Fields Medal Symposium Blog

Thursday October 2 was the fourth and last day of the Fields Medal Symposium. We had two lectures in the morning and two lectures in the afternoon. There was a stronger number-theoretical flavor in the lectures of this final day.

The speakers in the morning session were Yves Benoist and Tamar Ziegler. Benoist discussed equidistribution of orbits of irrational points in the torus. Ziegler gave a detailed account of the techniques involved in the current developments about simultaneous prime values of linear forms.

In the afternoon session the speakers were Hee Oh and Kannan Soundararajan. Oh discussed extensions of Ratner’s measure classification in the case of locally finite measures; this makes a crucial difference in the non-lattice case. The final lecture of the symposium was presented by Soundararajan, who gave a survey of conjectures and known results regarding estimates for central values and average size on the critial line of L-functions on families, and discussed some very new results in this subject.

The 2013 version of the Fields Medal Symposium was a total success. It was a unique oportunity for bringing the work of the Fields medalist Elon Lindenstrauss to the comunity (both specialists and non-specialists) and to get an accurate idea of what are the main problems and newest results in these matters.

If you were not able to attend the symposium, please don’t forget that video records of the lectures are available here. Enjoy.

In Wednesday October 2 activities started at 9:15 am at the Fields Institute. We had three lectures in the morning, by Hillel Furstenberg, Elon Lindenstrauss and Shimon Brooks.

Furstenberg’s lecture focussed on the following conjecture: for almost all x in the unit interval we have that Hdim(A_x)+Hdim(B_x) is at least 1, where A_x is the closure of the orbit of x under multiplication by 2 (modulo the integers) and B_x is the closure of the orbit of x under multiplication by 3 (Hdim is Hausdorff dimension). He showed how this conjecture would follow from another conjecture regarding fractals in the plane.

Lindenstrauss explained the notion of rigidity for orbits and for invariant measures in a dynamical system. Rigidity can happen on the presence of several dynamical conditions (higher rank semi-group actions) or under restrictions on the entropy of the system (maximality, positivity). The lecture covered classical results (such as Furstenberg’s rigidity of orbits, or Ratner’s classification of measures on unipotent dynamics) and explained some recent developments.

Brooks discussed Quantum Unique Ergodicity for quasimodes. This roughly means the following: you have a surface of negative curvature and look at quasimodes — approximate eigenfunctions for the laplacian operator. The question is whether or not these approximate eigenfunctions are equidistributed as the eigenvalue grows. The main theme was to understand to what extent large eigenspaces can be responsible for failure in equidistribution, and it seems that this phenomenon is easier to detect in the context of quasimodes rather than eigenfunctions.

There were no lectures in the afternoon, but there was a very nice field trip in the evening: the symposium banquette, held at the Integral House. Among the appetizers we had some delicious pistachio pizza, salmon crepes and a certain sup which I could not identify (but was definitely great). Our host was James Stewart, the famous mathematician and math textbook writer. It turns out that Stewart is a really charismatic and approachable person — he even gave us a tour through his Integral House!

In Tuesday October 1 the activities started in the morning, at 9:30 at the Fields Institute. We had two lectures in this morning session, the first by Nalini Anantharaman and the second by Jean Bourgain. Both of these lectures were focused on studying the eigenvalues of certain operators.

Anantharaman first recalled a classical theorem about how eigenfunctions of the laplacian distribute on manifolds with ergodic geodesic flow. Using this as inspiration, she described some recent work on a graph-theoretical analogue where the eigenvalues of the adjacency matrix play the role of the spectrum of the laplacian.

Bourgain discussed some progress towards a spectral gap theorem on the compact case. He used diverse techniques such as diophantine approximation, quite different to the classical hyperbolic approach.

After lunch, the afternoon session started at 1:30 pm. There were three lectures, by Masaki Tsukamoto, Manfred Einsiedler and Anatole Katok.

Tsukamoto considered the dynamics occurring on the space of Brody curves (1-Lipshitz holomorphic maps) when the complex numbers act by translations. The goal was to estimate the mean dimension of this action, and indeed, he gave good estimates in terms of more classical invariants.

Einsiedler used methods of ergodic theory (both in the real and p-adic case) to prove equidistribution of integer points in expanding spheres. Actually, he proved much more than this; the pair consisting of an integral vector and its complementary integer lattice gives a point in a suitable parameter space, and these points equidistribute.

