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This week I’m at the International Congress of Mathematicians  (ICM) 2014 in Seoul, and thought I would post a quick update from a TCS perspective. See Tim Gowers’s blog for a much more comprehensive account. There are several other TCS folks here, and I hope some would also post their impressions and recommendations as well.
For TCS the big news was of course that Subhash Khot won the Nevalinna award for his work on the Unique Games Conjecture. As Omer mentioned, this is a research topic that I am extremely interested in, and so am very happy about this well deserved choice. Subhash also gave a fantastic talk, which I highly recommend. Like many others, I was also excited to witness the first time a female mathematician, Maryam Mirzakhani,  is awarded the Fields medal and hope we won’t have to wait too long for the first female Nevalinna medalist.

All the plenary talks were videotaped, and I believe that sooner or later they would be available on this website, so I thought I would mention a few talks that TCS folks might want to look at. Every (plenary or section) talk also had an accompanying survey paper, which again I hope would be available online in the not too far future. (Some people, like many of the TCS folks, already posted the papers on the arxiv/eccc etc.., and I hope we will see some more blog posts about it.)

Two talks that I particularly recommend are Emmanuel Candes’s talk on the “Mathematics of sparsity” and Manjul Bhargava’s talk on “Rational points on elliptic and hyperelliptic curves”.

Candes’s talk was an amazing exposition of the power and importance of algorithms. He showed how efficient algorithms can actually make the difference in treating kids with cancer! Specifically, one of the challenges in taking MRI images is that traditionally they take two minutes to make during which the patient cannot make a single breath. You can imagine that this would be dangerous to nearly impossible to achieve for young children. The crucial difference is made by using a sublinear-samples algorithm (i.e. compressed sensing), which allows to recover the images from much fewer samples, reducing the time to about 15 seconds. Another approach of dealing with this issue is to allow the patient to breath but try to algorithmically correct for this movement. Here what they use [as far as I recall] is a decomposition to a low rank plus a sparse matrix which they achieve via a semidefinite program related to the famous Geomans-Williamson max cut algorithm.  Interestingly, the latter question is also related to the well known lower bound question of matrix rigidity, and the parameters they achieve roughly correspond to the best known values for this question– somewhat (extremely) speculatively, one can wonder if perhaps an improved rigidity lower bound would end up being useful for this application..

Hearing Candes’s talk I couldn’t help thinking that some of those advances could perhaps have been made sooner if the TCS community had closer ties to the applied math community, and realized the relevance of concepts such as property testing and tool such as the Geomans-Williamson to these kind of questions. Such missed opportunities are unfortunate for our community and (given the applications) also to society at large, which is another reason you should always try to go to talks in other areas..

Bhargava’s talk just blew me away. I can’t remember when I went to a talk in an area so far from my own and felt that I learned so much. I can’t recommend it enough and of course given the use of elliptic curves in cryptography, its not completely unrelated to TCS. I will not attempt any technical description of the talk [just watch it, or read the accompanying paper] but let me mention a TCS-related theme, which actually seems to appear in the works of some of the other Fields medalists as well.

One example of the “unreasonable effectiveness” of algorithms is that they often capture our notion of “mathematical understanding”. For example, a priori, the fact that the clique problem is NP hard, does not mean that we should not be able to figure out the clique number of a particular graph such as the Cayley graph, but it turns out that this actually a real obstacle to do it. Similarly, in other areas of mathematics, whether it is figuring out the solution of a differential equation, or the number of points on an elliptic curve, a priori the non existence of an algorithm should not preclude us from answering the quesiton, but it often does. (I am deliberately conflating here the notion of non-existence of an algorithm and the notion of the non existence of an efficient algorithm; indeed for any finite problem, there is some trivial brute-force algorithm, but the existence of it does not help at all in achieving mathematical understanding.)

While we have a difficult time determining the clique number of any specific graph, we do have many tools to determine the clique number of a random graph. Such problems are still by no means trivial: e.g., rigorously determining the precise satisfiability threshold of a random 3SAT is still open. Bhargava tackled the problem of trying to determine the number of rational points on a random elliptic curve. In particular he proved that with some nonzero constant probability this number is infinite, and with some nonzero constant probability the number is zero [at least I think so; perhaps its only guaranteed to be finite]. This can be viewed as progress towards the Birch and Swinnerton-Dyer conjecture, which is one of the Clay math problems.

