice as possible. However, we discovered that unfortunately, H0(18FPPHat,2K) contains strictly more than all degree two monomials in H0(18FPPHat,K), and furthermore, this also holds true for the C3xS3 invariant subset of H0(18FPPHat,2K) that we use for the quotients. Thus, after we make a good change of basis, we will still need to spend some time to find the "missing" basis elements of H0(18FPPHat,2K) before we have enough to take the quotient and get our explicit equations. We also began to think about how we will seperate our equations once we get them. Highest prioirity is to figure out which equations are part of the complex-conjugate pairs, and after that, it would be nice to know which pair corresponds to which FPP in the Cartwright-Steger classification (although this may not be possible for us to figure out). I also gave a presentation to the rest of the REU participants on Thursday on the work that I did over the summer on my project. A PDF of the slides for this presentation can be found under the Files section of this webpage.
I spent this week writing up my final report for the summer to submit to DIMACS. The report can be found under the Files section of this webpage. In terms of the project, we still need to make a suitable change of basis for the Ws to make our equations nicer when we get them, as well as find the "missing" basis elements, we are going to do this after the program has ended. Once we finish this, we are hopeful that we should have enough work to write up a paper and submit it for publication. I will continue to post updates on this website as we progress through these future steps.