Katok discussed the very general problem of completely describing all invariant measures on (reasonable) dynamical systems. Such a general problem is out of reach at present, but if we impose more conditions such as positive entropy, then some progress can be made. Katok also presented some ideas to attack the case of entropy zero.

The Student Event took place at 7:00 pm, at the Fields Institute. It consisted of two lectures followed by a panel discussion. See here for details, or here for the video.

The first lecture was given by Dmitry Jakobson, and the topic was spectral theory in the context of dynamical systems. Using very down-to-earth examples, Jakobson succeeded in making accessible to undergraduate students concepts such as the Quantum Unique Ergodicity problem (indeed, he proved a baby-case during his lecture!).

The second speaker was Ralf Spatzier. This beautiful lecture, of completely elementary nature, made understandable (even to high school students) the notion of a dynamical system and periodic orbits. He first explained that a dynamical system on a finite set always has periodic orbits. Later, after explaining the intermediate value theorem and the notion of continuity, he proved that every discrete-time dynamical system on the interval [0,1] with continuous transition function must have a fixed point. He used these example to illustrate more general principles about existence of fixed points and periodic orbits.

The panel discussion had the following participants (from left to right in the table): Elon Lindenstrauss, Ralf Spatzier, Zach Paikin (moderator), Dmitry Jakobson, Peter Sarnak. The topics discussed covered a variety of questions raised by the moderator, the audience and even some questions sent by twitter to the moderator during the discussion! Although it might not be in the records, the panelists were open to discuss with undergraduate and high school students after the panel. In particular, after the panel discussion, the Fields medalist Elon Lindenstrauss approached a kid sitting in front of me to give him a more detailed explanation about his question on Benford’s Law; this explanation was too long to be given during the panel discussion. It is very inspiring to see how professional mathematicians are so approachable, and are open to share their knowledge even with non-specialists.

As a side note which is completely irrelevant to the Symposium, 1st of October was my 25th birthday. I am not from Toronto, but fortunately I found a friend from Kingston who was at the Fields institute for the thematic program on Calabi-Yau varieties. We went for a beer after the panel discussion, which was a nice opportunity to get some insight of the non-mathematical activities in Toronto.

Here is a survey of the first day at the Fields Medal Symposium 2013.

Activities started in the afternoon at 1:30 pm, at the Fields Institute. Walter Craig (Director of the Fields Institute) gave some brief opening remarks about this scientific activity. He recalled the goals of this symposium: to celebrate the deep contributions of the Fields medalist Elon Lindenstrauss to mathematics, to bring together leading researchers in the fields relevant to Lindenstrauss’s work, and to encourage young generations to get interested in mathematics (specially, during the Student Event).

The speakers of this afternoon were Alex Eskin, Shahar Mozes and Benjamin Weiss.

Eskin took as starting point the problem of counting periodic trajectories of balls in a billiard table up to bounded length (and avoiding repetitions in a suitable way). It turns out that establishing an asymptotic formula for this problem in full generality is complicated issue, and it leads to the study of dynamics in moduli spaces of surfaces with a distinguished regular 1-form.

Mozes discussed new equidistribution results for integral points on expanding bodies. Current developments in ergodic theory are put together in a very elegant way, and the final outcome is a sharp refinement of a classical result of W. Schmidt about integer points in expanding domains.

Weiss discussed entropy as a tool for distinguishing non-isomorphic dynamical systems. Roughly speaking, entropy is a measure of the amount of information per time in a dynamical system. There are several definitions for this concept and in favorable cases these different definitions agree, providing fine invariants for studying the isomorphism problem among dynamical systems.

The evening program started with the Public Opening at 7:00 pm, at the Isabel Bader Theatre (UofT).  Details of the ceremony can be found here. The Public Opening was followed by a generous reception around 9:30, with refreshments and enjoyable discussions.

In this post we describe a precise achievement of Lindenstrauss and his co-authors, to get a more concrete idea of what is his research about. There are several deep contributions which one could try to explain, but it seems that the simplest to formulate is the one related to Littlewood’s conjecture, so let’s focus on this one.

Diophantine approximation.

