# Child Care at STOC 2018 + Riddle

I’ve been asked to post the following message by the STOC 2018 local arrangements chairs – Ilias Diakonikolas and David Kempe.

We are pleased to announce that we will provide pooled, subsidized
child care at STOC 2018. The cost will be $40 per day per child for regular conference attendees, and$20 per day per child for students.
For more detailed information, including how to register for STOC 2018
childcare, see http://acm-stoc.org/stoc2018/childcare.html

To have something slightly mathematical in this post, here’s a cute riddle: 100 passengers enter an airplane one at a time. The plane contains 100 seats and every passanger has a ticket with a seat number. The first passanger lost his ticket, so he randomly chooses a seat (uniformly). When any other passanger enters, if their seat is available they use it, and otherwise they randomly choose one of the available seats (uniformly). What is the probability that the last passanger got their correct seat.

Finally the name of this blog makes sense!

# Research and Willpower and Fools

Martin Hellman is one of the inventors of assymetric encryption – probably the biggest paradigm shift in the history of cryptography. I just stumbled upon a quote by him which goes well with my recent post about research and willpower:

“The way to get to the top of the heap in terms of developing original research is to be a fool, because only fools keep trying. You have idea number 1, you get excited, and it flops. Then you have idea number 2, you get excited, and it flops. Then you have idea number 99, you get excited, and it flops. Only a fool would be excited by the 100th idea, but it might take 100 ideas before one really pays off. Unless you’re foolish enough to be continually excited, you won’t have the motivation, you won’t have the energy to carry it through. God rewards fools.”

I wish I was more of a fool…

Martin Hellman

# A New Discrete Geometry Group!

This year I moved to Baruch College (which is part of CUNY – the City University of New York), and I’m constantly surprised about wonderful things that keep happening here. In this post I’d like to write about just a couple of these. First, we just hired two additional Discrete Geometers! Together with myself and Rados Radoicic who are already here, we will have quite a large Discrete Geometry group. We have plenty of plans for what to do with this group, and hope that we’ll manage to establish a strong and well-known Discrete Geometry center.

The Discrete Geometers who will be joining us are

Frank de Zeeuw, Pablo Soberon, Yumeng Ou, and Andrew Obus.

If the above is not enough for you, we just hired two additional amazing mathematicians! With these four new hires, the pure part of our math department is receiving a huge boost.

The two additional new hires are:

• Yumeng Ou – working in Harmonic Analysis and more specifically on restriction problems and related topics. Personally, I’m very interested in her works involving polynomial methods and Falconer’s distance problem.
• Andrew Obus – working in Algebraic Geometery and Number Theory. I can write more but I’m afraid of getting the details wrong. So just look at Andrew’s webpage to see his impressive works.

By some weird coincidence I am interested in the works of all four people, and I cannot wait to interact with all of them! The future here looks exciting!

# Research and Willpower

For years I have been mentoring undergraduate students (and others) in math research projects. For example, see the new REU program I recently posted about. I therefore spend time thinking about the issues that beginning researchers have to overcome. One issue stands out as the most common and most problematic problem that beginning researchers need to overcome: learning to think hard about the research for long stretches of time without being distracted. For lack of a better name, I will refer to this as the problem of willpower.

In this post I will write some of my own thoughts about the willpower problem. I am still struggling with it myself (who doesn’t?) and will be happy to hear your own opinion about handling it.

Even without getting into academic research, everyone is familiar with the issue of procrastination from school and work. While studying for an exam, the idea of peeling 200 grapes might become unusually tempting. One suddenly has to check online what happened with that friend of a friend they briefly met ten years ago (or is that just me?). Or perhaps there is a blog post about procrastination that must be written right now…

When one gets to working on academic research, they most likely already figured out how to overcome the above issues when doing homework or studying for an exam. But then they find out that the same problem pops up several orders of magnitude larger. There are several obvious reasons for that:

• Unlike learning material or doing an assignment, it is not clear whether what you are trying to do is possible. It might be that the math problem you are trying to solve is unsolvable. Or perhaps the problem is solvable but the tools for handling it would only be discovered in 200 years. Or perhaps it is solvable now but after several months of making slow progress, some renowned mathematician will publish a stronger result that makes your work obsolete. These scenarios are not that rare in mathematics and related theoretical fields. Are you still going to spend months of hard work on a problem with these possibilities in mind?
• Unlike exams and most jobs, there are no clear deadlines. It is likely that nothing horrible will happen if you will not work on research today, or this week, or this month. There might not be any short term consequences when spending a whole month watching the 769 episodes of “Antique Roadshow”.
• It’s hard! Working on an unsolved problem tends to require more focus and deeper thinking than learning a new topic. Also, part of the work involves trying to prove some claim for weeks/months/years and not giving up. It is surprising to discover that reading a textbook or doing homework becomes a way of procrastinating – it is easier than thinking hard on your research.

