Abstracts of Talks
Terry Tao
Public lectures
Structure and randomness in the prime numbers
"God may not play dice with the universe, but something strange is
going on with the prime numbers." - Pual Erdos
The prime numbers are a fascinating blend of both structure (for instance,
almost all primes are odd) and randomness. It is widely believed that
beyond the "obvious" structures in the primes, the primes otherwise behave
as if they were distributed randomly; this "pseudorandomness" then
underlies our belief in many unsolved conjectures about the primes,
from the twin prime conjecture to the Riemann hypothesis. This
pseudorandomness has been frustratingly elusive to actually prove
rigorously, but recently there has been progress in capturing enough
of this pseudorandomness to establish new results about the primes,
such as the fact that they contain arbitrarily long progressions. We
survey some of these developments in this talk.
The cosmic distance ladder
How do we know the distances from the earth to the sun and moon, from the sun to the other planets, and from the sun to other stars and distant galaxies? Clearly we cannot measure these directly. Nevertheless there are many indirect methods of measurement, combined with basic high-school mathematics, which can allow one to get quite convincing and accurate results without the need for advanced technology (for instance, even the ancient Greeks could compute the distances from the earth to the sun and moon to moderate accuracy). These methods rely on climbing a "cosmic distance ladder", using measurements of nearby distances to then deduce estimates on distances slightly further away; we shall discuss several of the rungs in this ladder in this talk.
Mathematical research and the internet
Some personal experiences on how the internet is transforming the way I, and other mathematicians, do research, from such mundane tools as email, home pages, and search engines, to blogs, preprint servers, wikis, and more.
Colloquium talks
Recent progress in additive prime number theory
Additive prime number theory is the study of additive patterns in the primes. We survey some recent advances in this subject, including the results of Goldston, Pintz, and Yildirim on small gaps between primes, the results of Green and myself on arithmetic progressions in the primes, and the results of Bourgain, Gamburd, and Sarnak for detecting almost primes in orbits.
Compressed sensing
Suppose one wants to recover an unknown signal x in R^n from a given vector Ax=b in Rm of linear measurements of the signal x. If the number of measurements m is less than the degrees of freedom n of the signal, then the problem is underdetermined and the solution x is not unique. However, if we also know that x is _sparse_ or _compressible_ with respect to some basis, then it is a remarkable fact that (given some assumptions on the measurement matrix A) we can reconstruct x from the measurements b with high accuracy, and in some cases with perfect accuracy. Furthermore, the algorithm for performing the reconstruction is computationally feasible. This observation underlies the newly developing field of _compressed sensing_. In this talk we will discuss some of the mathematical foundations of this field.
The proof of the Poincaré conjecture
In a series of three terse papers in 2003 and 2004, Grisha Perelman made spectacular advances in the theory of the Ricci flow on 3-manifolds, leading in particular to his celebrated proof of the Poincare conjecture (and most of the proof of the more general geometrization conjecture). Remarkably, while the Poincare conjecture is a purely topological statement, the proof is almost entirely analytic in nature, in particular relying on nonlinear PDE tools together with estimates from Riemannian geometry to establish the result. In this talk we discuss some of the ingredients used in the proof, and sketch a high-level outline of the argument.
Specialist talks
Discrete random matrices
The spectral theory of continuous random matrix models (e.g. real or complex gaussian random matrices) has been well studied, and very precise information on the distribution of eigenvalues and singular values is now known. But many of the results rely quite heavily on the special algebraic properties of the matrix ensemble (e.g. the invariance properties with respect to the orthogonal or unitary group). As such, the results do not easily extend to discrete random matrix models, such as the Bernoulli model of matrices with random +-1 signs as entries. Recently, however, tools from additive combinatorics and elementary linear algebra have been applied to establish several results for such discrete ensembles, such as the circular law for the distribution of eigenvalues, and also explicit asymptotic distributions for the least singular values of such matrices. We survey some of these developments in this talk.
Arithmetic progressions in the primes
A famous and difficult theorem of Szemeredi asserts that every subset of the integers of positive density will contain arbitrarily long arithmetic progressions; this theorem has had four different proofs (graph-theoretic, ergodic, Fourier analytic, and hypergraph-theoretic), each of which has been enormously influential, important, and deep. It had been conjectured for some time that the same result held for the primes (which of course have zero density). I shall discuss recent work with Ben Green obtaining this conjecture, by viewing the primes as a subset of the almost primes (numbers with few prime factors) of positive relative density. The point is that the almost primes are much easier to control than the primes themselves, thanks to sieve theory techniques such as the recent work of Goldston and Yildirim. To "transfer" Szemeredi's theorem to this relative setting requires that one borrow techniques from all four known proofs of Szemeredi's theorem, and especially from the ergodic theory proof.