Diophantus of Alexandria lived in the 3rd century ad, approx. He is best known by his Arithmetica, a collection of 13 books focused on algebraic manipulations and problems about whole numbers. It is now a common practice to use the term Diophantine to refer to any technique which is mainly concerned about integers. For instance, a Diophantine equation is an equation to be solved in the integers (or the rationals, or even number fields depending on the context). Diophantine approximation refers to a part of number theory where the main interest is in studying approximations that involve integers or rational numbers. Here is the most basic, elementary, and often useful result that can be labeled as a Diophantine approximation theorem:

Proposition. Let n be an integer. If n is not zero, then |n| is at least 1.

Approximating irrationals by rational numbers.

The set of rational numbers is dense in the set of real numbers. That is, for every real number x we can find rational approximations as close to x as we want. However, some rational approximations are better than others even if they have more or less the same error. For instance, take x=√2=1.4142135… and look at these two rational approximations:
99/70=1.41428571…
70707/50000=1.41414.
The absolute error of the first one is 0.000072… and the absolute error of the second one is 0.000073... so both approximations are (essentially) equally good in terms of accuracy. However, the first one is much simpler to write!

Are there simple, good approximations to any real number? A classical theorem of Dirichlet gives a positive answer.

Theorem (Dirichlet). Let x be a real irrational number. There are infinitely many rational numbers a/b (with a, b coprime integers, say b>0) such that |x-a/b|<1/b².

Can we get even better approximations? not in general. For instance, it is a fun exercise to show that there is no rational approximation a/b to √2 satisfying |√2-a/b|<1/(3·b²).

Littlewood’s conjecture.

For a real number x, we let (x) be the distance from x to the integers, that is, the minimum of the quantity |x-n| as n ranges over the integers. Of course, (x) is some number between 0 and 1/2. For instance
(√2)=0.4142…
(e)=0.2817…
where e=2.7182... is Euler’s constant. Another way to look at good simple rational approximations of x is to look for small values that appear early in the sequence
(x), (2x), (3x), (4x), …
Indeed, recall that there is some integer a such that |bx -a|=(bx) and hence
|x-a/b|=(bx)/b.
Therefore, we cannot expect (bx) to be significantly smaller than 1/b, as discussed in the previous paragraph. More precisely, the quantity b(bx) can very well be bounded away from 0 as b grows to infinity. For instance, b(b√2)>1/3 for every positive integer b.

Littlewood conjectured that a two variables version should avoid this problem.

Conjecture (Littlewood). For any given real numbers x and y, the sequence n(nx)(ny) takes arbitrarily small values as n grows to infinity.

The breakthrough.

In [EKL], Einsiedler, Katok and Lindenstrauss used techniques of Ergodic theory to produce an outstanding breakthrough towards Littlewood’s conjecture. They proved that the set of pairs (x,y) for which Littlewood’s conjecture fails is really tiny. Recall from last post that there are 3 steps in most applications of Ergodic theory to number theory: translation into dynamics, classification of measures and equidistribution.

The reduction of Littlewood’s conjecture to a dynamical statement is due to Cassels and Swinnerton-Dyer. Basically, to prove the conjecture for a pair of numbers (x,y) one needs to show that the orbit of certain point in an auxiliary dynamical system X is unbounded. To show unboundedness it would suffice to know that the invariant measures on this dynamical system X are of a very specific form. This classification of invariant measures is still an open question. Einsiedler, Katok and Lindenstrauss succeeded in proving enough of this conjectural classification of invariant measures, so that Littlewood’s conjecture follows in most cases.

References:

[EKL], Einsiedler, Katok, Lindenstrauss. Invariant measures and the set of exceptions to Littlewood’s conjecture. Annals of Mathematics (2006).

According to Lindenstrauss’s beautiful expository article [Lin], a common application of Ergodic Theory for solving a number theoretical problem involves three steps:

Step 1. Translate your number theoretical problem into a problem about dynamical systems.

Step 2. Classify the invariant measures for these dynamical systems.

Step 3. Use this measure classification to deduce a uniform distribution result, which hopefully implies a solution to your number theoretical problem.

If your number theoretical problem already asks for some uniform distribution result, then it is very natural (although not necessarily easy!) to follow this approach (see for instance the proof in [Fur] of the uniform distribution of fractional parts of n²x when x is irrational). The goal of this post is to work out a number theoretical problem which has no apparent relation with uniform distribution, giving an “ergodic proof” along the steps outlined above. Additionally, we will try to make most of the discussion accessible to advanced high school students.

