The trip to Ukraine, September 30, 2024- the final day of the journey, 8.56 a.m.

Posted on September 30, 2024 | Leave a comment

I am having breakfast in Terminal 4 and thinking about this wonderful trip. I miss my family terribly and I am more than ready to go home! I am full of impressions that are going to stay with me forever. The mathematical and personal connections I made on this trip are going to continue to develop in the coming days, months, and years. It would be a huge understatement to say that this has been a very successful trip. I want to thank Rostyslav Gryni, Taras Banach, Tetiana Zaharchenko, Maria Vlasenko, all the student participants, and many others for making this trip so wonderful! I also want to thank Oleksiy Klurman who initially recruited me for this wonderful adventure.

The English breakfast I am consuming is making me wonderfully sleepy, despite the copious amounts of coffee I am using to wash it down. I am going to sleep very well on the airplane, though I will probably wake up occasionally to work on the revision for a recent paper of mine. My co-author and I received the most detailed referee report in history! Seriously, I have never seen anything like it. The referee read every line and found every possible typo. This makes the revision process very simple and comfortable.

It is time to conclude this series of blog entries. I am going to continue my blogging in the coming days, focusing primarily on mathematics, but possibly other things as well. I never know what I am going to write until I open up the laptop screen and let it fly. See you soon!

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Trip to Ukraine, September 29, 2024, 11.44 a.m.

Posted on September 29, 2024 | Leave a comment

The final day of my trip to Ukraine was very pleasant. Rostyslav Hryniv picked me up at the train station and took to me a little market where I bought a small suitcase to take some presents back home, including a couple of bottles of horilka with pepper and honey, Lviv chocolate, and some souvenirs. There was an entertaining moment when the sales lady told me that her suitcases were imported from France, and that instantly reminded me of Bourgain’s wonderful quote: “…everybody has to make a living”.

After buying the suitcase, we walked to the city center, bought some horilka and coffee, discussed how the various parts of the city have changed, picked up a copy of the menu in “Krajivka” (more on that later), and went to have lunch at a traditional Ukrainian restaurant. I had borscht, salo, and pelmeni, and washed it down with some local beer. But this was not the end of the day’s adventures. Rostyslav and I walked to the Catholic University (UCU) and I was incredibly happy to walk through Striysky Park I remember from my childhood days. Right before going to the university, we got some ice cream and 150 grams each of “vishneva nalivka”, a kind of cherry liquor, incredibly delicious!

Upon arrival at UCU, I spent about 20 minutes arranging the souvenirs and horilka in my suitcase. This was a non-trivial optimization problem, and in the end, I had to leave a couple of small things behind. Rostyslav and I had some coffee, discussed some mathematics, and took a taxi to the train station from where I took a train to Przemysl. My plan was to stay in Przemysl, instead of Rzeszow, and then take a train to Warsaw in the morning, by-passing my connecting flight from Rzeszow to Warsaw. This plan did not work because there were literally no places to stay in Przemysl, so I took a taxi to Rzeszow, spent the night at a hotel, and arrived at the airport a little over an hour ago.

I am full of wonderful impressions from the trip! I will keep writing about it in the coming days.

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Trip to Ukraine, September 28, 2024, 7.31 a.m.

Posted on September 28, 2024 | Leave a comment

After giving my second talk in Lviv, at the Ivan Franco State University, I enjoyed a quiet evening. The next day I listened to a few education talks at the conference running at the Lviv Catholic University. It was very interesting to see that education research in Ukraine follows very similar patterns to those in the U.S. After having lunch and a very pleasant conversation with Tetiana Zaharchenko, I spent the whole day walking around the center of Lviv, doing work in various coffee shops, and having a very nice traditional Ukrainian dinner of borscht, vareniki, and salo. I finished off the meal with a pint of very nice dark beer, walked around the city a bit more, and headed over to the train station where I took an overnight train to Kyiv. The sleeping cot was not incredibly comfortable, but I was able to sleep for a few hours.

