Connections in Geometry and Physics 2010 http://pirsa.org/podcast/C10011 Science 2012 http://blogs.law.harvard.edu/tech/rss en-ca Thu, 09 Feb 2012 04:26:46 -0500 sbradwell@perimeterinstitute.ca Thu, 09 Feb 2012 04:26:46 -0500 G 180 pirsa-admin@perimeterinstitute.ca Steve Bradwell's - Podcast Generator I will discuss recent joint work with A. Ionescu and S. Klainerman on the black hole uniqueness problem. A classical result of Hawking (building on earlier work of Carter and Robinson) asserts that any vacuum, stationary black hole exterior region must be isometric to the Kerr exterior, under the restrictive assumption that the space-time metric shouldbe analytic in the entire exterior region. We prove that Hawking's theorem remains valid without the assumption of analyticity, for black hole exteriors which are apriori assumed to be ''close'' to the Kerr exterior solution in a very precise sense. Our method of proof relies on certain geometric Carleman-type estimates for the wave operator. Time permitting, some more recent developments will also be surveyed. Spyros Alexakis http://streamer.perimeterinstitute.ca/mp3/97f471f7-8dc1-46e3-a01b-bf7ba0db35c1.mp3 Science http://streamer.perimeterinstitute.ca/mp3/97f471f7-8dc1-46e3-a01b-bf7ba0db35c1.mp3 Fri, 07 May 2010 08:30:00 -0400 The Petrov classification of the Weyl tensor is an important tool in the study of exact solutions of the Einstein equation in 4d. For example, the Kerr solution was discovered in a study of spacetimes with algebraically special Weyl tensors. Algebraic classification of the Weyl tensor has been extended to higher dimensions. I shall review this classification and describe known families of algebraically special solutions. Recent progress towards obtaining a higher dimensional generalization of the Goldberg-Sachs theorem will be described. Harvey Reall http://streamer.perimeterinstitute.ca/mp3/8f5e0f1d-df92-46b4-8663-7733eb303604.mp3 Science http://streamer.perimeterinstitute.ca/mp3/8f5e0f1d-df92-46b4-8663-7733eb303604.mp3 Fri, 07 May 2010 10:00:00 -0400 There appear to be only two essentially distinct ways to understand intersection numbers on moduli spaces of curves --- via Hurwitz numbers or symplectic volumes. In this talk, we will consider polynomials defined by Norbury which bridge the gap between these two pictures. They appear in the enumeration of lattice points in moduli spaces of curves and it appears that their coefficients store interesting information. We will also describe a connection between these polynomials and the topological recursion defined by Eynard and Orantin. Norman Do http://streamer.perimeterinstitute.ca/mp3/ff73363b-8255-47d2-8915-302345351113.mp3 Science http://streamer.perimeterinstitute.ca/mp3/ff73363b-8255-47d2-8915-302345351113.mp3 Fri, 07 May 2010 11:15:00 -0400 We will discuss the notion of categorical Lie algebra actions, as introduced by Rouquier and Khovanov-Lauda. In particular, we will give examples of categorical Lie algebra actions on derived categories of coherent sheaves. We will show that such categorical Lie algebra actions lead to actions of braid groups. Joel Kamnitzer http://streamer.perimeterinstitute.ca/mp3/a6f4ae7d-ab4e-4f33-b7f2-838646190e8e.mp3 Science http://streamer.perimeterinstitute.ca/mp3/a6f4ae7d-ab4e-4f33-b7f2-838646190e8e.mp3 Fri, 07 May 2010 13:45:00 -0400 The topological recursion of Eynard and Orantin has found many applications in various areas of mathematics. In this talk I will discuss the recursion from the point of view of Hurwitz numbers and local mirror symmetry. I will explain the mathematics underlying the recursion, its relation with the cut-and-join equation, and explore first steps towards proving (and understanding geometrically) the appearance of the recursion in local mirror symmetry. Vincent Bouchard http://streamer.perimeterinstitute.ca/mp3/bd8011a0-43ec-436d-8b1e-8bb0532a1d7d.mp3 Science http://streamer.perimeterinstitute.ca/mp3/bd8011a0-43ec-436d-8b1e-8bb0532a1d7d.mp3 Fri, 07 May 2010 15:15:00 -0400 We introduce a projective hypersurface ''normal form'' for a class of K3 surfaces which generalizes the classical Weierstrass normal form for complex elliptic curves. A geometric two-isogeny relates these K3 surfaces to the Kummer