Research Report 2003
Research Fellows
Among the results of the CMI Fellows and Scholars was "The primes contain arbitrarily long arithmetic progressions" (math.NT/0404188) by Terry Tao and Ben Green. Tao, who holds a position at UCLA, is a Clay Research Fellow and was a recipient of the Clay Research Award in 2003. Ben Green, a fellow of Trinity College, Cambridge, received his Ph.D. under the direction of Tim Gowers in 2002 and is currently a postdoctoral fellow at the Pacific Institute for the Mathematical Sciences. Green and Tao show that for any integer k > 2, there exist infinitely many arithmetic progressions of prime numbers of length k.
Rather than study the structure of the primes directly, Green and Tao chose to cast them as a dense subset of a slightly larger set, namely the almost primes. Using some ingenious arguments heavily inspired by the ergodic theory proof of Szemeredi's theorem (which asserts that any subset of the integers of positive density contains progressions of arbitrary length), they deduce that any subset of a sufficiently pseudorandom set of positive relative density contains progressions of arbitrary length. The result then follows using recent work of Goldston and Yildirim to place the primes inside a pseudorandom set of "almost primes" with positive relative density. It is all the more remarkable that the final argument does not make any direct use of ergodic theory (in particular, it requires neither the axiom of choice nor any analysis on infinite measure spaces).
Daniel Biss
Daniel Biss has worked on decomposing the group of outer automorphisms of the free group of rank n, Out(F_n), and also the smoothing of combinatorical differential (CD) manifolds. His results will appear shortly in:
CD + PL implies smooth. L.M. Anderson, D.K. Biss.
Decomposing Out(F_n). D.K. Biss.
Maria Chudnovsky
Maria Chudnovsky, together with Ken-ichi Kawarabayashi and Paul Seymour, found a polynomial time algorithm for testing whether a graph has an induced subgraph that is a cycle of even length (i.e., an even hole).
Detecting even holes. M. Chudnovsky, K. Kawarabayashi, P. Seymour.
She is also finishing a paper "Non-zero A-paths in graphs with edges labeled by group elements", with J. Geelen, B. Gerards. L. Goddyn, M. Lohman and P. Seymour.
Dennis Gaitsgory
Dennis Gaitsgory was able to generalize his earlier work on the geometric Langlands conjecture, to prove a partial case of de Jong's conjecture (which concerns the finiteness of a continuous representation of the fundamental group of a projective curve). See:
arXiv:math.AG/0402184 On de Jong's conjecture. Dennis Gaitsgory
He also constructed the Uhlenbeck compactifications of the moduli space of G-bundles on a surface in:
arXiv:math.AG/0301176 Uhlenbeck spaces via affine Lie algebras. A.Braverman, M.Finkelberg, D.Gaitsgory
And continued his work with David Kazhdan on:
arXiv:math.RT/0302174 "Representations of algebraic groups over a 2-dimensional local field", D. Gaitsgory, D.Kazhdan
Together with Edward Frenkel he proved a version of the localization theorem:
arXiv:math.AG/0303173 "D-modules on the affine Grassmannian and representations of affine Kac-Moody algebras", E.Frenkel, D.Gaitsgory.
Mircea Mustata
Mircea Mustata used his earlier work with Ein and Lazarsfeld (see below) to compute multiplier ideals of hyperplane arrangements.
arXiv:math.AG/0402232 Multiplier ideals of hyperplane arrangements. Mircea Mustata.
arXiv:math.AG/0308116 Asymptotic invariants of base loci. Lawrence Ein, Robert Lazarsfeld, Mircea Mustata, Michael Nakamaye, Mihnea Popa
The authors define and study some asymptotic invariants associated to base loci of line bundles on smooth projective varieties.
In an attempt to find a "best" parametrization of a given curve or surface Mustata and collaborators introduce "universal rational Parametrizations" for projective toric varieties in:
arXiv:math.AG/0303316 Universal Rational Parametrizations and Toric Varieties. David Cox, Rimvydas Krasauskas, Mircea Mustata
The loci of arcs on a smooth variety defined by order of contact with a fixed subscheme is studied in:
arXiv:math.AG/0303268 Contact loci in arc spaces. Lawrence Ein, Robert Lazarsfeld, Mircea Mustata
A precise version of inversion of adjunction for varieties which are local complete intersections is proven in:
arXiv:math.AG/0301164 Inversion of adjunction for local complete intersection varieties. Lawrence Ein, Mircea Mustata.
