ICM 2018

 Day 0

Team NSF arrives in Rio de Janeiro on a rainy morning after mostly sleepless night on the plane. This year DMS is represented by Swatee, Joanna, Andy and Tomek and we are poised to tackle multiple responsibilities. Luckily the NSF mothership trained us in evading capture, detecting explosive devices in hotel rooms, blending with crowds in urban environments and such. We are numbed by immunizations against multiple exotic diseases and have suitcases full of clothes soaked in insect repelling potion.

We quickly register and receive our  badges. Organizers dispensed with  more traditional bits like the university/organization name and our badges say only our names and country.  Given our grant declination rates we treasure some anonymity.

This year International Congress of Mathematicians is held in a large business center on the outskirts of Rio de Janeiro.  The place is quite scenic and geared towards attending a lot of lectures because there does not seem to be  much else to do.

The rumors about Fields medals are rampant and we weed out some false positives against the list of attendees. There is also a rumor about a fire which is quickly confirmed. Our app with the ICM schedule is not populating with room numbers and such. Later in the evening during the  welcome reception we find out that the  hall that was supposed to hold the Fields medal ceremony has partially burned together with pricey backdrops, audio equipment and few other trinkets. Suddenly constant rain is no longer such a nuisance, and we admire organizers who seem to persevere and down caipirinhas with aplomb!

Wandering around we come across large throngs of people, mostly women, heading for activities of WM^2, that is World Meeting of Women in Mathematics, a satellite meeting of ICM that we did not register for. Bummer but I intend to find out more.

Eventually the ICM program app syncs with the new data and we start wading in the schedule. There is a huge number of talks even if one discounts almost infinite number of short communications. We start tomorrow with newly minted Fields medallists but the schedule has a significant number of past Fields medallists as well. Somehow it troubles me although I cannot quite put a finger on it. Only later an email from work with the  announcement of a slew of temporary appointments in our directorate makes it more clear. It is a bureaucratic myopia when the flow of ideas seems to originate from a small number of familiar sources while everything else appears distant and blurry. Apparently this may affect the Mathematical Union as well as a government agency.  

In the evening Team NSF attends welcome party. It is well attended, reportedly this year ICM has four thousand participants, and we mingle with the crowds and chat with our PIs and as well as some random people who look strangely familiar.

Day 1

Moderately rested and nourished by a light breakfast Team NSF heads for the ICM opening ceremony. Two security checks later we enter dark and musty auditorium filled with frigid air. For a moment I feel like shopping in a Costco giant fridge but loud jungle sounds bring me back to reality. Yes,   this is the ICM in Brazil and the Fields medals will be announced shortly. Shortly? It is a relative term ,and as hypothermia is setting in, I am urged by some members of Team NSF to scan the first row in hope for an early sighting of the award winners. It  is a correct decision and on a clandestine shot taken in a very challenging environment we see (from the left) the Nevanlinna Prize winner Constantinos Daskalakis, and Fields medalists Akshay Venkatesh, Peter Scholtze and Alessio Figalli. NSF Team instantly recognizes the middle two and beams the news to the mothership a solid 40 minutes before the official announcement!

When the ceremonies finally commence, apparently delayed by the late arrival of some officials, we get music, singing and dancing, introductions of various VIPs, and lastly the winners.  

Fields medals go to:

Peter Scholze (Bonn)

Akshay Venkatesh (IAS)

Caucher Birkar (Cambridge)

Alessio Figalli (ETH)

Nevanlinna Prize: Constantinos Daskalakis (MIT)

Gauss Prize: David Donoho 

Chern Prize: Masaki Kashiwara

Leelavati Prize: Ali Nesin

 The ceremony is quite uplifting and joyful and all goes fairly well until Birkar’s Fields medal gets stolen minutes after he received it. It is another embarrassing setback for the organizers still reeling from inadvertently burning the original ICM venue.

Caucher Birkar stands out among Fields medallist and I have high expectations regarding mathematical and societal impact of this recognition. He is a Kurd from Iran who succeed in becoming a mathematician against really long odds and then reached the absolute top of the profession in a stereotype shattering manner.

The rest of the day is devoted to laudations of the winners’ work. This is generally a tall order as in most cases the work is difficult and technical and at the cutting edge of this or that discipline, and as usual it was a mixed bag.

It starts with Eva Tardos presenting Costas Daskalakis work. Pretty good talk but  eclipsed by the Nevanlinna Prize winner speaking of his work himself later in the day. Next is Michael Rappaport speaking of Peter Scholtze work on perfectoid fields and spaces. All is well given the subject, and one has a glimpse of quite amazing connections, particularly between world of things  living in characteristic p  and those in characteristic zero.

This is followed by Peter Sarnak speaking of Vankatesh’s work. Excellent presentation even if the slides are hand written. Sarnak is a very good speaker and he easily convinced me that lowering some bound from 1/4 to 1/6 is not only a difficult challenge but a highly worthwhile one. Next Luis Cafarelli presents Figalli’s work and this one is a downer. The presentation dwells  on trivialities as in “if we assume that some function is twice differentiable we get a stronger estimate  of something” and is more boring than reading a wikipedia entry.  Luckily Figalli (as well as other Fields medalists) will speak about his work himself and this will likely set the record straight.

Lastly Chris Hacon speaks on Caucher Birkar’s work and it is quite exciting. He retraces last 50 years of algebraic geometry and focuses on the minimal model program and Birkar’s contribution to it.  At this point I suspect that Birkar’s backpack has already been found with all mathematics untouched by the perp and only his gold Fields medal still missing.

People presenting the last three prizes have easier job because the awardees do not seem to be around yet and may not speak themselves. This is not an incentive to butcher someones lifework but opportunity beckons. Enter Pierre Schapira presenting 50 years of Masaki Kashiwara work. He gives a truly dismal talk, clearly  a contender for the worst ICM 2018 presentation. Kashiwara’s work is deep and influential but Shapira rambles on with the subtlety of Godzilla spewing fire of incomprehensible jargon and gibberish and laying thick waste of minutiae. Sad! 