The problem.

We denote by Z/nZ the set of residue classes modulo n. This set has addition and multiplication operations inherited from the corresponding operations in the set of integers Z. For instance, when n=5 the elements of Z/5Z can be represented as 0,1,2,3,4. Inside Z/5Z we have the equality 3+3=1 (or, if the reader prefers to talk about congruences in Z, we have 3+3 congruent to 1 modulo 5).

This is the problem that we will consider.

Proposition 1. Let p>2 be an odd prime. Let b,c be elements of Z/pZ and consider the function f: Z/pZ → Z/pZ given by f(x)= x² + bx +c. Suppose that all the values f(0), f(1), …, f(p-1) are squares in Z/pZ. Then actually the function f(x) is a square, that is, there is v in Z/pZ such that f(x)=(x+v)².

In other words, the only reason for f(x) to take square values always is just the obvious reason, namely, that f(x) itself is a square. The proof that we give below is a very extended version of the proof of Proposition 12.1 in [PPV]. The reader can surely find alternative proofs or refinements; the point here is to give a toy example where the ergodic approach can solve a number theoretical problem which, a priori, has no obvious relation with dynamics.

Before going into the proof let us briefly recall some elementary facts about Z/pZ. If Z/pZ is an old friend of the reader, then she/he can safely skip the next paragraph.

Some facts about Z/pZ.

Let p be a prime. The first interesting fact about Z/pZ is that this set is actually a field, which means that every non-zero element has a multiplicative inverse. For instance, when p=5 the elements of Z/5Z are 0,1,2,3,4. The inverses of 1,2,3,4 are 1,3,2,4 respectively (for example, 2·3=6 which is 1 modulo 5).

In Z/2Z={0,1} every element is a square. When p>2 is an odd prime the situation is different, and exactly (p+1)/2 of the elements are squares (for instance, the squares in Z/5Z are 0,1,4). Here is an indication for the proof: since 0 is a square in Z/pZ we only have to show that there are exactly (p-1)/2 squares among the non-zero elements. Note that x and -x have the same square and deduce that there are at most (p+1)/2 squares in Z/pZ (which is enough for our purposes!). To get the equality, recall that a quadratic equation can have at most 2 solutions over a field.

Now we go into the proof of Proposition 1.

Step 1: Translation into dynamics.

Let p>2 be an odd prime and consider f(x)=x²+bx+c with b,c in Z/pZ. Completing the square, we can write f(x)=(x+v)²+a with v,a in Z/pZ. Under the hypothesis of Proposition 1, we would like to show that a=0, so that f(x)=(x+v)². For doing this, we will attach a “dynamical system” to each possible value of a in Z/pZ.

Given a in Z/pZ define the map  T_a (x)= x+a from Z/pZ to Z/pZ. We think about this function as acting on Z/pZ and moving its elements after we apply it once, twice, and again and again. The function is just a shift on the elements of Z/pZ, and this is precisely the dynamical system that we want to study.

When a=0, the dynamics of T_a is pretty boring: nothing moves, no matter how many times we apply T_a. When a is a non-zero element in Z/pZ there is a completely different dynamical behavior and we want to take advantage of that.

Step 2: Classification of invariant measures.

A measure in a set X is a way of assigning a (non-negative) mass to the subsets of X. For instance, in the interval [0,1] we can use the notion of length to measure (suitable) subsets of X=[0,1]. A probability measure is a measure with total mass 1 (like the natural length in [0,1]).

If T is a map from X to X, we say that a probability measure is invariant under T if the mass of S is the same as the mass of the pre-image of S under T, for all subsets S of X (omitting some technical restrictions on S and T). When T is bijective we can simply require that the measure of S and T(S) is the same; this is the case for our maps T_a.

It is easy to describe what are all the probability measures in Z/pZ: we only need to assign non-negative weights to the p elements of Z/pZ and make sure that all the weights add up to 1. Then, if you what to measure a subset S of Z/pZ you only need to add the weights of the elements that belong to S.

The next result classifies all the invariant probability measures for our functions T_a acting on Z/pZ.

