12.08 p.m.. We had a very pleasant flight and arrived, rested and happy, in Warsaw from where we are going to catch a connecting flight to Lviv. I used to call my hometown Lvov, but after recent Russian adventures in Donbass and Crimea I decided that terminology matters and made a switch.
The adventure begins today. In about an hour, I am going to fly to Chicago where my brother, my parents and I are going to embark on our first trip to Lvov since we left almost thirty-nine years ago. This is not just our first trip back as a family-none of us have been back since we had left in September of 1979. While I traveled to Kiev and Kharkov on business six years ago, I specifically avoided visiting Lvov because I was not yet ready to revisit the old memories in a truly comprehensive fashion. But today is the day and we are going to are going to board a flight to Warsaw at around 5.30 p.m., arrive at around 9.30 a.m. the next day, and catch a connecting flight to Lvov in mid-afternoon. By tomorrow night, we will be immersed in the world that seemed so far away for decades, yet started feeling much closer in recent years, culminating in the decision to visit.
As I sit here in the Rochester airport marveling at the slow pace of repairs to this facility and lamenting the shortage of electrical outlets, I experience flashbacks to the day when we left Lvov on September 29, 1979. Many family friends came to say goodbye, including several of my school friends. That was the last day I saw Vitaly Nefedov and Dima Zis. Vitaly’s family remained in the Soviet Union, though his mother and grandmother later moved to Israel. Vitaly is married and lives in Moscow. I found him on facebook a couple of years ago. Dima Zis and his family moved to the U.S. in the 80s, yet I never made contact with him for some reason in spite of opportunities to do so. There is no reason for that aside from the all-powerful inertia. Perhaps I will try to find him upon returning to the U.S. on June 18.
It is time to go to my gate. I will write more from Chicago!
The wonders of instant communication allow me to communicate with people all over the world at any time of day or night. I can sit behind my desk at the Department of Mathematics at the University of Rochester and think about ethnic tension in Indonesia or the upcoming elections, if you allow me such a loose use of terminology, in Russia and China. In principle, I should be thinking about mathematics, and much of the time I do, but my mind wonders and searches and takes me to faraway lands. But I am still much more likely to think about a place if I am actually there and my trip to Israel in the middle of March 2018 is no exception. Much of the trip was occupied with mathematics and political discussions for few and far between, but the trip ended with a major epiphany. I realized that all attempts to understand the current Israeli predicament in terms of the traditional Left-Right political divide is deeply flawed because it is based on the rather dubious idea that the two-state solution of the Arab-Israeli conflict in its current form is workable and beneficial to both sides of the conflict. I submit to you that any two-state solution with the goal of producing two viable nations, in the traditional sense, is doomed to failure and would only lead to ever-increasing misery and war.
Let us briefly review the perception one is likely to get about the two-state solution and its dangers from the mainstream media in the United States. The Left-wing writers focus on what they see as the Israeli rejection of the two-state formula, alleged oppression of the Palestinian people and the preferential treatment of Israel by the American government. The Right-wing writers focus on Palestinian terrorism, decades of rejection of Israel’s right to exist, the corruption and inefficiency of the Palestinian Authority and danger that any Palestinian State would inevitably pose to Israel. These are by no means exhaustive lists, but they will do for our purposes. With a healthy dose of nuance added, both sides have their points that are worth considering, but they tend to ignore the immediate day-to-day dynamics of the situation. The simple truth of the matter is that the Palestinian Arabs living in the West Bank and Gaza are economically dependent on Israel supply the labor force that cannot be easily obtained anywhere else. Whatever State emerges in West Bank in Gaza, assuming there is ever sufficient unity among the Palestinian Arabs to make this remotely possible, cannot exist without large-scale economic connections with the Jewish State that predate Israel’s existence. The inefficiency on the part of the Palestinian authority and lack of effort build the institutions of the putative State is, in part, a reflection of the realization on the part of the Palestinian Arabs that no viable State is possible under the formula that is currently being peddled.
I have previously expressed the belief that the only reasonable solution to the conflict is an autonomy for the Palestinian territories, political alliance with Jordan that would provide the citizens of the territories with a citizenship and continued economic partnership with Israel, with tighter standards and well-defined rights. But my main goal here is to shatter the myth that the two-state solution is a humanist ideal aimed at addressing the plight of the Palestinian Arabs in a compassionate and workable way. While I share the stated concern of the Israeli Left for the welfare of the Palestinian Arabs, I believe that the remedy they favor would lead to poverty and instability. The conversation needs to rapidly change from the pointless discussion of national rights of the Palestinian Arabs, for which there are no historical or practical bases, to the steady improvement of the human rights for these people in the context of the enduring economic and political realities.
