Colloquia, AY 2011-2012
Thursdays 4:10 pm in 1206 Stevenson Center, unless otherwise noted
Tea at 3:30 pm in 1425 Stevenson Center
September 22, 2011 |
Matilde Marcolli (Caltech) Quantum Statistical Mechanics, L-series and Anabelian Geometry This talk is based on joint work with Gunther Cornelissen. The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical system built from Artin reciprocity. While it is well known that two number fields with the same Dedekind zeta function are not necessarily isomorphic, we show using this quantum statistical mechanics point of view that isomorphism of number fields is the same as the existence of an isomorphism of character groups of the abelianized Galois groups that induces an equality of all corresponding L-series. This is in turn equivalent to the fact that number fields are isomorphic if and only if the associated quantum statistical mechanical systems are isomorphic. This can be seen as another version of Grothendieck’s “anabelian” program, much like the Neukirch-Uchida theorem that characterizes isomorphism of number fields by topological isomorphism of their associated absolute Galois groups. Contact person: |
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October 13, 2011 |
Tom Farrell (SUNY – Binghamton) The space of negatively curved metrics A well known problem in differential geometry is whether the space S(M) of all negatively curved Riemannian metrics on a closed smooth manifold M is always path connected. I will talk on joint work with Pedro Ontaneda motivated by this problem. We showed in particular that if the dimension of M > 9, then S(M) is either empty or has infinitely many path components. Contact person: |
October 20, 2011 |
Michael Anderson (SUNY – Stony Brook) Einstein metrics and minimal surfaces
The first part of the talk will discuss boundary value problems for the Einstein equations on manifolds with boundary. We’ll show, somewhat surprisingly, that the Dirichlet problem is not well-posed for the Einstein equations, and then discuss natural (i.e. geometric) boundary conditions. Global solvability of the Einstein equations with geometric boundary conditions is still a hard open problem. It becomes more tractable, and still interesting, in dimension 3, where Einstein metrics are of constant curvature. This study leads to a new perspective and results on embedded minimal surfaces in S3. Contact person: |
October 27, 2011 |
Antony Wassermann (CNRS, Marseille) Operator algebra aspects of conformal field theory Conformal field theory is a mathematical subject that has grown from the study of both string theory and statistical mechanics. It provides examples of quantum field theories in low dimensions which satisfy all the axioms proposed by physicists in the 1960s. The natural occurrence of the braid group suggested a link with the subfactors discovered by Vaughan Jones in the 1980s. This was made precise through the representation theory of loop groups and their Lie algebras, the Kac-Moody algebras. It turns out that many, but not all, families of interesting subfactors can be constructed using conformal field theory. One natural language for understanding these constructions is that of vertex algebras, due to Richard Borcherds. There is a dictionary from that algebraic theory to the analytic theory of von Neumann algebras. This approach is well adapted to more recent developments in open string theory, in particular M-theory and boundary conformal thery. Another important viewpoint, emphasized by Graeme Segal, is the holomorphic interpretation of conformal field theory, which provides a mathematical way to understand why Riemann surfaces appear in string theory. Applications in geometry have suggested that the von Neumann algebra theory, in particular Alain Connes’ notion of fusion, should arise as a limiting case of Segal’s holomorphic theory. Contact person: |
November 3, 2011 |
Thomas Strohmer (UC Davis) Phase Retrieval, Random Matrices, and Convex Programming Phase retrieval is the problem of reconstructing a function, such as a signal or image, from intensity measurements, typically from the modulus of the diffracted wave. Phase retrieval problems – which arise in numerous areas including X-ray crystallography, astronomy, and diffraction imaging – are notoriously difficult to solve numerically. They also pervade many areas of mathematics, such as numerical analysis, harmonic analysis, algebraic geometry, combinatorics, and differential geometry. In this talk I will introduce PhaseLift, a novel framework for phase retrieval, which comprises tools from optimization, random matrix theory, and compressive sensing. I will prove that for certain types of random measurements a signal or image can be recovered exactly with high probability by solving a convenient semidefinite program – a trace-norm minimization problem, without any assumption about the signal whatsoever and under a mild condition on the number of measurements. Our method is also provably stable vis-a-vis noise. This novel result demonstrates that in some instances, the combinatorial phase retrieval problem can be solved by convex programming techniques. I will then demonstrate how this approach carries over to the classical phase retrieval setting using multiple structured illuminations. I will discuss numerical and practical aspects of our approach, and present some open problems and conjectures. Contact person: |