I single out three presentations that simultaneously hit high marks on the  presenting great accomplishments, being understandable to a significant portion of the audience, and a flawless delivery.

First of those is Emmanuel Candes speaking of the work of his advisor, David Donoho, the Gauss Prize winner.  It is a beautiful talk that centers on the general theme of sparsity and discusses sparse signal recovery and compressed sensing. There is a good amount of very interesting math, great applications to tomography, geology etc., and the overarching theme that sometimes with the right tools we reduce seemingly complicated things to fairly simple ones.

The Leelavati Award is an award for outstanding contribution to public outreach in mathematics and the work of Ali Nesin and the story of Nesin Mathematics Village is presented by Gert-Martin Greuel. Nesin Mathematics Village (http://www.nesinkoyleri.org/eng/) is quite a place. The NMV was started a decade ago, and it welcomes kids 13 or older to visit and study mathematics.  NMV can host about 150 kids at a time, and has a  staff of fifteen and about hundred volunteers each year. Kids usually stay for two-week terms. The process is unlike any actual school and some supporters say that NMV has more impact on math education in Turkey than the entire Turkish Ministry of Education. With such helpers who needs enemies!

The last talk of the day was by the Nevanlinna Prize winner Costas Daskalakis and it concerns computational complexity of finding Nash equilibrium. To recap, a strategy is a Nash equilibrium if no player can do better by unilaterally changing it. Nash theorem asserts that if we allow mixed strategies (that is a selection of pure strategies with some probabilities) then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium. 

Unlike von Neumann theorem which gives poly-time algorithm for optimal strategies realizing Nash equilibrium for two-player zero sum games, the general case of finding Nash equilibrium does not seem computationally tractable. Yet, since Nash equilibrium always exists, finding it does not seem NP-complete either. Work of Daskalakis shows that finding Nash equilibrium for finite games (further narrowed to 2-player games) is PPAD-complete. PPAD is a class of problems that are computationally as hard as Brouwer fixed point theorem or Sperner’s lemma, and indicates that finding Nash equilibrium is generally intractable (unless P=NP).

That puts in question practical usefulness of the  entire concept of the Nash equilibrium if finding it is not really possible for any of the players.

Day 2

I wake up feeling blue and miserable and decide to skip morning lectures and go jogging instead. ICM is full of rumors of the dangers lurking outside but the first person that I ran into outside our compound is a former NSF program officer Sylvia Wiegand trotting merrily along. Not a threat, go figure.

The last portion of Costas Daskalakis talk yesterday concerned  applications in machine learning. I did not quite get the details, even if they were there, but the example had to do with neural networks and adversarial network games. What I got is that one can build a neural nework and train it on a data set,  for example pictures of celebrities or living rooms, in a way that when such network receives a random input, white noise really, it produces a “hallucination” - that is an object that looks like it belongs to the training set but is completely synthetic. Daskalakis showed a number of pictures to make this point and the whole thing is quite stunning. When I am jogging along this all comes back in a bothersome way. After all I am a neural network on two legs. Does it mean that if I watched Fox News all day along I would generate a conspiracy theory after spilling cofee on my lap?  Could it be that simple? I wonder, feeling like someone has given me a missing piece of a puzzle.

I am back an hour later safe and sound and feeling better. Meantime Joanna comes back impressed with Rahul Pandharipande's plenary lecture. Like Alessio Figalli, he is our former PI who chose Europe.

In the afternoon I choose among invited lectures. The first talk is by Jason Miller, a probabilist from Cambridge. It concerns the following question — how do you randomly sample an object that is homeomorphic to a sphere?

There are two ways -  Liouville quantum gravity which originates in conformal mapping theory, and SLE and Brownian map which is purely combinatorial. The main result is that both constructions are really the same, which is quite surprising.

The next talk is by James Mynard about gaps in primes,  big and small. It concludes with  the following result:  under some assumptions which are exceedingly unlikely to be true but  not yet known to be false  one of the following certainly holds,

A. There are infinitely many pairs of primes that are no more than 6 units apart,

B. Every even number is at most 2 units away from a number that is a sum of two primes.

Pretty cool even if not to be taken really seriously.

In the evening Team NSF goes a party organized by the Simons Foundations. There are lots of celebrities and gossip. We find out how it came to pass that the next ICM is in St. Petersburg. It turns out that in the final round St. Petersburg beat Paris with promises of low conference fees, Putin handing out Fields medals, and a dollop of funding for the Mathematical Union.  I recall that the last time ICM was in Russia was in 1966 and a Fields medalist ended up in jail but the ghosts of the past seem forgotten. 

I am happily consuming fair quantities of booze when Swatee summons me to help a fellow mathematician with an e-mail problem. Indeed, there is an unfamiliar and very old fellow surrounded by heavyweights of the mathematical world complaining bitterly about his inability to send mail. It is clear that there is no way that this can be helped but I bravely offer to fix the problem. As we depart to his room where his blasted laptop resides a throng of people lines up to bid him farewell and I overhear someone addressing my tormentor as “Sir Michael.” 

Indeed, it is Sir Michael Atiyah, the recipient of 1966 Fields medal in Moscow, whose ability to stay in touch with the world is now in my hands. Ouch!

We take the elevator to his room and he fires up his laptop. It seems to connect to internet and I gently sugest that he tries to send mail to himself. To everybody’s surprise it works! Next I urge him to send an e-mail to someone else and it works as well. 

“All is well,” I conclude greatly relieved.

“Indeed,” he says, “but what happened?”

“In ancient Egypt,” I explain, “there were people who could stop bleeding in a patient by merely being in the same room. It is one of those things.”

“So what happens when you leave?” Sir Michael is uncomforably fast.

“My room is six floors above yours, it should still work,” I assure him and go back to the party.