One interpretation of the “natural proofs” barrier is that to make progress on lower bounds, we need to develop more “non constructive” proof techniques, ideally going beyond those that apply only for “random” objects. Perhaps some of these advanced probabilistic tools can still be used in this effort. Also, there have been some “non constructive” results showing that deterministic objects have a certain “pseudo-random” property even in a setting where we don’t have algorithms to certify that a random object has that property. In particular, Bourgain (see this exposition by Rao) showed that a graph somewhat similar to the Payley graph has clique size at most $n^{0.499}$ even though we still don’t have an algorithm for the planted clique problem that can certify that a random graph has clique number $\ll \sqrt{n}$.

Two other plenary talks that I liked at ICM were János Kollár’s talk on “The structure of algebraic varieties” and James Arthur’s talk on “L-functions and automorphic representations”. I can’t say I understood much of the latter, but I am now slightly less terrified of the Langland’s program (though I still wouldn’t like to meet it in a dark alley..). While I also couldn’t follow much of the former, it still gave me a overview of the effort of classifying algebraic varieties, which I could imagine would have possible TCS applications.

I am delighted by the news that Subhash Khot was awarded the Rolf Nevanlinna Prize. I am reminded of a time (many years ago) when Robert Krauthgamer and I were arguing about one of Subhash’s papers if it is more of a Complexity Theory paper or more of an Algorithms paper. While this was a foolish argument then (and even more so now), it reflected our joint excitement by that work.

This is also a good opportunity to recall Boaz’ post on the unique game and other conjectures.

Guest post by Craig Gentry
The 2014 International Congress of Mathematicians (ICM 2014) is coming up in a few days, and (like Boaz said) we have a great collection of speakers in the “Mathematical Aspects of Computer Science” section. As it is the weekend, and I am sure that you are looking for excuses to avoid sunlight and socializing, let me point you to my survey on homomorphic encryption and obfuscation, intriguingly entitled “Computing on the Edge of Chaos: Structure and Randomness in Encrypted Computation“. Also, let this post also serve as a gentle and timely reminder to the other ICM speakers to hype their surveys.
As you read it, I think you will be surprised and delighted by the clarity of the concepts. Sadly (for me), this will not be due to the quality of my exposition (which is notoriously poor). Rather, despite everything you have heard, homomorphic encryption schemes have become embarrassingly simple. A couple of years ago, Boaz and Zvika remarked on this blog that homomorphic encryption schemes “have been simplified enough so that their description can fit, well, in a blog post…”. Since then, they have become even simpler. (As for obfuscation schemes, well, that’s a different story, and my survey keeps to the high-level concepts.) Enjoy!

As Boaz discussed, there is an excellent collection of papers to be presented at the upcoming FOCS in Philadelphia. These would be spread over three of the days of the conference.

Before this though, there will be an exciting day of workshops and tutorials. It is your chance to reach hundreds of people from across the community, and tell them about the latest developments in your exciting area. Organizing a workshop at FOCS takes away a bulk of the administrative work involved, and lets you concentrate on the more enjoyable scientific part. Below is the call for proposals for workhsops an tutorials. Send in your proposals.
http://www.cis.upenn.edu/~sanjeev/focs2014_workshops_call.html

Emanuele Viola started a new series of posts which is related to the research life-stories project. In his words:

“the series “behind the paper” collects snapshots of the generation of papers. For example, did you spend months proving an exciting bound, only to discover it was already known? Or what was the key insight which made everything fit together?
Records of this baffling process are typically expunged from research publications. This is a place for them. The posts will have a technical component.”

This is my last research life-story (at least for now), possibly concluding this project (though you are all very welcomed to share more as long as this blog lives). My main hope was to give legitimacy to all of us to acknowledge and discuss our uncomfortable feelings and the “non-scientific” challenges of our careers. My experience with myself and others is that many of these neuroses are quite universal. And they are not necessarily correlated with success, which sometimes only adds internal pressure. Paraphrasing what Russell Impagliazzo told me the first time we met (years ago): we really are competing with ourselves, and this is a hopeless competition (I’m sure he said it better). As for myself, I feel that I learned how to enjoy our profession much more over the years (mainly through becoming a little less childish with time). Still, at times, I do feel inapt. Such a period is the topic of my last story.

During my last postdoc year, we had our first child. This was a wonderful event that I had been craving for years. But it was also very demanding. My son was colicky and we were inexperienced and mostly alone in the U.S. In addition we had three house moves, one of which was back to Israel (a move that was surprisingly non-smooth). I was very content with putting my young family at the center and I realized that this is a period that will not repeat itself and should be cherished (turns out that with kids, many periods are like that). I also understood that I cannot expect to do too much research at this period. There was nothing concrete I was worried about: I had just landed my dream position at Weizmann, I wasn’t worried about getting tenure, and I already had many results that I was very proud of (including one with Irit Dinur on PCPs that was quite recent). I could allow myself to take it easy, but my ego was not ready for that. With time, internal pressure accumulated. “Is this it? Did my creativity dry up? Is it downhill from now on?”