(This might give the impression that theoretical research is a horrible career choice. It is much more stressful than one might expect, and requires a lot of mental energy. However, most people who have been doing this for a while seem to agree that it is one of the more satisfying, challenging, and fulfilling jobs that they can think of. I think that I am more excited about and happy with my job than most of my non-academic friends. But I digress…)

So how can one overcome the issue of willpower? While there are many good resources for similar academic issues (writing guides, career advice, etc.), I am not familiar with any good sources on this topic. I am also not an expert on this issue. All I will do here is write some of my current observations and personal opinions. I assume that some of these are naïve and will change over the years.

• Brainstorming is not a solution. For most people it is much easier to discuss a problem with others than to focus on it on their own. Sessions of working with someone obviously have many advantages, but they are not a solution for the willpower problem. One needs to spend time and frustration thinking hard on the problem on their own. Otherwise, they are unlikely to get a good understanding of the topic and get to the deeper issues. Brainstorming sessions become much more effective after first spending time alone and obtaining some deeper understanding and intuition.
• Collaborations do help. Unlike a brainstorming session, long-term collaborations do seem to help with the willpower problem. Not wanting to disappoint a collaborator that I respect, I will have extra motivation to work hard. Having someone else that is interested in the problems also helps keep the motivation high.
• Reserve long stretches of time for research work. Like most people, I constantly have a large amount of non-research tasks, from preparing lectures to babyproofing the house. It is tempting to focus on the non-research tasks since these require less focus and are easier to scratch of the to-do list. When this happens I try to place in my schedule long stretches of time dedicated to research. I try to find times when I am unlikely to be tired or distracted. Sometimes I turn off the wireless and phone during these times. To quote Terence Tao:

“Working with high-intensity requires a rather different “mode” of thought than with low-intensity tasks. (For instance, I find it can take a good half-hour or so of uninterrupted thinking before I am fully focused on a maths problem, with all the relevant background at my fingertips.) To reduce the mental fatigue of transitioning from one “mode” to another, I find it useful to batch similar low-intensity tasks together, and to separate them in time (or space) from the high-intensity ones.”

• Procrastination with writing tasks is a separate issue. While beginners often have a hard time sitting to write and revise their work, this seems to be a simpler problem. The magic solution seems to be writing a lot (not necessarily research work). After a lot of practice, writing becomes a task that does not require a lot of mental energy or deep concentration, is easy to do, and is mostly fun.
• Find the research environment that works best for you. This is an obvious observation, but I would still like to state it. Different people have different environments that work better for them: Some need a quiet environment while others focus better in a crowded coffee shop, some focus better in the morning while others prefer the middle of the night, and so on.
• Find ways to keep yourself highly motivated. Everyone seems to be at least somewhat motivated by being successful and by their ego. Everyone seem to be at least somewhat motivated by an urge to discover the mathematical truth. However, most people seem to need additional motivation when things are not going well. Some people get extra motivation by being surrounded with hard working people. Others become more motivated by reading biographies of successful mathematician and scientists. And so on.
So what are your thoughts? Do you have any tips? Any sources worth reading?

# Difference Sets with Local Properties

I recently attended a wonderful workshop about Algebraic methods in combinatorics, which took place in Harvard’s CMSA. There were many interesting participants from a variety of combinatorial fields, and a very friendly/productive atmosphere. My talk focused on a recent work with Cosmin Pohoata, and I also mentioned some distinct distances result that we derived. During the talk Zeev Dvir asked about an additive variant of the problem. After thinking about this variant for a bit, I think that it is a natural interesting problem. Surprisingly, so far I did not manage to find any hint of previous work on it (this might say more about my search capabilities than about the problem…)

Zeev Dvir and Cosmin Pohoata.

Let $\phi(n,k,\ell)$ denote the minimum size $A-A$ can have, when $A$ is a set of $n$ real numbers with the property that for any $A' \subset A$ with $|A'|=k$ we have $|A'-A'|\ge \ell$. That is, by having a local additive property of every small subset, we wish to obtain a global additive property of the entire set. For simplicity, we will ignore zero in the difference set. Similarly, we will ignore negative differences. These assumptions do not change the problem, but make it easier to discuss.