I just gave a series of lectures in Vietnam last week and one of the results I presented is one of my favorites, a result I proved with Derrick Hart ten years ago. The proof is not difficult, but the exponent has not been improved since then and appears to be a rather fundamental barrier. Let denote a finite field with elements. Given , let and .
Theorem: Let . Suppose that . Then .
This is an example of the so-called sum-product phenomenon. If is a multiplicative subgroup of , then . But then should be rather large because should not possess both good multiplicative and additive properties. I will write more posts on the sum-product phenomenon in the various settings in the near future.
Proof: We may assume that since does not contribute anything to , Let and define
, where is the indicator function of .
It follows that an upper bound on
will yield a lower bound on . This is called the second-moment method.
The reader should also note that we took a simple looking one-dimensional problem about and turned it into a problem about a two-dimensional object . We shall make an effort to explain why one might think that such an approach is useful at the conclusion of this argument.
Observe that by Cauchy-Schwartz,
It follows that
, i.e .
It follows that if , then
, which implies that
, or, in other words,
This completes the proof, but there is still the matter of why the approach of lifting the problem to the two-dimensional setting is a reasonable approach. This may seem like a philosophical point, but it is also the motivation behind the proof above. The operator implicit in the argument above is
. Its continuous analog is the Radon transform
, where is smooth cut-off function and is the Lebesgue measure on the hyperplane
It is well-known that if ,
The point of the proof above is that the discrete Radon transform also satisfies the suitable Sobolev bounds once the first eigenvalue is subtracted. This theme is ubiquitous in extremal problems in vector spaces over finite fields.
About a year ago, Azita Mayeli, Jonathan Pakianathan and I published a paper proving the Fuglede Conjecture in , prime. In this context, the conjecture says that the following. We say that tiles if there exists such that
for all , where is the indicator function of $E$.
We say that is spectral if there exists such that
, , is an orthogonal basis for in the sense that every function can be written as a finite sum
, for all .
In dimension and higher this result is known to be false. In five dimensions this was demonstrated by Terry Tao in one direction and by Kolountzakis and Matolcsi in the other. In dimension four the conjecture was disproved, in both direction, by Kolountzakis and Matolcsi. But the two and three dimensional cases remained open and it was widely speculated that the higher dimensional counter-examples will eventually apply there as well, However it turns out that something special is going on in lower dimensions and this is where we now turn out attention.
Theorem (Iosevich, Mayeli and Pakianathan) The Fuglede Conjecture holds in , prime.
The paper was published in Analysis and PDE last year (https://projecteuclid.org/euclid.apde/1508432238)
I will sketch the proof below and explain the basic ideas involved. We shall focus on the case because this allows us to avoid the technicalities involving non-zero vectors that are orthogonal to themselves.
Basic notation: if , define
. It is not difficult to check that
If , denotes its indicator function.
The set of directions determined by a set is the set of difference , , under the equivalence relation , , if for some .
Lemma 1 (number theory) Let . Suppose that there exists such that . Then
i) for all .
ii) is equi-distributed on lines perpendicular to in the sense that
To prove the lemma, we write
It follows that
Since the minimal polynomial of is
we conclude that
This proves part ii) and part i) follows at once.
Lemma 2 (combinatorics) Let of size . Then determines every possible direction in .
To prove this note that if a direction is missing, is a subset of a graph
, so ,
With the two lemmas in tow, we are ready to prove the theorem above. Suppose that is spectral. Then is an orthogonal basis and by elementary linear algebra, . The orthogonality hypothesis implies that Lemma 1 applies and we conclude that .
Suppose that .Then determines every possible direction since . Invoking orthogonality, we see that
for all , . By Lemma 1, for all and since determines every possible direction, for all . This implies that .
Thus we may assume that . The case is not very interesting, so we consider the case . Invoking Lemma 1 again, we see that for each , has exactly one point on the line , where is a fixed vector such . As we noted above, such a vector must exist since we are assuming that is spectral, so for all , .
By rotation, we may assume that has exactly one point on the lines given by the equation . But then it is not difficult to see that that tiles with the tiling set .
Let’s now assume that tiles by translation with a tiling set . Then must be a divisor of . The only interesting case is . Tiling means that
for all . Taking the Fourier transform of both sides, we see that
for all . We conclude from this that there exists such that . If not, then for all . It would follow that , which is not possible since , which implies that .