Proposition 2. If a=0, then any probability measure is invariant under T_a. If a is non-zero, then there is a unique invariant probability measure for T_a namely, the measure that assigns weight 1/p to each of the elements of Z/pZ (we call this the average measure).

Proof. The first part is obvious, because T_a(S)=S for all subsets of Z/pZ when a=0. For the second part, first note that the measure that gives mass 1/p to each element is indeed invariant under any shift, hence, under T_a. Conversely, let the positive integer n be the inverse of a modulo p. Then applying n times the function T_a to an element x gives
x+a+a+…+a=x+na=x+1.
Hence, all invariant measures must satisfy that the mass of any element x is the same as the mass of x+1. Therefore, all the elements of Z/pZ must have the same mass, and since the total sum must be 1 we see that each element has mass 1/p.

Step 3: Uniform distribution and conclusion.

A dynamical system with a unique invariant measure (and other technicalities) is called uniquely ergodic. As a general fact, when a dynamics is uniquely ergodic then most (if not all) points have orbits uniformly distributed under the unique invariant measure. That is, the frequency with which a given point visits a subset S under the dynamics is proportional to the mass of S (using the unique invariant measure for this dynamical system).

What Proposition 2 shows is that T_a gives a uniquely ergodic dynamic on Z/pZ exactly when a is a non-zero element of Z/pZ, and the unique invariant measure is the average measure. From this one can deduce uniform distribution of orbits under the average measure, but for our purposes the following corollary is enough.

Proposition 3. If a is a non-zero element of Z/pZ then for every element x of Z/pZ, the sequence x, T_a(x), T_a(T_a(x)), … contains all the elements of Z/pZ.

Proof. Let x in Z/pZ be given. Consider the sequence of probability measures m₁ , m₂ , m₃ , …, m_k , … where m_k is the measure that assigns mass to points t in Z/pZ as follows:

m_k(t)=N/k where N is the number of times that t appears among the first k iterates of x under T_a.

For given t, we claim that the sequence m_k(t) converges to some limit as k grows to infinity. If x visits at most once the point t then this is clear and the limit is 0. Otherwise, if x visits at least twice the point t then we let r be the number of iterates that takes from the first to the second visit. Since the iterates of x will follow the same path again, this periodicity shows that the limit of m_k(t) as k grows exists and it is 1/r. Therefore, the sequence of measures m_k converges to some limit measure. A shifting argument among the repeated compositions of T_a shows that the limit of the probability measures m_k will be invariant under T_a. But T_a has a unique invariant probability measure, namely, the average measure! therefore, the limit of the measures m_k is the average measure. Since the average measure is strictly positive at each point, we see that eventually some m_k will be positive at each point. Looking at the definition of m_k we see that the iterates of x under T_a will visit each point!

Remark: one can prove Proposition 3 directly without talking about invariant measures (the interested reader can do this as an easy exercise). However, the previous proof gives some idea of the general kind of arguments for deducing uniform distribution from classification of measures.

Finally, we can use this valuable information to prove Proposition 1.

Proof of Proposition 1. Recall that we just need to show that, if f(x)= (x+v)²+a has the property that f(0), f(1), …, f(p-1) are squares in Z/pZ, then necessarily a=0 in Z/pZ.
Suppose on the contrary that f(0), f(1), …, f(p-1) are squares in Z/pZ but a is non-zero. Let S be the set of squares in Z/pZ and note that the elements of S are (allowing repetitions):
(0+v)², (1+v)², …, (p-1+v)²
no matter the value of v in Z/pZ. Since f(0), f(1), …, f(p-1) are squares in Z/pZ, and since f(x)=(x+v)²+a, we see that for each t in S, the element t+a also belongs to S. Hence the function T_a(x)=x+a maps S to itself. Pick any element in S, say 1 (which is a square!). Then T_a(1) also is in S, hence T_a(T_a(1)) also is in S, and so forth. We conclude that the whole set of iterates of 1 under T_a is included in S. If a is non-zero Proposition 3 proves that the set of iterates of 1 under T_a is the whole Z/pZ, hence S=Z/pZ and every element of Z/pZ is a square. This is not possible because for p odd there are only (p+1)/2 squares in Z/pZ. This contradiction shows that a=0, and Proposition 1 is proved.