Day 3

Another rainy day in Rio but the schedule is packed with talks so it is no problem for always active Team NSF.

After quick breakfast we rush to the plenary talk by Andrei Okounkov. In 2006 he received the Fields Medal "for his contributions to bridging probability, representation theory and algebraic geometry,” and the talk builds on it in a spectacular way. Having already one Fields medal allows to experiment with the live audience and Okounkov takes this path with great deal of positive energy. The slides are nice and colorful and I feel like Forrest Gump staring into a box of chocolates. There are no theorems or conjectures, and Okounkov pledges to talk only about things that he believes are “reasonably true.” From where I sit it is all about tools, and ability to develop sophisticated mathematics for the purpose of understanding physical world. What makes it all so fascinating is how all these things are connected to one another, either by fixing some parameter, taking limits, scaling or whatever else. It is all quite visionary and Okounkov uses deep space pictures as a background so when the image of Roman Bezrukavnikov careening next to Sputnik comes up it is no longer such a big surprise. Naysayers scratch their heads but I say “Wow!”

Following such fireworks is a tall order and Greg Lawler who speaks next comes a bit weak. The talk is quite technical and gives an overview of random walks, Schramm-Loewner Evolution and finally Louiville Quantum Gravity.  It is all good but pretty dry and rather unfocussed. 

It is lunch time and Team NSF becomes a bit restless. So far we have not found a place that would merit a second try. We ate from food trucks, hotel restaurant, and two nearby places and in all cases the stuff was quite awful. The only reasonable thing is breakfast and now we gorge ourselves in the morning in hope that it will last through the day.  The food is simply very limited, tasteless and sometimes over- or undercooked. Of course there is great food here, we just haven’t found it yet. 

The first afternoon talk is Fields medal presentation from Akshay Venkatesh. It is perpendicular to Peter Sarnak’s laudation in that it focuses on a single topic and describes it very well. The talk starts with quadratic forms and then discusses arithmetic groups. Eventually number theoretic questions give rise to fascinating topological problems and even Joanna who is not known for fondness for number theory and usually starts playing with her phone the moment  primes enter the picture, gets quite excited and enthusiastic. It is a great lecture and very interesting mathematics.

I follow with invited lecture by Allan Sly, a probabilist from Princeton. The talk concerns random k-SAT satisfiability. Specifically, the existence of the strict threshold  alpha such that random  k_SATs  with density lower than alpha are almost surely satisfiable and those with density larger than alpha are almost surely not (density reflects the balance between the number of variables and clauses).  I note that the result is a joint work of Jian Ding, Allan Sly and Nike Sun, all three CAREER awardees in the Probability program. 

My last talk of the day is Laszlo Babai speaking about the graph isomorphism problem. This is a fantastic result answering the question when two finite graphs are isomorphic. The problem seemed to be solidly in the upper portion of the class NP and this  result shows that it is nearly in the class P.   This is the second time I hear Babai talk about it and it is really great result. I enjoy it all quite a bit when the disaster strikes. In a room filled with over  hundred mathematicians suddenly there is a sound of ABBA blasting  “Dancing queen.” People start looking around and ultimately all eyes are on me as the noise comes out of my pocket. I pull out my phone and eventually manage to shut it down. All is well until minute late ABBA comes back with renewed energy. I am completely mortified and manage to turn off my phone while pawing at the floor to dig a hole to bury myself in it.

Joanna and Swatee attend Emmy Noether lecture by Alice Chang  (quite ok they report later) while I retreat to my room to regroup. 

It is raining cats and dogs but eventually it is 7:30 and time for dinner. Andy, Mary, Joanna and myself decide to seek food outside  Riocentro. According to Google maps the recommended destination is only 2 miles away  but it is a swimming distance as we are separated by the giant lake. The restaurant is called CT Boucherie and is in the heart of a huge mall which feels very familiar because  it looks like any other mall. When we arrive at 8 the place is nearly empty but the restaurant crew seem to be pleased.

We are ravenous for  real food and for starters we decide to have our meat raw and cold while keeping ourselves warm from a round of caipirinhas. Score! The beef carpaccio and two orders of steak tartare are delightful and we start planning the main assault. The menu is a simple list of items and it provides a combinatorial map of a cow that gives us a chance to establish a bijection and learn some Portuguese in the process. Eventually three of us settle for slabs of undercooked meat while Joanna chooses grilled octopus. The food arrives some time later, precisely what  have we ordered — large plates with the requested items. It is a bit puzzling but the place vas recommended by Akshay and Sara Venkatesh so we suspect that this is just the combinatorial part from the map that we decoded, and topology and algebra will soon reveal themselves. Bingo! There is a parade of accoutrements that we can sample and this includes okra, white beans, potatoes, mushrooms and countless other items.  We cope with this onslaught sampling everything in the geometric series pattern, but when 10pm strikes, and the place is packed and in full swing, we surrender and ask for dessert menu.  Creme brulee and ice-cream later we decide to head back.

It is still raining when we get to our hotel but there is a profound change as scales have fallen from our eyes and we realize that we finally arrived in Brazil.

Day 4 and 5

This is the first sunny day since we arrived. Energized after a light  breakfast Team  NSF  heads for Peter Scholze lecture. There is a huge line of people waiting to enter the lecture hall. It is like a food line that I remember from my childhood except that in this case I fear that not many will fully digest their fare. Scholze is being introduced by Michael Rappaport who announces that the talk will be great because he learned the skill from him and I immediately start worrying. However it is not so bad. There is a pattern developing with the talks by Fields medalists, particularly Venkatesh, Okounkov and now Scholze. They speak little about their past work and instead focus on developing and advocating for new directions. These talks discuss mostly  new methods and connections and  use  results as an enticements and teasers. Scholze talk is precisely like that and it sounds a bit like an invitation to follow. The last result that he mentions is a theorem with ten names attached to it. 