Upon arriving in Kyiv, I was greeted by air raid sirens. Fortunately, I was able to get to my hotel quickly where I had some breakfast and I slept through two different rounds of air raid sirens before taking a shower and getting ready for a great day ahead! Masha Vlasenko picked me up a little after 1 p.m. and took me to a very cool restaurant that served rice bowls with various ingredients. When we took a trolly to the Kyiv School of Economics, we went past the circus where I once went with my father in 1978 when we were visiting my grandmother’s brother Misha.

The Kyiv School of Economics is a very interesting place. The building is very modern, equipped with all possible modern technologies, a nice coffee shop downstairs, a basketball court outside, and several ping-pong tables. The bomb shelter has several classrooms and plenty of electronic equipment. This came very handy at the end of my second lecture as we are going to see in a moment. My first lecture was very lively. The participants, mostly advanced undergraduate and graduate students asked many questions and worked very hard to understand the content. This made me think that it would be a real pleasure to teach a course for them sometime soon. Indeed, Masha Vlasenko and I discussed this possibility and will continue talking about it in the months to come.

In the evening, I walked around the city and had dinner at a very pleasant Georgian restaurant. After a long day, a bowl of Harcho was exactly what I needed, and I went to sleep fairly early after looking over the lecture notes one more time. Perhaps for the first time in my professional life, I added a theorem that I proved just in the past few days. I am going to have a post about it sometime in the next few days.

My last day in Kyiv was quite eventful. I spent the whole morning walking around the city, focusing on Independence Square and the surrounding area. The pictures of the fallen soldiers and civilians made a deep impression on me. Observing the war in Ukraine from afar is completely different than being here and feeling the atmosphere. I am determined to come back soon and contribute to the revival of this beautiful and determined nation.

While I was walking around downtown, I noticed that the percentage of people speaking Ukrainian has significantly increased. Twelve years ago, when I was here, most of the people spoke Russian. During this visit, my guess is that only about a third of the people spoke Russian. I did not hear a single young person speak Russian. At the same time, those who spoke Russian seemed comfortable doing so. The question I asked myself is what would have happened if someone spoke Ukrainian at a cafe in Moscow. The answer is obvious.

After arriving at the Kyiv School of Economics, I had lunch with Masha and another faculty member Giorgii. We then went to the mathematical institute on the second floor of the KSE building and discussed some mathematics before my lecture began. This was my final lecture in Ukraine and I was very inspired. The students were even more engaged and asked even more questions. When I was about to discuss the final result of the lectures, the air raid sirens went off and we had to complete the lectures in the bomb shelter, a very comfortable facility I mentioned earlier. When the lecture ended a mathematical discussion gave way to the discussion of my childhood in Ukraine. One of the participants, a local data scientist, presented me with a “smetannik”, a local desert I remember from my childhood in Lviv. This was a very nice gesture that I will always remember.

In the evening I went to a Tatat restaurant “Musafir” recommended by Masha and had a very nice dinner. I was quite tired and went to my hotel room soon after to pack and get a few hours of sleep before waking up at 5 a.m. to catch the train I am on right now to Lviv. Rostyslav is going to meet me at the train station and I will spend a few more hours in Lviv before taking a bus to Rzeszow from where I am going to fly to Warsaw, and then on to London, from where I am going to travel back to United States. I will never forget this trip!

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Trip to Ukraine 2024, September 24, 1.41 p.m.

Posted on September 24, 2024 | Leave a comment

The last couple of days just flew by! Taras Banach and Rostyslav Hryniv took me sightseeing on Sunday. We went to see the Ivan Franco University, the house where Stefan Banach lived, and Lychakiv cemetery, among other places. Along the way, we stopped by School No. 6 where I studied from 1975-1979, and the apartment building where I lived from 1967 until 1979 before moving to the United States. At the end of the day, we had dinner at Kryivka, a theme restaurant built on the site of the hideout for the Ukrainian Insurgent Army during WWII. The place is full of UPA memorabilia, and to enter one has to know the suitable password. Going there stretched my sense of humor to the limit, but the food was very good, the company was excellent, and the ambiance was unforgettable if somewhat disturbing. Beer and horilka were flowing like water and my plan to work on my grant proposal after returning to the hotel room had to be amended.

