# 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?

# 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!

# A First Draft of the Book “Incidence Theory”

You might have noticed that I did not post anything new for quite a while. The past months were unusually busy for me, due to personal reasons such as having my first child born(!), being on the job market, and several other things. I hope to return to my regular posting frequency around July.

The purpose of this post is to announce that I just uploaded the first draft of my book “Incidence Theory”. This book is about our current understanding of incidences (with a focus on the polynomial method), and their applications in other fields. I am trying to achieve two goals in this book: To have a clear and basic introduction of this subfield, while also creating a repository of results and techniques which may be used as a reference to experts. The current draft already contains several folklore results that I have not seen written before. It contains only the first seven chapters. I predict that the final version would contain about 15 chapters, and plan to gradually release the remaining ones.

Comments would be very appreciated, preferably by email. These can point out mistakes, typos, unclear formulations, suggestions for style changes, additional topics, simpler arguments, exercises, or anything else that might help improve the draft. The acknowledgements section is way too short. Please help me to extend it!

# The Sum-Product Bound of Konyagin and Shkredov

In Solymosi’s famous 2009 paper, he proved that every finite set $A\subset {\mathbb R}$ satisfies

$|A+A||AA| = \Omega\left(|A|^{4/3}/\log^{1/3}|A|\right).$

In the past couple of years, Konyagin and Shkredov published two papers that extend Solymosi’s argument, obtaining a slightly stronger sum-product bound (one and two). These papers derive several additional results, and apply a variety of tools. I just uploaded to this blog my own exposition to the sum-product proof of Konyagin and Shkredov (a link can also be found in the pdf files page). This exposition ignores the additional results that are in the two papers, and tries to explain in detail every step that is part of the sum-product proof. In this aspect, the document would hopefully also fit beginners. As usual, I’m happy to receive any comments and corrections.

Ilya Shkredov and Sergei Konyagin.

I recently started teaching an “Additive combinatorics” class, and am writing lecture notes for it. So far I put the first three chapters online. I’d appreciate any comments about these, from pointing out serious mistakes, to pointing out minor typos, or even a recommendation for the final topic (which I have not chosen yet). The chapters that are already up are:

• In Chapter 1 we start to study the principle that sets with small doubling must have structure. We prove some basic results such as Ruzsa’s triangle inequality, Plünnecke’s inequality, and variants of Freiman’s theorem.
• Chapter 2 studies the sum-product problem over the reals. In addition to showing the proofs of Elekes and Solymosi, we see how the same techniques can be applied to several other problems.
• Chapter 3 discusses the The Balog-Szemerédi-Gowers theorem. Specifically, we present the variant of Schoen and the variant by Sudakov, Szemerédi, and Vu.

# John von Neumann by Norman Macrae

So far, in my posts I have only written about popular science books that I liked. In this post I am making an exception, writing a rather negative review for a book that somewhat bothered me. I hope that writing such a post is not a big mistake. I guess I’ll soon find out…

John von Neumann.

This post is about the biography of the mathematician John von Neumann by Norman Macrae. Perhaps I should start by stating that this is not really a popular science book, in the sense that it seems to avoid any discussion that is even a bit technical. The mathematical research of Von Neumann is stated very vaguely, and one mostly finds statements such as “… number theory, a branch of mathematics that is at once fascinating and frustrating” without any further details. When mentioning various scientists, the focus is usually about anecdotes, political opinions, and how smart they were. This last point usually involves some sort of ranking. According to Macrae, Aristotle was definitely smarter than Von Neumann, which was in turn smarter than any other mathematician of the 20th century. As another example, Von Neumann’s tutor Gabor Szego “was later to become one of the half dozen most distinguished Hungarian mathematicians of the twentieth century.” No explanation is provided for what these rankings are based on (occasionally they appear together with a statement by one person saying that another is very smart).

After reading my previous paragraph, you might claim that having a non-scientific biography of a mathematician can be a good thing, especially for readers who are not mathematicians. So instead of describing what the book does not contain, I now move to describe what it does. One thing that the book is filled with is strong opinions by the author, sometimes stated as facts. These opinions concern which government systems are better, which countries have the best school systems (apparently Japan is at the top), various historical speculation (“If Haber had not found a way to fix nitrogen from the air to make nitrates for explosives, blockaded Germany would have had to surrender from the 1914-18 war in about 1915”), among other topics. Some of these opinions concern someone being right or wrong about a subjective issue, without any further explanation (“… Eckert, whom in 1945 he rightly still wanted in his Princeton project as chief engineer”).