After Peter Scholze talk there is another ceremony. Fields medal stolen from Caucher Birkar is being replaced, making him the first mathematician with two Fields medals. It follows the same format — swelling music and gold medal flipping in space followed by announcements from the officials and handshakes. The difference here is that he  gives a very nice thank you speech which puts the whole incident in perspective. “This is hardly the worst that happened to me,” he assures the audience, “and overall pretty minor in a large scheme of things.”

After the lunch break I ran into Birkar in the elevator and congratulate him on the Fields medal and his speech. He is rather shy but smiles mischievously at these complements without saying much. We walk together toward lecture hall chatting about rainy weather and I comment that he must be used to it living in England. He nods and asks me what I do.  I mention NSF.

“DMS?” he queries surprising me a bit, “which program?”

I am profoundly impressed with the depth of his information about our agency. 

“By the way,” he says when we reach the lecture hall, “I work at Tulane University.”

I know that the day when I  loose my marbles is coming but did not expected that it is today. Rapidly sinking in depression I google two mathematicians from Tulane attending the ICM and download their pictures.  I need to do a test. I shove the picture of alt-Birkar in the colleague’s face and ask who  is it. 

“Birkar,” comes back instantly.

It is just a parity check but it lifts my mood tremendously and I hum “we will all go together when we go” as I sit to listen  to the talk is by Alessio Figalli. The subject is Stefan problem which concerns temperature distribution in a homogeneous medium undergoing phase change such as melting ice. He describes ideas and tools that lead to the solution, much of it work of Cafarelli form the 80’s and 90’s on classification of points at the interface boundary. 

Next talk that I attend is by Andras Mathe who speaks on Banach-Tarski paradox and Tarski ’s problem of squaring the circle.  Both problems concern equidecompositions - breaking one shape into disjoint pieces and using isometries to reassemble them (still disjoint) into another. Squaring the circle asks for doing it for a circle and square of the same area. This  was shown to be possible almost 30 years ago and recently Mathe and his collaborators showed that the pieces can be measurable. It is a nice result but eclipsed by two Foundations PIs, Andrew Marks and Spencer Unger, who showed just recently that the pieces could just be Borel.

Next invited  talk that I attend is by Virginia Vassilevska Williams. She is the  daughter of a former colleague. I come a bit late and sit next to a women who looks quite familiar  and snaps scads of pictures as the talk progresses.   Could it be my colleague and the speakers’ mother? After my senior moment I no longer dare to confirm  by poking  her in the side.

The lecture concerns fine-grained complexity, that instead of putting all problems in P together allows to distinguish algorithm requiring O(n^3) steps from those with O(n^2) steps. This is quite interesting because class P is really only a theoretical concept.

My last talk of the day is by Neeraj Kayal from Microsoft who speaks about hard to compute polynomials. As it turns out, most multivariate polynomials are hard to compute and require exponential number of  additions and multiplications. However finding specific examples is elusive. Best known multivariate polynomial is the determinant which in principle requires n! operations but in practice Gaussian eliminations brings it down to low degree polynomial. Permanent, which is like determinant but does not alternate using the sign of permutation, is conjectured to be hard but it is not proved that it is. The goal of the talk is to build a framework for  finding a polynomial of some intermediate hardness.

In the meantime Joanna decides to crash the Math Institute Director’s meeting. It is by invitation only but Team NSF has a vested interest in the event. When she shows up there is nobody to be seen except a lonely attendant checking the credentials. Soon directors of Oberwolfach and Banach Center show up, and start chatting. As it happens none of them were invited and they came drawn by curiosity and a sense of mission. Friendly the attendant proposes that they enter the room and start working.  Can it be that nobody has been invited in a ruse to  identify the leaders and bury the deadweight?

The lectures are over and Team NSF heads to the official banquet that features Gauss and Chern Prizes ceremonies. The amphitheater where we have plenary lectures is converted to a dining hall with probably close to hundred 8-person tables. Team NSF elbows its way to a place with a good visibility of the stage and proximity to the food station. We generously give two vacant seats to orphaned invites speakers: Bjorn Poonen and Pham Tiep, in return for the promises of quality panel service and timely submissions of annual reports.

The celebrations start with a short speech about Brazil by one of the organizers. It emphasizes openness and blending of the cultures sweeping slavery and genocide perpetrated on these lands under the rug. 

We anxiously await medal ceremonies as food and alcohol are embargoed until these formalities are over.

First is David Donoho who receives the Gauss Prize of 10,000 euro plus a gold  medal. The Simons Foundations movie about him is narrated by the medalist himself and is a powerful endorsement of mathematics as a tool to cope with emerging phenomena of modern civilization.

Next is the Chern medal for Masaki Kashiwara who receives half million dollars plus a gold medal with a stipulation that half of the award becomes a donation to an institution of his choice. Kashiwara readily makes the donation to his own institution whose director is already there to cash the check. It looks like a poor choice, at least from political and historical perspective, unless it is not really a choice.

I am surprized to hear that the Leelavati Prize will be given at the closing ceremony.

Finally food stations open up and wine and champagne start trickling in. Trickling? Sadly organizers chose to have a small and inefficient crew of pourers that is no match for our thirst. 

I joke that Jesus turned water into wine in a biggest miracle of pre-science era, but if he decided  to pour it all by himself as well,  it would have been remembered as the biggest rip-off in the entire history of humanity.  So the banquet is not living to its full potential and badly needs a jolt of energy. Alas, organizers have a surprise planned and it is in the  form of  a carnival group playing and dancing samba. Suddenly the temperature in the room goes up, some Fields medalists get restless and  the rest of the audience follows the lead. The dancers are looking great, have outlandish costumes, and I never felt better making a fool of myself! 

And as you see I was not alone.

—————————

Sunday is free from lectures and Team NSF decides on an all-day  tour of Rio de Janeiro covering the essentials. We are supposed to start at 7:10am forcing us to a really early breakfast. It does not quite work as planned but around 9am we are on the way. Our group consists of Team NSF, Team Zentralblatt consisting of the director and two of his subordinates, one of our PIs who is an invited speaker, and a couple of random mathematicians.