At the end of that year at Weizmann (with my son being just a bit over a year), I headed with my family to a summer trip to Berkeley (to work with Luca Trevisan and Irit Dinur) and to Cambridge (to Work with Salil Vadhan). I decided to invest all of the effort in problems related to RL vs. L and felt that this is a test for me. If I’ll fail, then I will scale down my expectation of myself. With this shaky (and so very silly) state of mind, I came to a complexity-theory workshop that started the trip. Though my talk about the work with Irit was very well received, I felt quite depressed. It felt like everyone have been doing these wonderful research and only I was idle. I especially remember one of these talks, with a speaker (who I knew to be very nice) that had an over-confident demeanor. Such individuals always put me off, but at this strangely vulnerable state of mind, it was a challenge to keep the tears inside.

The summer continued quite differently. Spending time with wonderful friends (who happen to be brilliant researchers), having a lot of time for vacationing with my family (thanks to Luca’s great life balance), and ending up with a result that exceeded all of my hopes (undirected connectivity in log-space). I remember very vividly the exact moment when two ideas that I had for many years suddenly clicked in an exciting new way. I remember pacing back and forth in our hotel room, going over the proof that then only existed in my mind. Could it be true? Surely not! But where could the bug be hiding? I remember going out to find a store that would have a notepad and pen for me to start writing the proof down and the slowly growing confidence that came from writing it down and every session of verification (Luca, Irit, Salil, …). And most of all, I remember all of the colleagues being happy with me and for me.

I am not sure if there is a lesson to be learned here. Perhaps, don’t believe everything you are feeling. Or at least – if you are neurotic, you are not the only one here.

* title inspired by Aretha .

The accepted papers list for FOCS 2014 is now posted online.

I am always amazed by the depth and breadth of works in the TCS community, and this FOCS is no exception. Whether you are a physicist interested in the possibility of general “area law” governing entanglement between different parts of systems, a geometer interested in Gromov’s topological notion of expanders, an optimization expert interested in the latest and greatest in interior point methods, a game theorist interested in Karlin’s longstanding conjecture on convergence of fictitious play, a complexity theorist interested in the latest efforts to separate the determinant from the permanent, or simply a dog owner or triangle lover, you will find something of interest in the conference. And of course FOCS is not just a collection of unrelated papers. A quantum computing expert would want to check the paper on topological expanders, as similar concepts have arose in the context of topological approaches to quantum error correction. An optimization expert might want to understand the convergence of “fictitious play” which is a very natural algorithm for solving linear programs, and of course since STOC 2014 we all know  that circuit lower bounds are tightly connected to improving the exponents of algorithms for combinatorial problems. This is just a taste and I could have chosen many other such examples, all strengthening Avi Wigderson’s point why we should all go to talks in areas other than our own.

I was also amazed by the effort reviewers and program committee members have put in the selection process. Conference reviewing sometimes get a bad reputation as being superficial. I did not find this to be the case at all. People have invested an amazing amount of work reading the papers, checking proofs, chasing down references, verifying technical points with the authors and other experts, and generally doing the best job they can to have an informed selection process and assemble the best program we can for the TCS community. I am sure we made mistakes, and the final program, as a product of a committee, is not fully consistent with any particular PC member’s taste, including my own. In particular, there were many submissions that some of us personally found novel and interesting, but were not included in the final program. But I do feel good about the process and believe that while some of our decisions may have been wrong, they were not wrong because we were superficial or lazy or cut corners due to the time pressure. Many times during this process I asked the PC members to go above and beyond what is typically expected, and they have more than risen to this challenge, often making heroic efforts to understand very complex (and sometimes not so greatly written) papers, and trying to get to the bottom of any misunderstanding. I am deeply grateful to them all.

Finally, some statistics. We accepted 70 papers, which is about 26% of the 268-273 submissions (depending on whether you count withdrawn ones). Aside from 9 submissions that were judged to be out of scope and received minimal reviewing, on average each submission had 3.3 reviews and 11.7 comments (including both comments by the PC and short comments/opinions by outside experts that were solicited in addition to the reviews.) Of course these numbers varied greatly based on how much attention and investigation we felt each submission needed and there was also extensive discussion on some of the papers during our two long days of the physical PC meeting. Finally, a very personal statistic for me  is that there are about 2800 emails in my “FOCS14″ folder.  As many past chairs told me, the best thing about this job is that you only do it once…