Alexei Borodin
Alexei Borodin and Grigori Olshanski study the asymptotics of certain measures on partitions in:
arXiv:math-ph/0305043 Random partitions and the Gamma kernel. Alexei Borodin, Grigori Olshanski.
And compute the dynamical correlation functions of certain Markov chains related to the Plancherel measure in:
arXiv:math-ph/0402064 Continuous time Markov chains related to Plancherel measure (announcement of results). Alexei Borodin, Grigori Olshanski.
Sergei Gukov
Sergei Gukov worked on various topics in theoretical physics related to string theory, including super Yang-Mills theory, the AdS(3)/CFT(2) correspondence, fractional Chern-Simons invariants and three-dimensional quantum gravity. His most recent articles include:
arXiv:hep-th/0404085 Equivalence of twistor prescriptions for super Yang-Mills. Sergei Gukov, Lubos Motl, Andrew Neitzke
arXiv:hep-th/0404023 An Index for 2D field theories with large N=4 superconformal symmetry. Sergei Gukov, Emil Martinec, Gregory Moore, Andrew Strominger.
arXiv:hep-th/0403225 Chern-Simons Gauge Theory and the AdS(3)/CFT(2) Correspondence. Sergei Gukov, Emil Martinec, Gregory Moore, Andrew Strominger.
arXiv:hep-th/0403090. The Search for a Holographic Dual to AdS(3)xS(3)xS(3)xS(1) Authors: Sergei Gukov, Emil Martinec, Gregory Moore, Andrew Strominger
arXiv:hep-th/0312208 Flux Backgrounds in 2D String Theory. Sergei Gukov, Tadashi Takayanagi, Nicolaos Toumbas.
arXiv:hep-th/0310159 [abs, ps, pdf, other] : Title: Heterotic Moduli Stabilization with Fractional Chern-Simons Invariants. Sergei Gukov, Shamit Kachru, Xiao Liu, Liam McAllister.
arXiv:hep-th/0306165. Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial. Sergei Gukov
Elon Lindenstrauss
Elon Lindenstrauss and Klaus Schmidt proved that the restriction of an invariant probability measure under a nonhyperbolic, ergodic and totally irreducible automorphism of a compact connected abelian group, to the leaves of the central foliation is severely restricted in:
arXiv:math.DS/0303121 Invariant sets and measures of nonexpansive group automorphisms. Elon Lindenstrauss, Klaus Schmidt
And in:
arXiv:math.DS/0402165 Rigidity of multiparameter actions. Elon Lindenstrauss
Lindenstrauss reviewed recent developments and applications of the study of the rigidity properties of natural algebraic actions of multidimensional abelian groups as initiated by Hillel Furstenberg.
Igor Rodnianski
Igor Rodnianski worked on a varety of topics in the analysis of PDE's related to Einstein's equations, differential geometry, the Maxwell-Klein-Gordon equation, and the asymptotic stability of N-soliton states of nonlinear Schrodinger equation. His latest works include:
arXiv:math.AP/0312479 Global existence for the Einstein vacuum equations in wave coordinates. Hans Lindblad, Igor Rodnianski.
arXiv:math.AP/0309463 A geometric approach to the Littlewood-Paley theory.Sergiu Klainerman, Igor Rodnianski.
arXiv:math.AP/0309459 Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux. Sergiu Klainerman, Igor Rodnianski.
arXiv:math.AP/0309353 Global regularity for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in high dimensions. Igor Rodnianski, Terence Tao.
arXiv:gr-qc/0309115 A proof of Price's law for the collapse of a self-gravitating scalar field. Mihalis Dafermos, Igor Rodnianski.
arXiv:math.AP/0309114 Asymptotic stability of N-soliton states of NLS. I. Rodnianski, W. Schlag, A. Soffer.
arXiv:math.AP/0309112 Dispersive analysis of the charge transfer models. I. Rodnianski, W. Schlag, A. Soffer.