Rio de Janeiro is the most scenic city that I have ever seen and after being interned in the barracks of Riocentro it feels like a trip to Disneyland. We visit all iconic sites such as Christ the Redeemer, Sugar Loaf, Copacabana nad Ipanema beaches. The lunch is in a churrascaria where we consume enormous amounts of meat washing it down with caipirinhas.

On top of that it is a sunny and cloudless day, what else can you hope for?

Zentralblatt people tell us that originally ICM was supposed to be in a prime downtown location but eventually the locals were bullied into accepting Riocentro to the dismay of the Mathematical Union. This is quite understandable as mathematicians do not have much negotiating power in such matters, and so I am really puzzled by the choice of Russia as the next venue.  What if the ICM 2022 is held in a renovated gulag and Fields medals are applied by Putin using a branding iron? Impossible? I do not think so. Exaggerated? Perhaps slightly.

Day 6

Another day with constant rain anticipated for another 24 hours. I feel fast approaching melancholy that only a good dollop of exercise can chase away. So I skip the morning talks and go for a run. Other than spotting a wild capybara there is not much to report. On the other hand I am missing a great talk and that earns me a scoldation from some members of Team NSF. 

What did I miss? Well rehearsed tag team of Peter Kronheimer and Tom Mrowka brought a powerful red tool-box of representations and PDEs to identify the unknot. They have shown in the past that Floer homology (which is computationally intractable) does the job and now use this result to  show that combinatorially defined Khovanov homology  does the job as well and allows accessible computational tools. They use  similar methods to formulate a conjecture that would give the first non-computer assisted proof of a Four Color theorem for planar graphs. Great math and super delivery, I am told while nursing a short glympse of wet capybara in my memory.

The last of the Fields medalists’ talks is by Caucher Birkar. The amphitheater is packed and  the talk is good but not  great. He discusses the basics of algebraic geometry for what seems like interminably long time, and then dives into the discussion of the minimal model program. His work represents a significant advance, but the talk underscores the fact that  Chris Hacon could have also been recognized eight years ago. 

Afterward I saunter to the invited lecture by June Huh, a Korean-American mathematician who is a Clay Fellow at Princeton.

He (together with Karim Adiprasito and Eric Katz (CAREER award from Combinatorics)) has proven that sequences of coefficients of characteristic polynomial of matroids are log concave (a finite sequence is log-concave if every entry other than the first and last is bigger than the geometric mean of the adjacent entries).

This talk however is not about this result but instead puts out a more general question of why so many naturally defined finite sequences are log-concave. The answer appears to be the holy grail of enumerative combinatorics, because these sequences are the ghosts of structures satisfying some fundamental and equally ubiquitous laws. These laws are (do not scratch your head, you are reading correctly) Poincare Duality, Hard Lefschetz Theorem and Hodge-Riemann Relation. What follows is the list of examples constituting a body of evidence, with their matroid theorem being one of them. Holy Molly, is what I say to this.

The next lecture is by Josi Balogh and Robert Morris speaking about the hypergraph container method they developed (with Wojciech Samotij) for proving better estimates for the number of objects with forbidden structures. They share the talk, with Morris speaking about the container method and Balogh about applications, in particular to the (3,4) problem, which asks for the estimate of the size of  the collection of subsets of an  n-element set on the plane with no 4 points on the same line, that have no 3 colinear points. 

The best attended talk of the day seems to be the Abel Lecture by Sir Michael Atiyah. He got the Abel prize 15 years ago and now at 89 he was coaxed into delivering  one hour lecture at ICM. For many in the audience he has a god-like stature and this talk cuts him to human size which  is very tough for many of us.

Measured by ordinary standards of mathematical lectures the talk is just awful. The presentation is lucid, logical  and quite entertaining but its mathematical content is negligible. There is one slide that uses latin, Cyrillic and arabic letters to allude to the  existence of deep connections between mathematical and physical constants, particularly higher analogs of the Euler formula, derivation of fine-structure constant and more. Sir Michael hints at having found all these things. All remaining slides are pictures of about 20 famous mathematicians from Pythagoras to Edward Witten. The presentation is somewhat sexist and this is not  because all personae are male but because of the exclusion of Hypathia and Noether who clearly belong in this pantheon.  At the same time the talk shows how arbitrary stereotypes can be. The running joke is that mathematicians with beards are more serious and powerful (Pythagoras, Riemann, Poincare,…) but when Atiyah’s favorites are beardless (Abel, Weyl, Witten,…)  it is ok as well.  Team NSF is quite unhappy and heads for a culinary experience that will shift our attention elsewhere.

Day 7

Tuesday is  wet and cloudy but we get a short break from the rain. The morning has three plenary talks and all of them look very interesting. Overall, there is plenty of  more and more interesting talks and Team NSF is working so hard that we  all start looking forward to the end of ICM 2018.

The first talk of the day is by Alex Lubotzky who speaks about high-dimensional expanders. It is a beautiful talk where combinatorial statements are proved using sophisticated methods from representation theory, group theory and much more.  Graph expanders are well studied and by now we have a number of explicit constructions  including Ramanujan graphs which are expanders with some extreme properties.  Hypergraph expanders are a different story as there are many different (not necessarily equivalent) ways to define them.  It is work in progress with many amazing mathematical connections. The talk is very funny, Lubotzky shows on Google maps Ramanujan Complex in the residential area of Kharagpur and notes that it appeared whole 15 years before its existence was finally proved by mathematicians!

Nalini Anantharaman is a star responsible for recent progress on Quantum Unique Ergodicity Conjecture. She is a very good speaker and gives a survey of the field from the paper of Einstein from 1919 which started the field to the latest result by Diatlov (this year’s CAREER award). 