As a first example, note that $\phi(n,3,3)$ is the minimum number of differences determined by a set of $n$ reals with no 3-term arithmetic progressions. Behrend’s construction is a set $A$ of positive integers $a_1< a_2 < \cdots < a_n$ with no 3-term arithmetic progression and $a_n < n2^{O(\sqrt{\log n})}$. Thus, $\phi(n,3,3) < n2^{O(\sqrt{\log n})}$.

For another simple example, Consider a constant $k\ge 4$. Since we consider only positive differences, any set of $k$ reals determines at most $\binom{k}{2}$ differences. If a specific difference $d$ repeats $\lfloor k/2 \rfloor$ times, then by taking the numbers that span $d$ we obtain $A'\subset A$ such that $|A'|\le k$ and $|A'-A'| \le \binom{k}{2}- \lfloor k/2 \rfloor+1$. Thus, by asking every subset of size $k$ to span at least $\binom{k}{2}- \lfloor k/2 \rfloor+2$ differences, we obtain that no difference repeats $\lfloor k/2 \rfloor$ times in $A$. In other words

$\phi\left(n,k,\binom{k}{2}-\lfloor k/2 \rfloor +2 \right) = \Omega\left(n^2\right).$

Repeating a simple argument of Erdős and Gyárfás gives

$\phi\left(n,k,\binom{k}{2}-\lfloor k/2 \rfloor +1\right) = \Omega\left(n^{4/3}\right).$

That is, when moving from $\ell = \binom{k}{2}-\lfloor k/2 \rfloor +2$ to $\ell = \binom{k}{2}-\lfloor k/2 \rfloor +1$, we move from a trivial problem to a wide open one. My work with Cosmin Pohoata leads to the following result.

Theorem 1. For any $d\ge 2$ there exists $c$ such that

$\phi\left(n,k,\binom{k}{2}-k\frac{d}{d+1}+c\right) =\Omega\left(n^{1+1/d} \right).$

For example, when $d=2$ we get the bound

$\phi\left(n,k,\binom{k}{2}-\frac{2k}{3}+c\right) =\Omega\left(n^{3/2} \right).$

When $d=3$ we get a significant improvement for the range of the Erdős-Gyárfás bound:

$\phi\left(n,k,\binom{k}{2}-\frac{2k}{3}+c\right) =\Omega\left(n^{3/2} \right). \qquad \qquad \qquad (1)$

Since not much is known for this problem, it seems plausible that additional bounds could be obtained using current tools. Our technique does not rely on any additive properties, and holds for a more abstract scenario of graphs with colored edges. Hopefully in the case of difference sets one would be able to use additive properties to improve the bounds. Moreover, so far I know nothing about much smaller values of $\ell$, such as $\phi(n,k,100k)$.

Proof sketch for Theorem 1. For simplicity, let us consider the case of $d=3$, as stated in $(1)$. Other values of $d$ are handled in a similar manner. Let $A$ be a set of $n$ reals, such that any $A'\subset A$ of size $k$ satisfies $|A'-A'|\ge \binom{k}{2}-\frac{3k}{4}+13$. We define the third distance energy of $A$ as

$E_3(A) = \left|\left\{(a_1,a_2,a_3,b_1,b_2,b_3) \in A^6 :\, a_1-b_1=a_2-b_2=a_3-b_3 >0 \right\}\right|.$

The proof is based on double counting $E_3(A)$. For $\delta\in {\mathbb R}$, let $m_\delta = \left|\left\{(a,b)\in A^2 : a-b = \delta\right\}\right|$. That is, $m_\delta$ is the number of representations of $\delta$ as a difference of two elements of $A$. Note that the number of 6-tuples that satisfy $a_1-b_1=a_2-b_2=a_3-b_3$ is $m_\delta^3$. A simple application of Hölder ‘s inequality implies

$E_3(A) = \sum_{\delta>0} m_\delta^3 \ge \frac{n^6}{|A-A|^2}.$

To obtain a lower bound for $|A-A|$, it remains to derive an upper bound for $E_3(A)$.

For $j\in {\mathbb N}$ let $k_j$ denote the number of differences $\delta \in {\mathbb R}^+$ such that $m_\delta \ge j$. A dyadic decomposition gives

$E_3(A) = \sum_{\delta>0} m_\delta^3 = \sum_{j=1}^{\log n} \sum_{\substack{\delta>0 \\ 2^j \le m_\delta < 2^{j+1}}} m_\delta^3< \sum_{j=1}^{\log n} k_{2^j} 2^{3(j+1)}. \qquad \qquad \qquad (2)$

For $j\in {\mathbb N}$ let $\Delta_j$ denote the set of $\delta>0$ with $m_\delta\ge j$ (so $|\Delta_j| = k_j$). For $\delta >0$, let $A_\delta$ be the set of points that participate in at least one of the representations of $\delta$. If there exist $\delta_1,\delta_2, \delta_3$ such that $|A_{\delta_1} \cap A_{\delta_2} \cap A_{\delta_3}| \ge k/4$, then there exist a subset $A'\subset A$ with $|A'|=k$ and $|A'-A'|< \binom{k}{2}-\frac{3k}{4}+13$ (see the paper for a full explanation). Thus, for every $\delta_1,\delta_2, \delta_3$ we have that $|A_{\delta_1} \cap A_{\delta_2} \cap A_{\delta_3}| < k/4$.