Well, that was a lot of work for proving Proposition 1 but I hope you enjoyed it. I am not so good saying good bye, so here is an exercise instead:

Do we really need to check that f(0), f(1), …,f(p-1) are squares in Z/pZ for concluding that f(x) itself is a square? Use the same type of arguments to get an improved bound. Use any kind of arguments to get the optimal bound.

References:

[Fur] Furstenberg, Recurrence in ergodic theory and combinatorial number theory. Princeton University Press (1981).

[Lin] Lindenstrauss, Some examples how to use measure classification in Number Theory. Equidistribution in number theory, an introduction. Springer (2007)

[PPV] Pasten, Pheidas, Vidaux, A survey on Buchi’s Problem. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 377 (2010)

Welcome to the 2013 version of the Fields Medal Symposium blog! I am Hector Pasten and I will be writing for you about the 2013 Fields Medal Symposium during the next couple of weeks.
This year the symposium will take place at the Fields Institute, Toronto, from September 30 to October 3, in honor of Professor Elon Lindenstrauss who was awarded the Fields Medal in the International Congress of Mathematicians (ICM) held in India, 2010.

To get started, let’s have a quick look at some of main topics related to this year’s event. This information will be expanded in forthcoming posts.

The Fields Medal.

This is the top honor that a young researcher in mathematics can aim for. If you are under 40 years old and have produced revolutionary advances in mathematics then you may qualify for it.
Originally, it started under the name of “International Medal for Outstanding Discoveries in Mathematics” and was first awarded at the ICM held in Norway, 1936, to Lars Ahlfors and Jesse Douglas. It later became known as Fields Medal in memory of John Charles Fields, a Canadian mathematician who established this award (who even designed the medal itself!).

Elon Lindenstrauss.

Professor Lindenstrauss is a mathematician working in Ergodic Theory, Dynamical systems and applications to Number Theory. He was born in 1970, Jerusalem, Israel, and obtained his PhD at the Hebrew University in 1999. His thesis has the title “Entropy properties of dynamical systems”.
Because of his deep contributions to mathematics, he was awarded the Fields Medal at the ICM held in India, 2010, along with Ngo Bao Chau, Stanislav Smirnov and Cedric Villani. More precisely, according to the web-page of the ICM 2010, Professor Lindenstrauss was awarded the 2010 Fields Medal for his results on measure rigidity in ergodic theory, and their applications to number theory.

Ergodic Theory.

If somebody on the street asks you “what is Ergodic Theory about?” here is a short answer which is not too far from the truth: is the theory that explains why if you put 10 litres of white paint and 1 litre of black paint in a container and you shake it, then you can expect to get 11 litres of grey paint.
A main topic in Ergodic theory is understanding (using invariant measures, group actions, test functions, equidistributed sequences, etc.) when is it true that the space average in a dynamical process equals the time average.
In our example of white and black paints, the “space average” is the initial information of having white and black paint in a proportion 10:1, while “time average” corresponds to the final color (grey) that you get after the dynamical process of shaking the container for a while.

Number Theory.

Again, if the same individual on the street asks you “what is Number Theory?”, here is a short answer that might make him happy: it is the theory that investigates the properties of whole numbers and their close relatives (rational numbers, algebraic numbers, polynomials over finite fields, holomorphic functions, modular forms, etc.).
Of course, the trick is what we mean with “close relatives” but let’s not discuss this point here. Instead, let us mention that several areas in Number Theory have been greatly benefited by the developments in Ergodic Theory. This is the case for instance with quadratic forms, Diophantine approximation, Modular forms and special values of L-functions.

Connections between Ergodic Theory and Number Theory.

Here is a simple statement that can give an idea of the connection between Number Theory and Ergodic Theory: if x is an irrational real number, then the sequence x, 2x, 3x, … contains terms that are arbitrarily close to integers. If you think in terms of Ergodic Theory, then you would like to prove this in a dynamical way: the fractional parts of these numbers are uniformly distributed in the interval [0,1] with respect to the Lebesgue measure (in particular, there are fractional parts arbitrarily close to 0, hence the result). Of course, there are other ways to approach this simple problem, but in some cases Ergodic Theory really gives the key for opening doors that remained closed for a long time. For instance, this is the case with the Oppenheim conjecture about quadratic forms, which we hopefully will discuss in a later post.

Stay tuned, and let’s look forward to the 2013 Fields Medal Symposium!