The last talk before lunch is by Sanjeev Arora. He talks about machine learning with particular emphasis on deep learning (which is an iterative variant). The talk is quite elementary and explains  the theoretical underpinning and key algorithms. I like it quite a bit although I am most impressed with specific examples which illustrate the power of these methods (go, chess, language translation). His talk is aimed at drawing mathematicians into the field. It is a pity that so many subjects including this one could have been mainstream statistics but instead they went up for grabs by others and split off.

For  lunch break we go with Joanna to a nearby shopping center to buy water and we devour a small pizza in the process. I cannot shake the feeling that we are overeating.

The first talk after lunch break is by David Donoho who is the recipient of this year’s Gauss Prize. His work is a great example of applied mathematics of the highest caliber. Year ago at the urging of David Eisenbud and AMS he gave a briefing to the US Congress titled “From blackboard to bedside.” Americans spend about 20 billion dollars a year on tomography and 10 times speedup achieved using compressed sensing that already revolutionized the industry is a great benefit to patients and tremendous improvement of the technology. His Congress briefing had a lot of personal stories, cool pictures and minimal amount of mathematics. I guess that it lasted about 15 minutes and was quite successful. For his one-hour ICM presentation Donoho repeated his Congress speech approximately four times, showing the same few slides in various permutations and scrubbing any evidence of mathematical ideas that might turn a congressman off. Not surprisingly it was less successful as the ICM crowd has somewhat different expectations. 

Afterwards the Team NSF splits up for various  invited lectures as usual. Hugo Dominil-Copin was rumored to be a contender for the Fields medal and he talks about percolation. After historical overview he focusses on recent developments, particularly results concerning various thresholds and phase changes. It is all interesting but lacks a strong highlight and becomes quite technical towards the end.

During the longer coffee break I make an observation that was at the back of my head for a few days now. For a number of  years I was maintaining the database of DMS CAREER recipients and this has given me some awareness to their presence. DMS CAREER awardees seem to be bread and butter among US speakers. When conditioned on being affiliated with   US institution, being in the right career stage  and giving an invited talk, the odds are very high that the person is or was a DMS CAREER PI. Suddenly I feel a bit like Robert Muller  looking for Russian operatives in american political life, CAREER recipients clearly infiltrate the ICM organization and are pulling on many strings. I make a mental note to study this issue better upon coming home.

Peter Keevash is a former PI who recently proved the existence of arbitrary Steiner systems, a long standing open problem (Steiner system S(t,k,n) is an n-element set and a family of its k-element subsets such that every t-element subset is contained in exactly one of them.)  The talk is about partitions of hypergraphs which yields the existence of Steiner systems. The talk is quite good but I depart when he gets into details and run to the lecture by Richard Haydon and Spiros Argyros on the Invariant Subspace Problem. When I enter, Haydon just finishes talking about their construction of a Banach space where every operator is compact plus lambda times identity. Argyros continues saying that such space has invariant subspace property (because every compact operator does) and goes on to discuss a construction (with our PI Pavlos Motakis)  of a reflexive space whose every closed subspace has and invariant subspace property. This is state of art at the moment.

Andy, Joanna and Swatee come back from a lecture by John Pardon, a Waterman recipient. They report that unlike his presentation to  NSF this was quite beautiful, and involved very classical problems, and used long forgotten tools in general topology. He showed that every wild group action on a 3-manifold is a limit of tame actions (question is still open in higher dimensions). Just like for his NSF presentation he was waring shorts but his pressed polo shirt indicated that he is moving up in the world. He is also more aware of his surroundings, and this was confirmed by his recognition of NSF proggies who came up to congratulate him after the talk. 

Team NSF jumps into two taxis and heads for dinner at the churrascaria. As before there are mountains of meat and buffet that rivals the best I have ever seen. When we reach the point when we can eat no more, we are offered banana flambe that tops the meal with another 1000 calories. We barrel out of the restaurant and head home.

Day 8 and the last day

We wake up to see the sun and blue sky. This is so unusual that some members of Team NSF linger in beds and skip the first plenary talk. Slackers!

The first lecture is by Assaf Naor who speaks of embeddings of finite metric spaces and dimension reduction. The starting point is an observation that if we have n vectors in R^k then we can find n vectors in R^l with approximately the same distances between then and l=log(n). The situation changes quite a bit when the initial vectors are from some arbitrary metric space.

Another nice example concerns the situation  when we have n points moving in space and we want to know the average distance between them. In this case we can monitor the distance between 3n/2 pairs of preselected in advance points (linear in n) and know the average with a negligible error. How? The selected pairs represent edges in an expander 3-graph with n-vertices.

We follow with plenary talk by Geordie Williamson who was rumored to be a serious Fields medal contender. The talk is great, and given the subject — representation theory — I would be willing to upgrade to fantastic. Suddenly a long list of things that I was always afraid to ask about is laid out in a clear and lucid manner. He outlines the basic motivation and progresses to describe what looks like solutions to fundamental questions many of which he has  solved himself (with collaborators). We overhear Andrey Okounkov and Yuri Tschinkel fuming about missed opportunity to reward this work with a Fields medal. 

During lunch we wander with Andy outside Riocenter zone but return for Masaki Kashiwara Chern medal presentation. I am expecting the worst and I am stunned when it turns out that I am completely wrong. The talk is very good and the slides are slick and professional. Kashiwara starts by discussing quantum groups, and follows to crystal bases that he discovered by searching for an object corresponding to quantum groups when q=0. 

I recall Pierre Shapira laudation and get irritated. How could they have chosen such a dufus! 

I follow through with invited talks in the logic section. The first is by Maryanthe Malliaris  and it concerns Keisler order, a way to compare theories by relating saturation of ultrapowers of their models. This is something that Malliaris started in her thesis work and have subsequently worked on with Shelah solving a number of outstanding problems along the way. The talk is very nice and the only weir thing is that it was delivered in  near darkness!