November 10, 2011 |
Dani Wise (McGill University, Montreal, Canada) Cube complexes CAT(0) cube complexes are high-dimensional generalizations of trees that have emerged as increasingly central objects in infinite group theory. I will indicate their prominent geometric properties and survey some of their appearances within mathematics and then describe how these cube complexes arise from infinite groups. Contact person: |
November 17, 2011 |
Matt Valeriote (McMaster University) The Mathematics of Constraint Satisfaction Many important problems from combinatorics, logic, and computer science can be expressed as instances of the constraint satisfaction problem (CSP). For example instances of graph k-colorability for some k and instances of boolean satisfiability can be regarded as CSPs. While the class of constraint satisfaction problems is known to be NP-complete, there are many natural subclasses of it that are tractable. It is conjectured by Feder and Vardi that in fact any such naturally defined subclass is either tractable or is NP-complete. In my talk I will introduce the class of CSPs and then discuss an approach via algebra to settle the Feder-Vardi Dichotomy Conjecture. Contact person: |
December 8, 2011 |
Alex Eskin (University of Chicago) Rational billiards and the SL(2,R) action on moduli space. I will discuss ergodic theory over the moduli space of compact Riemann surfaces and its applications to the study of polygonal billiard tables. There is an analogy between this subject and the theory of flows on homogeneous spaces; I will talk about some successes and limitations of this viewpoint. This is joint work with Maryam Mirzakhani. Contact person: |
January 19, 2012 |
George Metcalfe (University of Bern) Consequences Logic may be understood as the formal study of consequences: what follows from what and how. Consequences, however, may be characterized in different ways: semantically, via classes of algebraic structures, or syntactically, via proof systems. The aim of this talk is to explore the interplay between these different characterizations and to explain how proof surgery (manipulating derivations in proof systems) can be used to establish important theorems about classes of algebraic structures such as generation results for quasivarieties.
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January 26, 2012 |
Faculty meeting, no colloquium |
February 2, 2012 |
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February 9, 2012 |
Adebisi Agboola (University of California at Santa Barbara) Elliptic curves and the conjecture of Birch and Swinnerton-Dyer The problem of finding integer solutions to Diophantine equations is one that has fascinated mathematicians for thousands of years. Although we now know (thanks to the work of Davis, Matiyasevich, Punam and Robinson resolving Hilbert’s 10th problem in the negative) that it is impossible to do this in general, it ought to be possible to say a great deal in special cases. For example, when the equation in question defines an elliptic curve, a remarkable conjecture due to Birch and Swinnerton-Dyer implies that the behaviour of the solutions is governed by the properties of an analytic object (whose very existence is a deep problem in and of itself), namely the L-function attached to the elliptic curve. In this talk, I shall explain some of the ideas that go into the formulation of the Birch and Swinnerton-Dyer conjecture, and I shall discuss some aspects of what is currently known about the conjecture. I shall not assume any previous knowledge of this topic.
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February 16, 2012 |
Alexandru Oancea (IRMA, University of Strasbourg and IAS, Princeton) Symplectic topology and free loop spaces Symplectic geometry is the study of symplectic manifolds endowed with a closed nondegenerate 2-form. Unlike Riemannian manifolds, symplectic manifolds have no local invariants. Instead, they feature a rich global theory which is commonly referred to as “symplectic topology”. The main idea that I wish to convey in this talk is that spaces of free loops play a fundamental role in symplectic topology. On the one hand, algebraic invariants of symplectic manifolds are often extensions to free loop spaces of invariants familiar from differential topology. On the other hand, this is a manifestation of the close and somewhat mysterious relationship between symplectic rigidity phenomena and dynamical properties of Hamiltonian systems.