There are two contexts in which I found this abundance of opinions rather problematic. The first is when they appear in relation to mathematics or other exact sciences. My possibly-false impression is that Macrae does not have much of a scientific background. This is of course fine, and many wonderful popular science books were written by non-scientists. However, it makes Macrae’s many unusual assertions about science seem rather awkward. For example, after mentioning how important Von Neumann was to the development of mathematics comes the sentence “happily, and this is not sufficiently realized, it should be possible to get more like him”. The following sentence then explains that this is due to the lower financial needs of a mathematician (that is, paper and pencil). Claiming that von Neumann had a wider conception about computers, Macrae writes “He did not envisage these machines as today’s glorified typewriters. He hoped that experimenting scientists could use computers to start scientific revolutions that would change the planet.” Many such statements gave me a somewhat awkward feeling while reading the book.

The second case in which I found the many opinions in the book somewhat troubling is in the context of people with left-wing opinions. The book is PACKED with anti-left-wing statements, sometime without any visible connection to Von Neumann. One example (that also involves an awkward “technical” statement) is “He thought that those who believed in communism or socialism deserved the same sympathy as other simpletons who could not understand even linear equations.” After mentioning that various scientists supported Stalin or did not sufficiently oppose Hitler, Macrae writes “A common feature of clever men who occasionally supped on too-short spoons with Nazism or Communnism was that as kids they had never adequately learned to laugh”. It is clear that Macrae feels strongly about the subject, as the book contains statements such as “Among people who wanted to believe optimistic nonsense about western Europe or Soviet Russia…” and “… speaks in volumes to those who want to hear.” Such statements appear throughout the book and are possibly the most common topic that is discussed. Often such opinions are attributed to Von Neumann, although almost never with a reference or an actual quote (unlike many of Von Neumann’s statements on other topics). In fact, one finds sentences such as “I cannot find anything in his papers that suggests that he advocated that, although a lot of honest people thought that he did.”

Another issue is the extreme glorification of Von Neumann. I don’t think that many people doubt that Von Neumann was a genius that made a large impact on many fields, but the book takes this to the next level. It actually contains statements such as “the cleverest man in the mathematical world” and “Because Johnny was the world’s leading mathematical logician, he could clearly improve its logical design” (of the EDVAC computer).

I have more to say but I probably wrote too much already, so let me conclude this post. If you would like to hear how the smartest man of the 20th century had hawkish opinions, and that these were much better than all of those silly left-wing scientists such as Einstein, definitely read Macrae’s book. Personally I would appreciate a recommendation for a more standard biography of Von Neumann.

# Incidences: Lower Bounds (part 7)

In this post we continue our survey of the known lower bounds for incidence problems (click here for the previous post in the series). In the current post we finally start discussing incidences with objects of dimension larger than one. The simplest such case seems to be incidences with unit spheres in ${\mathbb R}^3$ (i.e., spheres of radius one). We present two rather different constructions for this case. I am not aware of any paper that describes either of these constructions, and they appear to be folklore. If you are aware of a relevant reference, I will be happy to hear about it.

First construction: Inverse gnomonic projection. Our first construction is based on lower bounds for the Szemerédi-Trotter problem (e.g., see the first two posts of this series). It shows that $m$ points and $n$ unit spheres in ${\mathbb R}^3$ can yield $\Omega(m^{2/3}n^{2/3}+m+n)$ incidences.

For the requested values of $m$ and $n$, consider a set $\cal P$ of $m$ points and a set $\cal L$ of $n$ lines, both in ${\mathbb R}^2$ and with $I({\cal P},{\cal L})=\Theta(m^{2/3}n^{2/3}+m+n)$. We place this plane as the $xy$-plane in ${\mathbb R}^3$. We then perform an inverse gnomonic projection that takes $\cal P$ and $\cal L$ to a sphere in ${\mathbb R}^3$, as follows. We set $p=(0,0,1/\sqrt{2})$ and denote by $S$ the sphere that is centered $p$ and is of radius $1/\sqrt{2}$. Notice that $S$ is tangent to the $xy$-plane at the origin. The inverse projection takes a point $q$ in the $xy$-plane to the intersection point of the line segment $pq$ and $S$. One can easily verify that this mapping is a bijection between the $xy$-plane and the lower half of $S$; e.g., see the following figure.