Next talk is by Steve Jackson and it concerns Axiom of Determinancy. It is a nice lecture but a wide range of topics makes it quite unfocussed.

Last talk in this section is by Ulrich Kohlenbach and it concerns proof-mining, technique that he developed in early 2000’s. It is a way to extract explicit bounds from seemingly non-constructive proofs. There is a lot of interest in this topic. Even Terry Tao joined the effort end used these methods for proving some effective Ergodic Theorem estimates and reinventing the logic techniques in the process.

I am slowly fading after whirlwind of talks but I head for the Ali Nesin’s Leelavati Prize presentation to the amphitheater. I have met Nesin in 1985 in Berkeley and took over his model theory class when he returned to Turkey and promptly got arrested for refusing military service. Now he is being recognized for creation of the Nesin Mathematical Village and I am very proud. The presentation starts with a movie about the village and Nesin and the focus is on tremendous difficulties that the creators, Ali Nesin and Sevan Nisanyan have endured. Next both of them give speeches but the presentation seem to be falling apart a little. It turns out that their computers were lost and it is a bit of an improvisation. Still their undertaking is quite amazing. In little over 10 years of its existence the village hosted over 60 thousand kids who spent time studying math in a unique and stimulating environment. The Village survived blockades, curfews, court orders requesting closure and arrests of the leaders. In fact Nisanyan spent 3 years in jail, escaped and now lives in exile on a Greek island just off the Turkish coast.

Swatee and Warren have already departed and the Team NSF heads to Pipo restaurant, an establishment recommended by locals for contemporary Brazilian cousine. We reach the target after interminably long taxi ride but we are not disappointed. With help of Google translator we decipher the menu and order. At last the quantity of food is within our abilities. We return dead tired and go to sleep.

The following morning is sunny again. We plan to attend only the first talk and pack for the long planned field excursion. We are heading to the heart of the Amazon jungle and thoughts of piranhas, sloths, and poisonous snakes occupy our minds (we are already immune to most tropical diseases known to humanity.)

The first talk of the Last Day of ICM is by Gil Kalai and very quickly it is clear that we made the right choice. The talk concerns noise stability and noise sensitivity, and starts with the following problem: what is the probability that the results of an  election are incorrect because of random errors in vote counting? 

Kalai segways to quantum computing and soon develops the following paradigm:

1. Quantum supremacy (that is building quantum computer surpassing standard computers) requires quantum correction algorithms.

2. Developing quantum correction algorithms is harder than achieving quantum supremacy.

The consequence of both statement seems to be in contradiction with the existence of a quantum machine of large size and this is precisely what Kalai is saying. 

The evidence is not overwhelming but significant, and next few years will provide the answer. By ICM 2022 or ICM 2026 ,and after spending 10 billion dollars, Kalai promises with a great deal of confidence.

Telenovela

On the last day of the ICM 2018 I become aware of a cultural phenomenon that has a chance of bringing issues facing mathematics and mathematicians to the attention of society at large. As it turns out, the local organizers decided, on  a shoestring budget and with amateur actors, to produce a 7-episode  telenovela that was streaming on youtube each day of the ICM. The finale was posted on the last day and enormous number of downloads attest to the popularity of  the series.  I have not seen any of the episodes yet, but was able to piece the story together after chatting with several  ICM 2018 participants. 

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ICM 2018 Telenovela (Synopsis)

Episode 1 begins with Carmen and Paulo bumping into each one day before the ICM 2018 (International Congress of Mathematicians) starts in Rio de Janeiro in the business center called Riocentro. Carmen is heading for the WM^2 (World Meeting of Women in Mathematics) meeting and Paulo is looking for a place in Riocentro where you can have a decent lunch. Both are proud participants of ICM 2018, which is for the first time in history happening on the southern hemisphere.

The camera shows the scene in slo-mo,  spies on them apologizing to each other,  and  finally zooms in on Carmen to tell her story.

Carmen is a mathematician working in a provincial university somewhere in Sao Paulo. She is battling low self-esteem resulting from the negative perception of female mathematicians in a male dominated profession. Bias, micro-aggression, harassment by peers and students, workload and  performance expectations exceeding what her male colleagues have to deal with,  and crippling pay disparity —- they all contribute to her stress, frustration, and the abandonment syndrome. The camera gives evidence to each and every one of these issues with brief snapshots giving a glimpse of her tortured life. Positives are  illustrated by flashbacks from the past and especially scenes from WM^2 meeting that she has just attended. Viewers see multiple working groups that discuss issues facing female mathematicians across countries and cultures, several technical lectures and joyous camaraderie between over four hundred predominantly female participants.

Episode 2 traces back to the encounter between Carmen and Paulo in front of the hall holding WM^2 meeting and focusses on Paulo.

He is a Brazilian mathematician working at one of the  top American universities. In spite of being at the pinnacle of his profession, he is facing a real threat of deportation under the Trump regime. Heavy handed broadening participation policies force him to be on enormous number of university committees and serve on multiple NSF panels each year. This is all a consequence of being tagged as “hispanic,” a label that he finds offensive given his Portuguese roots. In addition he and his male colleagues live under a constant threat of being accused of sexual harassment, and Paulo worries that steamrolling MeToo campaign can upend his life at any moment. He is a couple of strong papers away from tenure but constant stress severely undercuts his productivity. The episode shows flashbacks to happier times when Paulo in diapers plays in a sandbox or when he is being fed a tapioca omelet by his doting mother.