We have $k_j$ sets $A_\delta$ with $|A_\delta| \ge j$. These are all subsets of the same set $A$ of size $n$, and every three intersect in fewer than $k/4$ elements. We now have a set theoretic problem: How many large subsets can $A$ have with no three having a large intersection. We can use the following counting lemma (for example, see Lemma 2.3 of Jukna’s Extremal Combinatorics) to obtain an upper bound on $k_j$.

Lemma 2. Let $A$ be a set of $n$ elements and let $d\ge 2$ be an integer. Let $A_1,\ldots,A_k$ be subsets of $A$, each of size at least $m$. If $k \ge 2d n^d/m^d$ then there exist $1\le j_1 < \ldots < j_d \le d$ such that $|A_{j_1}\cap \ldots \cap A_{j_d}| \ge \frac{m^d}{2n^{d-1}}$.

Lemma 2 implies the bound $k_j = O(n^3/j^3)$ for large values of $j$. Combining this with $(2)$ and with a couple of standard arguments leads to $E_3(A) = O(n^{10/3})$. Combining this with $E_3(A) \ge \frac{n^6}{|A-A|^2}$ implies $|A-A|=\Omega(n^{4/3})$. $\Box$

# A New Combinatorics REU

I am excited to announce the beginning of the CUNY Combinatorics REU, which I am organizing together with Radoš Radoičić. For the past three years I have been mentoring Caltech undergraduates in research projects, and before that students in Tel-Aviv University. These often led to papers and almost all of the students continued to grad school or are applying now. This REU is our way of continuing this work in CUNY. Many more details can be found here.

Please send us strong students! Also, if you are a mathematician with some interest in combinatorics, might be around NYC at some point during the summer, and might be willing to give a talk or just come to chat with the participants, let me know!

I’m happy to hear any comments and questions. Now let’s work hard and get some impressive research done in this program!

# Incidences Outside of Discrete Geometry (part 2)

You may have noticed that I have a bit of an obsession with geometric incidences. I do believe that incidences are a natural mathematical object that is connected to many different parts of math. This belief seems at least partially justified by the developments of the recent years. Less than two years ago I posted a list of some applications of incidences outside of Discrete Geometry. The purpose of that list was to show how incidences are becoming useful in a variety of fields, such as Harmonic Analysis, Theoretical Computer Science, and Number Theory. It seems that this process did not slow down in the past two years – incidences have continued to demonstrate their usefulness in the aforementioned fields, and there is even a new interest in the subject by model theorists.

This post is part 2 of the list of incidence uses outside of Discrete Geometry. It is just a list of references, and does not include many details. In future posts I might focus on specific applications and provide actual explanations. Hopefully new applications will continue to appear and I’ll have to keep adding more and more parts to this list!

The Kakeya conjecture. Katz and Zahl derived an improved bound for the Kakeya conjecture. Specifically, they improved Wolff’s longstanding bound for the Hausdorff dimension problem in ${\mathbb R}^3$. This is the latest in a sequence of Harmonic Analysis works that have strong connections to incidences (see the first part of this list and also below).

Model theory. In Logic, a group of model theorists generalized incidence results to Distal structures. Another similar work extended incidence results to o-minimal structures. In general, there seems to be some interest in generalizing incidence-related problems to various models.

Number theory. A very recent number theoretic result is relying on incidence bounds. Admittedly, I still do not understand what this paper is about, and am hoping to learn that soon.

Algorithms. Moving to Theoretical Computer Science, a recent work analyzes point covering algorithms using incidence results.

Quantum Information. A few years ago, incidences in spaces over finite fields were used to study a problem in Quantum Information. This result is not from the past two years, but I was not aware of it before (thanks Ben Lund).

More Harmonic Analysis. An older survey of Łaba contains a nice review of previous appearances of incidences in Harmonic Analysis. I write “older” although not even a decade have passed. I just mean that this survey appeared before the new era of polynomial methods in Discrete Geometry.

There are obviously more results that are still missing from this list. If you noticed anything that I missed, I would be happy to hear about it.