The next day passed by like a blur. I met many interesting people, worked on some mathematics with Rostyslav, gave the first of my LMS lectures, and went out to dinner with participants of the technology conference that is taking place this week at the Lviv Catholic University. It is remarkable that during wartime so much thirst for knowledge remains! My talk was a truly warm experience. Rostyslav gave a very kind introduction and the students in the audience listened attentively and asked many questions. More than 30 minutes of questions followed my talk and the discussion spilled over the hallway where coffee and snacks were waiting for us. I was deeply impressed by the fact that the students waited for me to appear from the lecture hall before grabbing any of the snacks or drinks. At some point, I was asked about my childhood in Lviv and my stories about being thrown out of school two weeks before we left the country in 1979 seemed to have made a deep impression on my conversation partners.

This morning I attended a plenary talk at the technology conference where the speaker, a very prominent scientist, talked about the need to improve the “knowledge graph” technology to combat the exponential growth of publications. The use of Artificial Intelligence in general, and large language models in particular, was certainly interesting, but it left me wondering whether the overarching paradigm is sustainable. I asked, very politely, whether a low-tech approach to the problem in the form of discouraging flooding the journals with papers should be considered, the speaker agreed but expressed the belief that even if the number of publications were to decrease drastically, the problem would remain. I did not press the point because this is not my field, but have the feeling that if a scientific field produces more papers than it can absorb, no amount of technology is going to help.

This afternoon I am giving the second talk of my lecture series, this time at the Ivan Franco University. The audience should be more or less the same, though some people who watched the first lecture on Zoom may be coming in person today. Yesterday, I covered the basics of exact signal recovery a la Matolcsi-Szhucks/ Donoho-Stark, introduced and proved some basic properties of the discrete Fourier transform, and proved the classical uncertainty principle on {\mathbb Z}_N^d. Today, I am going to discuss how the uncertainty principle and, consequently, the recovery mechanism can be improved is the set of missing frequencies S satisfies a non-trivial restriction estimate.

I will certainly have more to report later today!

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Trip to Ukraine 2024, September 22, 10.42 a.m.

Posted on September 22, 2024 | Leave a comment

My bus finally reached Lviv at 8.55 p.m. yesterday. It was scheduled to arrive at 7 p.m., but the border crossing was long and entertaining. The passports were collected and returned twice, once by the Polish border guards, and once by the Ukrainian border guards. This is quite normal, but then the fun began. After a long wait, we were asked to disembark and take some of our luggage with us. If we only had one bag, we had to bring it with us, but if we had two, we could leave one behind on the bus. The logic of this escapes me, but perhaps this is why I am in a different line of work. When we got off the bus we had to pass through a security line. The folks in front of me and behind me, all with Ukrainian passports, had to answer a few questions. I went through without a single question after the guard saw my American passport. Once again, I do not understand the logic of this, though I suspect there is a banal explanation that the guard did not speak English and did not realize that I speak Ukrainian.

The people on the bus were incredibly friendly. Two very nice people, a brother and sister inquired about my visit to Lviv, and told them about the lecture series. In a small-world twist, it turned out that both have applied mathematics degrees from the university I am visiting and know my host very well. After talking to them for a while, I took and nap and woke up when the bus entered Lviv. I felt an immediate wave of warmth after seeing my hometown roughly five years after I came here with my brother and my parents. That trip was the first since we left in September 1979.

Rostyslav Hryniv, an excellent mathematician and a very friendly person, met me at the bus stop located right next to the train station and took me to my hotel near the Catholic University of Lviv. We grabbed some food and chatted about a variety of mathematical and non-mathematical topics before I returned to my hotel room and went to sleep. The room is very comfortable, the internet works very well, and there is always hot water in the shower, unlike the situation when we left the Soviet Union in 1979.

I am not sitting in the Lviv Croissants, a very pleasant coffee shop where I bought a nice cup of coffee and raspberry croissant for roughly 95 cents. I saw virtually no signs of wartime walking around the center, with some minor exceptions that I will describe in more detail later. My plan is to work for a couple of hours and then have lunch with friends and do some sightseeing. The first place I am going to visit is School No. 5 on Zelanaya Street I attended for five years from 1975 till 1979. I will then walk over to the apartment building where we used to live. I am full of pleasant anticipation.

