## On humanism and the two-state solution

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.

## From 1933 to 2018

For those who were wondering how Trump’s approach to governance is different from Hitler’s, you have a partial answer in the new budget. Instead of an across the board Hitler-style deficit spending plan that drastically increased both the military and domestic spending, knowing that the war of expansion will write it all off, Trump is proposing to increase military spending, trash domestic programs and launch a largely fraudulent infrastructure plan based on non-existent funding by the state and local governments.

The point is that the German electorate in 1933 was more sophisticated than the American electorate in 2016 and, I fear, in 2020 and beyond. The former supporters of the left and center parties accepted Hitler’s rule without very much resistance under a combination of crude intimidation and immediate economic and infrastructure benefits. This approach inevitably led to a large scale war, almost independently of the ideological pronouncements of the Nazi regime.

Trump appears to be banking on the fact that his lower middle class supporters do not even need to be paid off. Race bating, misogyny, military expenditures and mortal fear of progressive thought are more than enough to keep the Trumpist hoards in line, as his current %40+ approval rating indicates. To put it simply, Trump never had to win over much of the “proletariat” in this country. They were on his side from the beginning, giving Trump a tremendous advantage in his quest to turn this country into a totalitarian cesspool for the benefit of the rich and powerful.

With all the instability in the current political process that many of my friends point to, Trump is executing his agenda with much greater skill than most give him credit for. He contradicts himself and changes his mind on many things, but the bottom line generally remains the same. He wants a country where a small group of white male elites run everything and keep the masses bamboozled with a combination of militaristic and chauvinistic rhetoric. In spite of all the incompetence, Trump is making steady progress towards achieving his goal, especially since it runs hand in hand with the long standing Republican agenda.

## Sum-product estimates in finite fields

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 ${\Bbb F}_q$ denote a finite field with $q$ elements. Given $A,B \subset {\Bbb Z}_p$, let $A+B=\{a+b: a \in A, b \in B\}$ and $A \cdot B=\{ab: a \in A, b \in B\}$.

Theorem: Let $A \subset {\Bbb F}_q$. Suppose that $|A|>q^{\frac{2}{3}}$. Then $|A \cdot A+A \cdot A|>\frac{q}{2}$.

This is an example of the so-called sum-product phenomenon. If $A$ is a multiplicative subgroup of ${\Bbb Z}_p$, then $A \cdot A=A$. But then $A+A$ should be rather large because $A$ 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 $0 \notin A$ since $0$ does not contribute anything to $A \cdot A+A \cdot A$, Let $E=A \times A$ and define

$\nu(t)=\sum_{x \cdot y=t} E(x)E(y)$, where $E$ is the indicator function of $E$.

Observe that

${|E|}^4={\left( \sum_t \nu(t) \right)}^2 \leq |\Pi(E)| \cdot \sum_t \nu^2(t),$ where

$\Pi(E)=\{x \cdot y: x,y \in E\}$, $x \cdot y=x_1y_2+x_2y_2$.

It follows that an upper bound on

$\sum_t \nu^2(t)$ will yield a lower bound on $\Pi(E)$. This is called the second-moment method.

The reader should also note that we took a simple looking one-dimensional problem about $A \subset {\Bbb Z}_p$ and turned it into a problem about a two-dimensional object $E=A \times A$. 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,

$\nu^2(t) \leq |E| \cdot \sum_{x \cdot y=x \cdot y'=t} E(x)E(y)E(y')$ and

$\sum_t \nu^2(t) \leq |E| \cdot \sum_{x \cdot y=x \cdot y'} E(x)E(y)E(y')$

$=|E| \cdot q^{-1} \sum_{s \in {\Bbb F}_q} \sum_{x,y,y'} \chi(sx \cdot (y-y')) E(x)E(y)E(y')$

$={|E|}^4q^{-1}+q^{-1} |E| \cdot \sum_{s \not=0} \sum_{x,y,y'} \chi(sx \cdot (y-y')) E(x)E(y)E(y')$

$={|E|}^4q^{-1}+R(E).$

Now,

$R(E)=q^3|E|\sum_{s \not=0} \sum_x E(x) {|\widehat{E}(sx)|}^2$

$=q^3|E| \sum_x {|\widehat{E}(x)|}^2 \left\{ \sum_{s \not=0} E(sx) \right\}$

$=q^3|E| \sum_x {|\widehat{E}(x)|}^2 |E \cap l_x|$, where

$l_x=\{tx: t \not=0\}$.

Since $E=A \times A$,

$|E \cap l_x| \leq |A|={|E|}^{\frac{1}{2}}$.