Since there exists such that , we can invoke Lemma 1 again to see that contains exactly one point on the lines perpendicular to . By rotation, we may assume that has exactly one point on the lines given by the equation . But then one can check by a direct calculation that the set is an orthogonal basis for , where .
This completes the proof. Here are some exercises for the interested reader.
Exercise: Work out the details of the case .
Exercise: Prove that if and tiles by translation, then is spectral.
It is not known whether spectral implies tiling in .
This was my second trip to Vietnam, but it was the first time I had the chance to think about it carefully. My first visit lasted two and a half days and all my concentration went into staying awake, delivering my lectures and making it back to the airport on time. During this trip I was able to combine mathematics with sightseeing and, more importantly, with a healthy dose of soul-searching. I was not swayed by the skillful propaganda of the Ho Chi Minh museum or the grandeur of his mausoleum. The statue of Lenin and the plethora of red flags did not fill me with warmth or confidence. The moment of clarity came when I was sitting on a boat floating down the Tom Coc River, going through a series of enchanting grottos and surrounded by scenic mountains serenely drowned in the morning fog. As we passed the first grotto, an old cemetery appeared on the left bank of the river and my heart sank because for the first time, the enormity of the crime committed by our country against the people of Vietnam became palpable and deeply painful. For the next hour or so, I could not focus on the complex history of the Vietnam war that occupied my thoughts during my museum visits in the previous days. All I could think about was the millions of Vietnamese who died because our elected leaders decided that this was a small price to pay in the struggle against communism. Or whatever the real reasons were. The bottom line is, we murdered these people and this bitter fact should be acknowledged by every American individually and our nation as a whole. But are we in any position to undertake such a journey?
Some may wonder how rehashing the events of almost fifty years is useful or helpful, an old and tired routine of evasion of responsibility. The same arguments have been made against Holocaust education or the discussion of slavery and voter suppression. Let me share a bit of my personal experience. About six months ago, an old judo sense of mine, the man I liked and respected for over twenty years posted on facebook a photo of himself from 50 years earlier in Vietnam, taken moments after he “made his bones” by killing his first Vietnamese soldier. A swarm of congratulatory comments followed, many from the judo people I have known for years and considered to be decent human beings. I do not fault a soldier, drafted at a young age, for performing his duty. I do not find it realistic to demand that every person in a situation of this type take a stand and refuse to follow orders. As much as I would like to believe otherwise, I am not at all certain that I would have done so myself. But celebrating killing a human being fifty years later, long after the horror of what the United States armed forces had done in Vietnam has become public knowledge, is deeply perverse. The fact that such an act is, without any doubt, considered normal by millions of our fellow citizens indicates that our country is very sick.
The efforts by many in our education establishment to balance patriotism with a healthy dose of moral responsibility and realism have largely failed. The country is now divided between those who mindlessly waive the flag, even on behalf of the five time draft dodging hypocrite in the White House, and the equally uncritical proponents of the liberal faith. Yes, they are, on average, better educated and they are not supporting a political party that based its electoral strategy for the past fifty years on racism, voter suppression and making mockery of women’s rights. But they are not free of hypocrisy either, having largely taken advantage of the African-American voters without delivering much that is tangible and clamoring for women’s rights while embracing and covering for William Jefferson Clinton. The number of people who think, analyze and change their views in accordance with the facts is very small and they have less and less influence over the course of the country. Thoughtful people exist throughout the American political spectrum, barring the unbearable extremes, but fewer and fewer of them resonate with the voting public. Shrill voices on the Left are clamoring for us to turn to half-witted demagogues like Bernie Sanders. On the Right only a handful of decent people had the courage to stand up to the man who is making mockery of the office of the President and degrading the American democracy beyond what most of us could have imagined. There is little reason to hope that pragmatism will prevail in the near term.
Returning to the subject of Vietnam, there is little I can do to influence the current state of our country, but my personal journey continues. I have been working with several Vietnamese mathematicians and I hope to make more contacts over there. There is tremendous scientific potential in Vietnam and it will take decades before it reaches its full potential. The inevitable evolution of the political climate will probably contribute positively to this process. Beyond work, I would like to go back to Vietnam with my wife and kids someday, travel deep into the country’s interior and truly get a feel for how the people live and think. I got small glimpses of these things during my visit, but this only made me hungry for more. This trip was truly one of the most fulfilling experiences of my life.