Episode 3 opens up with the Fields medal ceremony that marks the beginning of ICM 2018. We see Carmen and Paulo entering the large amphitheater through different entrances and searching for a good seat. As if guided by a powerful magnet  they converge on the same space trying to claim it. When they bump into each other again there is an unmistakable flash of recognition but just as viewers wonder what will happen next, the narration takes a detour and  skips to a brief description  of the Fields medal (which is equivalent to a Nobel Prize in mathematics) We see vignettes of some of the 59 men and one women who have received this recognition so far and get a sense of genius, stamina and willpower it takes to get it. Afterwords, we quickly return to our protagonists who apparently did strike a conversation and now perambulate in the dreary vistas of Riocentro, a fitting metaphor for their bleak and precarious lives. During an increasingly passionate discussion Paulo agrees  that the MeToo movement is rooted in real and serious issues that women face in nearly every profession, while Carmen concedes that her excessive workload might be a sign of appreciation for her professionalism rather than a wanton exploitation by her department chair.  When Carmen mentions that she as well as most female mathematicians make less money than her male colleagues, Paulo does not disagree but insists that it is more proper to say that her male colleagues simply make more money. Carmen gets increasingly agitated and proclaims that this is exactly the same thing, while Paulo is clearly unwilling to elaborate on what exactly men do to earn more compensation for the same job. Soon after Carmen finds Paulo “talking to her breasts” and threatens to terminate the budding friendship ending the episode in a cliffhanger. 

Episode 4 continues where we left off and viewers sigh with relief to see Carmen and Paulo still talking to each other.  With growing anxiety Carmen and Paulo circle the confines of Riocentro discussing issues that face men and women of mathematics. Joint collaborations, interactions at the workplace, gender stereotyping, this is just the  tip of an iceberg. There is growing enthusiasm coming from identifying large areas of mutual agreement and understanding, and when she finds him again “speaking to her breasts” there are no bad consequences. Carmen brings up the topic of the “talent gap,” a popular view purporting that women’s ability to do math does not match that of man. This time Paulo is lucky since his life experiences  prove otherwise, and he says so. In elementary school, highs school and at the university he has  encountered women whose mathematical powers he found truly intimidating to the point of  making him want to start looking for a new day-job. Carmen is visibly pleased, but of course Paulo has some skeletons in the closet. A flashback to elementary school shows him screaming at a girl who rather effortlessly beat him in a math competition. It is only later that he persevered, and we see him at some point in time as a teacher’s pet and a math prodigy,  while the girl who bested him at math is sitting in the back row painting her fingernails and not showing much interest in schoolwork. But some things are best left unsaid and eventually Paulo brings up topics dear to many mathematicians such as climate change, pollution, threats to the environment as well as acute shortage of decent blackboard chalk, particularly the Hagoromo variety. Paulo raves against Trump’s government but Carmen quiets him down reminding him of his pending green card application.  

Episode 5 is called “Escape from Riocentro” and starts with our protagonists getting in a taxi and heading towards Copacabana. This episode is  pure entertainment as we watch them working out mathematical problems in a cafe, walking on a beach gesticulating wildly and drawing figures in  the sand. Disneyland like scenery of Rio de Janeiro brings a possibility of a happy ending  in form of intellectual as well as physical intimacy. The  samba scene towards the end of the episode brings powerful imagery fleshing out these possibilities. Everything hangs in balance as the episode ends with  Carmen and Paulo taking an uber back to Riocentro. Another cliffhanger awaits anxious viewers. Will Carmen and Paulo split up and go to their separate rooms? Will they work on a joint paper in the hotel bar? Or will they have a few caipirinhas  and through some serendipitous move Paulo will manage to get to the second base or better?

Episode 6 begins with fado music and viewers see immediately  that there was no  joint paper let alone physical intimacy. Carmen and Paulo walk around Riocentro which now, after a fresh breath of seeing the world outside,  looks positively apocalyptic. A short distance from Riocentro they  come across dilapidated Olympic Village, rusting, abandoned and forgotten after 2016 Summer Olympics. Is this the future of mathematics? is the question that they do not dare to ask. They wander in a restless and desperate manner unable to free themselves from the bondage imposed by the suffocating culture of mathematical community and political forces that are bent on destroying mathematical sciences.  They agree on things that made the past so difficult for them and their peers, but the road forward seems precarious and littered with massive obstacles.  At some point they encounter a wild capybara stranded in a maze of concrete at a badly designed interchange. They wade in the polluted canal to save  the poor creature and this small gesture prolongs  hope for a bright future ahead of them, and possibly a blessing for the entire planet as well. Yet again, telenovela creators bait  viewers whose anxiety reaches boiling point as most issues, including a long overdue joint math paper and physical intimacy,  remain unresolved when the episode ends.

Episode 7 opens up in a hospital with the relentless fado soundtrack signifying total doom. Apparently Carmen and Paulo contracted a fatal disease from  the rescued capybara and now they lay in adjacent beds connected to medical machinery. They are  devastated that their relationship did not progress to the next step when they would write a math paper together. They also seem to undress each other with smoldering looks as they confess eternal love and respect for each other. However, IV drips and various sensors attached to their bodies make physical contact impossible. Furthermore,  their medical condition deteriorates quickly, as well as  the condition of the world around them.  Producers give us unflinching view of these horrors.  We see flashes of Mick Mulvaney clamoring  to eliminate US government funding for science, right-wing sympathizers  in Poland burning  books from the mathematical library of the Polish Academy of Sciences, and wildfires in California being fanned by bloodthirsty liberals who allow rivers to wastefully drain into the Pacific. This is interspersed with boils, rashes and bloating visible on  Carmen and Paulo’s bodies. Eventually the camera zooms back to the ECG machine showing Carmen’s and Paulo’s heartbeats slowing down and flatlining. They expire simultaneously as Amalia Rodrigues blasts the viewers with the most depressing and melancholic fado, and the closing ceremony ending the ICM 2018 commences.

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What can I say?  The ICM 2018 telenovela is the biggest public relations event in the history of mathematics. Multiple times Google servers crashed under the enormous number of simultaneous download requests. Episodes of telenovela were downloaded over billion times beating the number of Gangnam style downloads as well as the size of crowds at Trump’s presidential inauguration.  For one brief moment love story of two mathematicians seeking truth, justice and life free of harassment became a focal point of the entire global internet community enshrining ICM 2018 in history forever.