It follows that

$|R(E)| \leq q^3 \cdot {|E|}^{\frac{3}{2}} \cdot \sum_x {|\widehat{E}(x)|}^2$

$=q^3 \cdot {|E|}^{\frac{3}{2}} \cdot q^{-2} \cdot |E|$

$=q {|E|}^{\frac{5}{2}}$.

Note that

$q {|E|}^{\frac{5}{2}} \leq {|E|}^4 q^{-1}$ if

$=|E| \ge q^{\frac{4}{3}}$, i.e $|A| \ge q^{\frac{2}{3}}$.

It follows that if $|A| \ge q^{\frac{2}{3}}$, then

$\sum_t \nu^2(t) \leq 2{|E|}^4q^{-1}$, which implies that

$|\Pi(E)| \ge \frac{q}{2}$, or, in other words,

$|A \cdot A+A \cdot A| \ge \frac{q}{2}$.

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

$T_tf(x)=\sum_{x \cdot y=t} f(y)$. Its continuous analog is the Radon transform

$R_tf(x)=\int_{x \cdot y=t} f(y) \psi(y) d\sigma_{x,t}(y)$, where $\psi$ is smooth cut-off function and $\sigma_{x,t}$ is the Lebesgue measure on the hyperplane

$\{y: x \cdot y=t\}$.

It is well-known that if $t \not=0$,

$R_t: L^2({\Bbb R}^d) \to L^2_{\frac{1}{2}}({\Bbb R}^d)$.

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.

## Fuglede Conjecture in vector spaces over prime fields

About a year ago, Azita Mayeli, Jonathan Pakianathan and I published a paper proving the Fuglede Conjecture in ${\Bbb Z}_p^2$, $p$ prime. In this context, the conjecture says that the following. We say that $E \subset {\Bbb Z}_p^2$ tiles if there exists $T \subset {\Bbb Z}_p^2$ such that

$\sum_{t \in T} E(x-t)=1$ for all $x \in {\Bbb Z}_p^2$, where $E(x)$ is the indicator function of $E$.

We say that $E$ is spectral if there exists $A \subset {\Bbb Z}_p^2$ such that

${\{\chi(x \cdot a)\}}_{a \in A}$, $\chi(t)=e^{\frac{2 \pi i t}{p}}$, is an orthogonal basis for $L^2(E)$ in the sense that every function $f: E \to {\Bbb C}$ can be written as a finite sum

$f(x)=\sum_{a \in A} c_a \chi(x \cdot a)$ and

$\sum_{x \in E} \chi(x \cdot (a-a'))=0$, for all $a \not= a' \in A$.

In dimension $4$ 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 ${\Bbb Z}_p^2$, $p$ 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 $p=3 \mod 4$ because this allows us to avoid the technicalities involving non-zero vectors that are orthogonal to themselves.

Basic notation: if $f: {\Bbb Z}_p^2 \to {\Bbb C}$, define

$\widehat{f}(m)=p^{-2} \sum_{x \in {\Bbb Z}_p^2} \chi(-x \cdot m) f(x)$. It is not difficult to check that

$f(x)=\sum_{m \in {\Bbb Z}_p^2} \chi(x \cdot m) \widehat{f}(m).$

If $E \subset {\Bbb Z}_p^2$, $E(x)$ denotes its indicator function.

The set of directions determined by a set $E \subset {\Bbb Z}_p^2$ is the set of difference $x-y$, $x,y \in E$, under the equivalence relation $z \sim z'$, $z,z' \in E-E$, if $z=tz'$ for some $t \in {\Bbb Z}$.

Lemma 1 (number theory) Let $E \subset {\Bbb Z}_p^2$. Suppose that there exists $m \not=(0,0)$ such that $\widehat{E}(m)=0$.  Then

i) $\widehat{E}(rm)=0$ for all $r \not=0$.

ii) $E$ is equi-distributed on lines perpendicular to $m$ in the sense that

$n(t)=\sum_{x \cdot m=t} E(x)=const.$

To prove the lemma, we write

$0=\sum_{x \in {\Bbb Z}_p^2} \chi(-x \cdot m) E(x)=\sum_t \sum_{x \cdot m=t} E(x) \chi(-t)$

$\sum_t \zeta^t n(t),$ where $\zeta=\chi(-1)$.

It follows that

$n(0)+n(1)\zeta+\dots+n(p-1)\zeta^{p-1}=0.$

Since the minimal polynomial of $\zeta$ is

$1+\zeta+\zeta^2+\dots+\zeta^{p-1}=0,$ we conclude that $n(t)=const.$

This proves part ii) and part i) follows at once.