My lectures begin tomorrow. I am seriously considering using the blackboard instead of recycling my slides from the Hausdorff Institute lecture series. This would allow me to interact with the audience a bit more. I will make the final decision when I talk to Rostyslav this afternoon.

It is time to prove some theorems, or at least try!

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Trip to Ukraine 2024, September 21, 12.59 p.m.

Posted on September 21, 2024 | Leave a comment

My first attempt to catch an earlier bus failed. I anticipated the bus being full, but the problem was more basic – the bus did not show up. I am now back in the airport terminal, eating a Polish sausage and drinking it down with apple juice. The next reasonable bus is around 3 p.m. In the meantime, I am sitting around thinking about one of my favorite problems. Let S be a compact subset of {\mathbb R}^d of Lebesgue measure 0. Suppose that f is a locally integrable function such that the Fourier transform \widehat{f}(\xi)=\int e^{-2 \pi i x \cdot \xi} f(x) dx is supported in S. What is the critical value p_{critical} such that if f \in L^p({\mathbb R}^d) for p<p_{critical}, then f is identically equal to 0.

Agranovsky and Narayanan proved that if S is a smooth k-dimensional submanifold of {\mathbb R}^d, then if \widehat{f} is supported in S, and f \in L^p({\mathbb R}^d) for p \leq \frac{2d}{k}, then f is identically 0. Their result was extended to sets of packing dimension k (not necessarily an integer) by Senthil Raani. For manifolds, the exponent \frac{2d}{k} is known to be sharp when k \ge \frac{d}{2}, but it is never sharp when k<\frac{d}{2}. For example, if S=\{(t,t^2, \dots, t^d): t \in [0,1]\}, Guo, Iosevich, Zhang, and Zorin-Kranich proved that the critical exponent is \frac{d^2+d+2}{2}.

In the case when the packing dimension is k, Senthil Raani showed that one can construct a set of packing dimension k such that the exponent \frac{2d}{k} is sharp.

In the coming days, I am going to describe some recent progress on this problem and applications to exact signal recovery.

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Trip to Ukraine, 2024

Posted on September 21, 2024 | Leave a comment

September 21, 2024, 11.30 a.m.

I just landed in Rzeszow a few minutes ago, and soon I will be taking a bus to Lviv to give a series of lectures as the LMS Distinguished Visiting Fellow at ICMU. The city of Rzeszow seems peaceful and carefree, but the anti-aircraft batteries are clearly visible at the airport – a clear reminder that Russian missiles are not very far away and occasionally fly into Polish airspace. There are Ukrainian-speaking people at the cafe. Some of them will probably be on the bus with me.

The sense of anticipation I am experiencing is very pleasant. I cannot wait to see the Scottish Cafe. When I was here five years ago, it was a wonderful family trip and I did not do much mathematics. This time, I would like to sit at the Scottish Cafe for a few hours, sip on coffee, and try to prove a theorem in the same place where Banach and Steinhaus worked their magic 85 years ago. I want to wander around the streets in the center of Lviv and let the memories pour in, both from the trip five years ago and from my time here as a child. We left Lviv for Chop on September 29, 1979, and crossed the border into Hungary the following day. Coincidentally, I will be departing Lviv at the end of this trip on September 28, 2024, exactly 45 years later.

I have a ticket on the bus to Lviv at 7.30 p.m., but I am going to try to catch an earlier bus at 12.20 p.m. I will walk over to the bus stop in a few minutes to see if they have open seats. I was originally planning to spend the day working at the airport, but I want to get to Lviv as quickly as possible and officially begin this wonderful adventure.

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The trip to my hometown: Day 2

Posted on June 10, 2018 | Leave a comment

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.

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The trip to my hometown: Day 1

Posted on June 9, 2018 | Leave a comment

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!

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On humanism and the two-state solution

Posted on March 18, 2018 | Leave a comment

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.

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