K3 surfaces of principally polarized abelian surfaces, with the normal form coefficients naturally identifying with the Igusa basis of Siegel modular forms of degree two. These results are reinterpreted through the lens of the Kuga-Satake Hodge Conjecture, and seen as a prediction coming from mirror symmetry. Charles Doran http://streamer.perimeterinstitute.ca/mp3/dbafaa70-368b-4421-8edb-63e6bc6c4995.mp3 Science http://streamer.perimeterinstitute.ca/mp3/dbafaa70-368b-4421-8edb-63e6bc6c4995.mp3 Fri, 07 May 2010 16:30:00 -0400 There have been many attempts to define quasilocal mass for a spacelike 2-surface in a spacetime by the Hamilton-Jacobi method. The essential difficulty in this approach is the choice of the background configuration to be subtracted from the physical Hamiltonian. The quasilocal mass should be positive for general surfaces, but on the other hand should be zero for surfaces in the flat spacetime. In this talk, I shall describe how to use isometric embeddings into the Minkowski space to overcome this difficulty and propose a new definition of gauge-independent quasi-local mass that has the desired properties, in addition to other natural requirements for a mass. This talk is based on a joint work with Shing-Tung Yau at Harvard. Mu-Tao Wang http://streamer.perimeterinstitute.ca/mp3/8b244a1b-ce96-4663-aac6-0978a3eda33c.mp3 Science http://streamer.perimeterinstitute.ca/mp3/8b244a1b-ce96-4663-aac6-0978a3eda33c.mp3 Sat, 08 May 2010 08:30:00 -0400 I will talk about the work that I did with Jixiang Fu and Jun Li on the Strominger system and their role in string theory. Shing-Tung Yau http://streamer.perimeterinstitute.ca/mp3/cf177175-894a-42d7-9a6c-d4e1379c3822.mp3 Science http://streamer.perimeterinstitute.ca/mp3/cf177175-894a-42d7-9a6c-d4e1379c3822.mp3 Sat, 08 May 2010 10:00:00 -0400 We discuss recent progress on the rigorous description of the dynamics of the energy concentration sets in the abelian Higgs model. This is joint work with R. Jerrard. Magdalena Czubak http://streamer.perimeterinstitute.ca/mp3/dfb43ffb-83ea-48cd-bcd9-96eaed9320ac.mp3 Science http://streamer.perimeterinstitute.ca/mp3/dfb43ffb-83ea-48cd-bcd9-96eaed9320ac.mp3 Sat, 08 May 2010 11:15:00 -0400 Clifford algebras arose in Dirac's work on the relativistic wave equation in quantum mechanics. Using the Clifford algebra associated to a quadratic form on a finite dimensional vector space, one can reduce the relativistic wave equation, a PDE of order two, to a system of linear PDEs. Similarly, one can use matrix representations of generalized (i.e. higher degree) Clifford algebras to reduce a PDE of higher degree. These generalized Clifford algebras have been the subject of ongoing research since late 1980s. In this talk, we will discuss generalized Clifford algebras, known results about their representations, and results of ongoing work in this direction. Emre Coskun http://streamer.perimeterinstitute.ca/mp3/3496d9d4-9f6c-46e6-b8b8-d0fec6e1e086.mp3 Science http://streamer.perimeterinstitute.ca/mp3/3496d9d4-9f6c-46e6-b8b8-d0fec6e1e086.mp3 Sat, 08 May 2010 13:00:00 -0400 The Hilbert scheme X[n] of n points on variety X parameterizes length n, zero dimensional subschemes of X. When X is a smooth surface, X[n] is also smooth and a beautiful formula for its motive was determined by Gottsche. When X is a threefold, X[n] is in general singular, of the wrong dimension, and reducible. However if X is a smooth Calabi-Yau threefold, X[n] has a canonical virtual motive --- a motification of the degree zero Donaldson-Thomas invariants. We give a formula analogous to Gottsche's for the virtual motive of X[n]. The key computation gives a q-refinement of the classical formula of MacMahon which counts 3D partitions. Jim Bryan http://streamer.perimeterinstitute.ca/mp3/f153a9ca-40f4-4df2-be19-d4b9369c66a4.mp3 Science http://streamer.perimeterinstitute.ca/mp3/f153a9ca-40f4-4df2-be19-d4b9369c66a4.mp3 Sat, 08 May 2010 13:45:00 -0400 The Hilbert scheme X[n] of n points on variety X parameterizes length n, zero dimensional subschemes of X. When X is a smooth surface, X[n] is also smooth and a beautiful formula for its motive was determined by Gottsche. When X is a threefold, X[n] is in general singular, of the wrong