Lemma 2 (combinatorics) Let $E \subset {\Bbb Z}_p^2$ of size $>p$. Then $E$ determines every possible direction in ${\Bbb Z}_p^2$.

To prove this note that if a direction is missing, $E$ is a subset of a graph

$\{(x,f(x)): x \in {\Bbb Z}_p \}$, so $|E| \leq p$,

With the two lemmas in tow, we are ready to prove the theorem above. Suppose that $E$ is spectral. Then ${\{ \chi(x \cdot a)\}}_{a \in A}$ is an orthogonal basis and by elementary linear algebra, $|A|=|E|$. The orthogonality hypothesis implies that Lemma 1 applies and we conclude that $|E|=kp$.

Suppose that $k>1$.Then $A$ determines every possible direction since $|E|=|A|$. Invoking orthogonality, we see that

$\widehat{E}(a-a')=0$ for all $a \not=a'$, $a,a' \in A$. By Lemma 1, $\widehat{E}(r(a-a'))=0$ for all $r \not=0$ and since $A$ determines every possible direction, $\widehat{E}(m)=0$ for all $m \not=(0,0)$. This implies that $E={\Bbb Z}_p^2$.

Thus we may assume that $k=0,1$. The case $k=1$ is not very interesting, so we consider the case $k=1$. Invoking Lemma 1 again, we see that for each $t \in {\Bbb Z}_p$, $E$ has exactly one point on the line $\{x \in {\Bbb Z}_p^2: x \cdot m=t\}$, where $m \not=(0,0)$ is a fixed vector such $\widehat{E}(m)=0$. As we noted above, such a vector must exist since we are assuming that $E$ is spectral, so $\widehat{E}(a-a')=0$ for all $a \not=a'$, $a,a' \in A$.

By rotation, we may assume that $E$ has exactly one point on the lines given by the equation $x_2=t$. But then it is not difficult to see that that $E$ tiles with the tiling set $T=\{(\lambda,0): \lambda \in {\Bbb Z}_p\}$.

Let’s now assume that $E$ tiles by translation with a tiling set $T$. Then $|E|$ must be a divisor of $p^2$. The only interesting case is $|E|=p$. Tiling means that

$\sum_t E(x-t)T(t)=1$ for all $x \in {\Bbb Z}_p^2$. Taking the Fourier transform of both sides, we see that

$\widehat{E}(m)\widehat{T}(m)=0$ for all $m \not=(0,0)$. We conclude from this that there exists $m \not=(0,0)$ such that $\widehat{E}(m)=0$. If not, then $\widehat{T}(m)=0$ for all $m \not=(0,0)$. It would follow that $T={\Bbb Z}_p^2$, which is not possible since $|E|=p$, which implies that $|T|=p$.

Since there exists $m \not=(0,0)$ such that $\widehat{E}(m)=0$, we can invoke Lemma 1 again to see that $E$ contains exactly one point on the lines perpendicular to $m$. By rotation, we may assume that $E$ has exactly one point on the lines given by the equation $x_2=t$. But then one can check by a direct calculation that the set ${\{\chi(x \cdot a)\}}_{a \in A}$ is an orthogonal basis for $L^2(E)$, where $A=\{(0,\lambda): \lambda \in {\Bbb Z}_p\}$.

This completes the proof. Here are some exercises for the interested reader.

Exercise: Work out the details of the case $p=1 \mod 4$.

Exercise: Prove that if $E \subset {\Bbb Z}_p^3$ and $E$ tiles by translation, then $E$ is spectral.

It is not known whether spectral implies tiling in ${\Bbb Z}_p^3$.

## Some thoughts on my trip to Vietnam

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.

## Conference Hotel

My eyelids are heavy, the blanket is cold

as the light dances in through the shades.

I am laying alone, feeling frightfully old

On the carpet below, a crumpled up sheet

holding thoughts that were brutally crushed.

As I turn to the side, I can hear the beat

of ideas- simplistic and rushed.

Seeking warmth for the body and soul I rush in

to the shower and turn up the heat.

Then I sit on the floor all bewildered but clean,

getting ready to face my defeat.

I pick up the chalk and the fear is gone ,

washed away by the coffee and soap.

Just a few more remarks and I will be done

As the night settles in, I am back on the bed

feeling happy, relaxed and awake.

All this will be replaced by the onset of dread

for the future, for sanity’s sake.