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February 23, 2012 |
Peter Sarnak (Princeton) SC 4309 Mobius Randomness and Dynamics The Mobius Function mu(n) is minus one to the number of prime factors of n, if n has no square factors and is zero otherwise. Understanding the randomness (often referred as the Mobius randomness principle) in this function is a fundamental and very difficult problem.We will explain a precise dynamical formulation of this randomness principle and report on recent advances in establishing it and its applications.
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March 1, 2012 |
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March 15, 2012 |
Lewis Bowen (Texas A&M University) Entropy for sofic group actions In 1958, Kolmogorov defined the entropy of a probability measure preserving transformation. Entropy has since been central to the classification theory of measurable dynamics. In the 70s and 80s researchers extended entropy theory to measure preserving actions of amenable groups (Kieffer, Ornstein-Weiss). Recent work generalizes the entropy concept to actions of sofic groups; a class of groups that contains for example, all subgroups of GL(n,C). Applications include the classification of Bernoulli shifts over a free group. This answers a question of Ornstein and Weiss. Contact person: |
March 22, 2012 |
Faculty meeting, no colloquium |
March 29, 2012 |
Van Vu (Yale) Random matrices: Getting inside the bulk! Random matrix theory is a well developed area of mathematics with strong links to various other areas such as mathematical physics, probability, number theory, combinatorics, to mentioned a few. The main question in the field is to get information about eigenvalues inside the bulk of the spectrum. For a long time, it seemed out of reach and was done in very special cases, such as for matrices with gaussian entries. The situation changed substantially in the last few years, due to the works of Erdos et. al. and Tao and the speaker. In this talk, I am going to present a new method developed by Tao and myself. This method is motivated by the Lindenberg replacement method in probability theory and enabled us to get limiting distribution of every single eigenvalue in the spectrum. This gives a new way to attack many long standing problems. For instance, combining our method with other recent results, one can show the universality of the sine-kernel, confirming a famous conjecture of Mehta concerning random hermitian matrices. Contact person: |
April 5, 2012 |
Vladimir Temlyakov (University of South Carolina) Lebesgue type inequalities for greedy approximation
While the ℓ1 minimization technique plays an important role in designing computationally tractable recovery methods in compressed sensing, its complexity is still impractical for many applications. An attractive alternative to the ℓ1 minimization is a family of greedy algorithms. We will discuss some greedy algorithms from the point of view of their theoretical performance. We will discuss Lebesgue type inequalities for greedy algorithms in both Hilbert and Banach spaces. By the Lebesgue type inequality we mean an inequality that provides an upper estimate for the error of a particular method of approximation of f by elements of a special form, say, form A, by the best-possible approximation of f by elements of the form A. Contact person: |
April 12, 2012 |
Ian Sloan (University of New South Wales) Numerical Integration in Unboundedly High Dimensions – Theory and Application
Richard Bellman coined the phrase “the curse of dimensionality” to describe the extraordinarily rapid increase in the difficulty of most problems as the number of variables increases. An example is numerical multiple integration, where the cost of an integration formula of product type obviously rises exponentially with the number of variables. Nevertheless, problems with a large or even an unbounded number of variables are now being tackled successfully with non-product integration formulas. In this talk I will outline recent developments attributable to many people, in which within a decade the focus has turned from existence theorems to concrete constructions of rules that can handle an unbounded number of variables. I will then sketch a recent application to the computation of expected values of an elliptic PDE with a random parameter, perhaps describing the flow of a liquid through a porous material whose permeability is an infinite-dimensional random field. Contact person: |
April 19, 2012 |
Ionut Chifan Rigidity theory in von Neumann algebras By now rigidity phenomena are present in many areas of Mathematics. The general idea is to show that if two mathematical objects are equivalent in some weak sense, which ignores some parts of the structure, then the two objects are isomorphic as objects with their full structure. Over the past decade we have witnessed an intense activity in the study of W*-rigidity for measure preserving group actions on probability spaces. This concept, which is key for the classification of group-measure space von Neumann algebras, has generated a fruitful interplay between operator algebras, functional analysis, group theory, and ergodic theory. In my talk I will explain in detail this notion and its importance, and I will survey some recent exciting developments. The last part of the talk is devoted to some open problems. Contact person: |
Colloquium Chair (2011-2012): Denis Osin
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