Wave maps
Wave maps are one of the fundamental geometric wave equations, being on the one hand the dynamic analogue of harmonic maps, and a simplified model for the Einstein equations and gauge field theories such as the Yang-Mills equations on the other. In recent years there has been substantial progress in understanding basic questions such as global regularity and singularity formation for this equation, using new tools such as the induction-on-energy strategy of Bourgain, the concentration-compactness technology of Kenig and Merle, a geometric gauge fixing arising from the harmonic map heat flow, and even some limiting arguments used by Perelman in his proof of the Poincare conjecture. We will survey some of these developments in this talk.
Recent progress on the Kakeya problem
The Kakeya needle problem asks: is it possible to rotate a unit needle in the plane using an arbitrarily small amount of area? The answer is known to be yes, but analogous problems in higher dimensions (where one now seeks to find sets of small dimension that contain line segments in each direction) remain open, and are related to many other important conjectures in harmonic analysis, PDE, and even number theory and computer science. There have been many partial results on this problem, using such diverse techniques as geometric measure theory, incidence combinatorics, additive combinatorics, and PDE; more recently, algebraic geometry, and even algebraic topology have been used to obtain new breakthroughs in this subject. We will discuss many of these new developments in this talk.
Mohammed Abouzaid
Colloquium talks
Understanding hypersurfaces through tropical geometry
Given a polynomial in two (or more variables), one may study the zero locus from the point of view of different mathematical subjects (number theory, algebraic geometry, ...). I will explain how tropical geometry allows to encode all topological aspects by elementary combinatorial objects called "tropical varieties."
Functoriality in Homological Mirror Symmetry
Kontsevich's original version of the Homological Mirror Symmetry Conjecture was a statement about pairs of Calabi-Yau manifolds, with no indication of any connection between mirrors of varieties which are related to each other. I will describe recent progress which reveals situations in which Homological mirror symmetry exhibits more "functorial" properties. This conjectural functoriality is clearest for the case of the inclusion of an anticanonical divisor in a Fano variety. The talk will focus on examples starting in dimension 1, and on explaining the geometric source of these phenomena.
Specialist talks
A mirror construction for hypersurfaces in toric varieties I, II
The Strominger-Yau-Zaslow conjecture gives an intrinsic explanation for Homological Mirror Symmetry in the case of Calabi Yau manifolds. I will explain that by extending the SYZ conjecture beyond the Calabi-Yau case, one may associate a Landau-Ginzburg mirror to generic hypersurfaces in toric varieties. The key idea is to use tropical geometry to reduce the problem to understanding the mirror of hyperplanes.
String topology and the Fukaya category of cotangents bundles
The most interesting version of the Fukaya category of cotangent bundles includes Lagrangians which are allowed to be non-compact. I will explain how this category is essentially equivalent to the category of modules over the based loop space. The "classical" equivalence between symplectic homology and the homology of the based loop space (with the pair of pants product on one side and the Chas-Sullivan product on the other) follows from this story.
A topological model for the Fukaya category of plumbings
The simplest examples of symplectic manifolds beyond cotangent bundles are obtained by plumbings these. I will explain a topological model for the Fukaya categories of these manifolds; the model is given in terms of classical invariants from algebraic topology (the cochain complexes on the skeleta ...).
Danny Calegari
Colloquium talks
Faces of the stable commutator length norm ball
It often happens that a solution of an extremal problem in geometry has more regularity and nicer features than one has an a priori right to expect. I will show how a simple topological problem - when does an immersed curve on a surface bound an immersed subsurface? - is unexpectedly related to linear programming in nonseparable Banach spaces, and gives rise to geometric and dynamical rigidity and discreteness of symplectic representations.
Specialist talks
Surface subgroups from homology
Faces of the scl norm ball
Scl, sails and surgery
Faces of the scl norm ball
Scl, sails and surgery
Scl answers the question: "what is the simplest surface in a given space with prescribed boundary?" where "simplest" is interpreted in topological terms. This topological definition is complemented by several equivalent definitions - in group theory, as a measure of non-commutativity of a group; and in linear programming, as the solution of a certain linear optimization problem. On the topological side, scl is concerned with questions such as computing the genus of a knot, or finding the simplest 4-manifold that bounds a given 3-manifold. On the linear programming side, scl is measured in terms of certain functions called quasimorphisms, which arise from hyperbolic geometry (negative curvature) and symplectic geometry (causal structures). In these talks we will discuss how scl in free and surface groups is connected to such diverse phenomena as the existence of closed surface subgroups in graphs of groups, rigidity and discreteness of symplectic representations, bounding immersed curves on a surface by immersed subsurfaces, and the theory of multi-dimensional continued fractions and Klein polyhedra.
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