## Travel blog, May 3-5, 2017

This is going to be a long trip! The original plan was to arrive in Atlanta at 2 pm on Wednesday, May 3, rent a car and drive to Athens, GA to hang out with my friends and attend Hans Parshal’s Ph.D. defense on Thursday. On Friday I was suppose to drive to Atlanta to speak at the analysis conference at the Kennesaw State University, then fly to NYC on Friday evening in order to speak at the AMS Special Session at Hunter College on Saturday morning. On Sunday morning the plan is to fly to Seoul, South Korea, to speak at the conference in honor of Peter Jones. The following Sunday I am due to fly from Seoul to San Francisco to speak at the analysis workshop at MSRI. This is an insane plan and it was certain from the beginning that logistical obstructions are inevitable.

Small problems began on Day 1, namely Wednesday, May 3. My flight out of Rochester was delayed and I did not arrive in Atlanta until after 3 p.m.. The folks at Payless Car Rental, all wearing the unmistakable “I don’t give a fuck” grin on their faces claimed that I arrived too late and economy cars were no longer available. They offer to rent me a slightly more expensive vehicle, which I rejected as a matter of principle. After a short exchange of pleasantries I retreated to a nearby bench and reserved a car at roughly the same price with Alamo on Priceline. Just after 4 p.m. I was finally on the way to Athens, GA.

Since I left after 4 p.m., I was driving into the teeth of the Atlanta rush hour. It was nearly 6.30 p.m. when I finally reached Yorgis and Lenore’s house, tired yet deeply amused by the number of gun shops I passed along the way. Yorgis and Lenore treated me to a wonderful spaghetti carbonara and some very pleasant wine. I followed this experience with a double single malt scotch with Neil and Akos and the nearby bar. There is nothing quite so satisfying as catching up with old friends, especially when mathematical ideas try to penetrate the murky haze induced by sleep deprivation and a light buzz.

In the morning Yorgis and revisited some of our favorite problems that we have been beating our heads against wall over for the past few years. The one problem that has been the absolute bane of my existence is the question of how large $A \subset {\Bbb Z}_p$, $p$ prime, needs to be to ensure that

$|A \cdot A+A \cdot A|>\frac{p}{2}$.

Derrick Hart and I proved almost ten years ago that the conclusion holds if $|A|>p^{\frac{2}{3}}$ and I fully expected the problem to be solved by now. But no such luck! The exponent has not moved below $\frac{2}{3}$ at all, not even by a logarithm. I have not heard a plausible idea to advance this problem in a very long time.

After a very pleasant lunch at the Royal Peasant, Yorgis and I returned to the department, chatted with Neil and Akos and then went downstairs to attend Hans Parschal’s Ph.D. defense. Hans did a fantastic job and the presentation smoothly transitioned to a trip to the local beer garden followed by an excellent dinner at an authentic Central American restaurant.

The plan to speak at the Kennesaw State conference began to unravel on Thursday, though it was not clear until late that day that it was a corpse. Due to weather issues in NYC, my early evening flight from Atlanta to NYC was very likely to materialize, so I was advised to take a 6.45 a.m. flight to NYC instead. This forced me to cancel the Kennesaw State talk and get up at 3 a.m. to drive from Athens, GA to Atlanta, turn in the rental car and fly. AM radio kept me awake during the drive and the shrill voice of our glorious leader woke me up faster than any coffee ever could.

Upon arrival at La Guardia, I took an Uber to Azita Mayeli’s place, dropped off my back, gave little Carla a little stuffed kitty with glasses and took the Long Island Railroad train with Azita to Manhattan. There we met up with Chun-Kit Lai and did a bit of mathematics. Azita went off to teach while Chun-Kit and chatted a while longer. But soon after I collapsed on the couch in the lounge of the CUNY Graduate Center and Chun-Kit went to his hotel room. When I woke up, Azita was done teaching and it was time to listen to an analysis seminar lecture delivered by my Rochester colleague Kazuo Yamazaki. This was truly an excellent talk on the Navier-Stokes equation with lots of background material. But by the end of the talk I understand that I can no longer process anything resembling complicated information until I get a few hours asleep.

The rest of the day and evening was a blur. Azita and I picked up some groceries and I made an eggplant, garlic and tomato pasta which Reza, Azita and I proceeded to consume with great vigor along with several glasses of nice red wine. I am now on the couch typing away and thinking that I should be sleeping instead. But writing is a real pleasure, so I continue even though my eyelids are getting droopy and my thought processes have long passed the state of coherence.

Tomorrow I am supposed to be speaking the AMS Special Session at Hunter College and after that I am getting together with Sinan Gunturk for lunch followed by some beer drinking with David Karapetyan later in the evening. This is the plan anyways and in my next blog entry I will describe how closely the reality followed it in the end.