dimension, and reducible. However if X is a smooth Calabi-Yau threefold, X[n] has a canonical virtual motive --- a motification of the degree zero Donaldson-Thomas invariants. We give a formula analogous to Gottsche's for the virtual motive of X[n]. The key computation gives a q-refinement of the classical formula of MacMahon which counts 3D partitions. Chiu-Chu Liu http://streamer.perimeterinstitute.ca/mp3/bb4dc078-603f-41ca-aa91-5bc979b9b460.mp3 Science http://streamer.perimeterinstitute.ca/mp3/bb4dc078-603f-41ca-aa91-5bc979b9b460.mp3 Sat, 08 May 2010 15:15:00 -0400 The study of D-branes at singular points of Calabi-Yau threefolds has revealed interesting connections between certain noncommutative algebras and singular algebraic varieties. In many respects, the choice of an appropriate noncommutative algebra is analogous to finding a resolution of singularities of the variety. We will explain this connection in detail, and outline a program for studying such ''noncommutative resolutions'' globally, for compact algebraic (Calabi--Yau) threefolds. David Morrison http://streamer.perimeterinstitute.ca/mp3/71f5a87a-d704-48bf-9461-8425cee6b23f.mp3 Science http://streamer.perimeterinstitute.ca/mp3/71f5a87a-d704-48bf-9461-8425cee6b23f.mp3 Sat, 08 May 2010 16:30:00 -0400 I will give an overview of recent work with Davide Gaiotto and Greg Moore. This work relates the phenomenon of ''wall-crossing'' for BPS states in four-dimensional N=2 theories to a new construction of hyperkahler metrics. These metrics include in particular the metrics on moduli spaces of solutions to Hitchin equations. I will also briefly describe some extensions of this work to incorporate line and surface operators in the N=2 theory (in progress). Andy Neitzke http://streamer.perimeterinstitute.ca/mp3/262d5fd5-c02b-4507-89b4-1135abb5a5cf.mp3 Science http://streamer.perimeterinstitute.ca/mp3/262d5fd5-c02b-4507-89b4-1135abb5a5cf.mp3 Sun, 09 May 2010 08:30:00 -0400 In the past year, motivated by physics, a rich structure has emerged from studying certain contour integrals in Grassmannians. Physical considerations single out a natural meromorphic form in G(k,n) with a cyclic structure. The residues obtained from these contour integrals have been shown to be invariants of a Yangian algebra. These residues also control what happens deep inside collisions of protons taking place at colliders like the Large Hadron Collider or LHC at CERN. Applications of the Global Residue Theorem give rise to relations among residues which ensure important physical properties. Freddy Cachazo http://streamer.perimeterinstitute.ca/mp3/38eaf278-c8d7-4ec7-adc1-c6b18ae52a4c.mp3 Science http://streamer.perimeterinstitute.ca/mp3/38eaf278-c8d7-4ec7-adc1-c6b18ae52a4c.mp3 Sun, 09 May 2010 10:00:00 -0400 I'll give an introduction to twistor-string theory, which is an attempt to reformulate supersymmetric gauge theory in four-dimensional space-time in terms of a certain generalisation of Gromov-Witten theory in twistor space. The resulting theory is closely related to the multi-dimensional residue calculus in G(k,n) (introduced in Cachazo's talk). David Skinner http://streamer.perimeterinstitute.ca/mp3/f117bc56-f6ef-49df-b434-59e898c51183.mp3 Science http://streamer.perimeterinstitute.ca/mp3/f117bc56-f6ef-49df-b434-59e898c51183.mp3 Sun, 09 May 2010 11:15:00 -0400 I will discuss a hybrid between Chern-Simons and Rozansky-Witten models. In particular, Wilson loops in this topological field theory are objects of a quantum deformation of the equivariant derived category of coherent sheaves. Natalia Saulina http://streamer.perimeterinstitute.ca/mp3/590fdc10-fce8-41b1-8d55-88a2a1b6fbe3.mp3 Science http://streamer.perimeterinstitute.ca/mp3/590fdc10-fce8-41b1-8d55-88a2a1b6fbe3.mp3 Sun, 09 May 2010 13:00:00 -0400 We show that the generating function of the equivariant (generalized) Donaldson invariants of ${bf R}^2 X {Sigma}$ is captured by the solution of a thermodynamic Bethe ansatz equation. Based on a joint work with S. Shatashvili. Nikita Nekrasov http://streamer.perimeterinstitute.ca/mp3/7ea7a7e0-bece-4aaf-993f-fac5a6ea3e36.mp3 Science http://streamer.perimeterinstitute.ca/mp3/7ea7a7e0-bece-4aaf-993f-fac5a6ea3e36.mp3 Sun, 09 May 2010 13:45:00 -0400