arXiv:math.AP/0308123 Causal Geometry of Einstein-Vacuum Spacetimes with Finite. Sergiu Klainerman, Igor Rodnianski
Terence Tao
Terry and his collaborators also wrote many other articles, the latest of which include:
arXiv:math.AP/0402130 Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrodinger equation for radial data. Terence Tao. 24 pages.
arXiv:math.AP/0402129 Global well-posedness and scattering for the energy-critical nonlinear Schroedinger equation in R^3. Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka, Terry Tao. 83 pages.
arXiv:math.AP/0312225 A Strichartz inequality for the Schroedinger equation on non-trapping asymptotically conic manifolds. Andrew Hassell, Terence Tao, Jared Wunsch. 43 pages.
arXiv:math.AP/0311227 Instability of the periodic nonlinear Schrodinger equation. Michael Christ (UC Berkeley), James Colliander (U Toronto), Terence Tao (UCLA). 11 pages.
arXiv:math.CA/0311181 Recent progress on the restriction conjecture. Terence Tao. 63 pages.
arXiv:math.AP/0311048 Ill-posedness for nonlinear Schrodinger and wave equations. Michael Christ (UC Berkeley), James Colliander (Toronto), Terence Tao (UCLA). 29 pages.
arXiv:math.CA/0311039 On multilinear oscillatory integrals, nonsingular and singular. Michael Christ (1), Xiaochun Li (2), Terence Tao (2), Christoph Thiele (2) ((1) UC Berkeley (2) UCLA). 24 pages.
arXiv:math.CA/0310367 Bi-parameter paraproducts. Camil Muscalu, Jill Pipher, Terence Tao, Christoph Thiele. 26 pages.
arXiv:math.AP/0309428 On the asymptotic behavior of large radial data for a focusing non-linear Schroedinger equation. Terence Tao. 51 pages.
arXiv:math.AP/0309353 Global regularity for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in high dimensions. Igor Rodnianski, Terence Tao. 49 pages.
arXiv:math.NA/0309285 An Algorithm for Optimal Partitioning of Data on an Interval. Brad Jackson, Jeffrey D. Scargle, David Barnes, Sundararajan Arabhi, Alina Alt, Peter Gioumousis, Elyus Gwin, Paungkaew Sangtrakulcharoen, Linda Tan, Tun Tao Tsai. 3 pages.
arXiv:math.CA/0308286 An uncertainty principle for cyclic groups of prime order. Terence Tao. 6 pages.
arXiv:math.AP/0307289 Global well-posedness of the Benjamin-Ono equation in H^1(R). Terence Tao. 21 pages.
arXiv:math.CO/0306274 A positive proof of the Littlewood-Richardson rule using the octahedron recurrence. Allen Knutson, Terence Tao, Christopher T. Woodward. 15 pages.
arXiv:math.CO/0306134 Fuglede's conjecture is false in 5 and higher dimensions. Terence Tao. 8 pages.
arXiv:math.CA/0303136 Some recent progress on the Restriction conjecture. Terence Tao. 26 pages.
arXiv:math.CO/0301343 A sum-product estimate in finite fields, and applications. Jean Bourgain, Nets Katz, Terence Tao. 29 pages.
arXiv:math.AP/0301260 Global existence and scattering for rough solutions of a nonlinear Schroedinger equation on R^3. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao.
Research Scholars
In 2003 CMI appointed Manindra Agrawal, Jorgen Andersen, Dietmar Bisch, Alexander Braverman, Michael Finkelberg, Kamal Khuri-Makdisi, Nicolas Monod, Pierre van Moerbeke and Shou-Wu Zhang as CMI Scholars.
Jørgen Ellegaard Andersen
Jørgen Ellegaard Andersen worked with Kenji Ueno on:
arXiv:math.QA/0304135 "Abelian conformal field theories and determinant bundles", J. Andersen, K.Ueno (MPS-preprint 2003-5).
They construct the modular functors and topological quantum field theories associated with certain chiral conformal field theories (with the gauge symmetry of a simple affine Lie algebra) as developed by Tsuchiya, Ueno and Yamada, and in:
arXiv:math.DG/0306235 "Geometric construction of modular functors from conformal field theory", J. Andersen, K.Ueno (MPS-preprint 2003-13).
They employ these ideas to give a geometric construct of a modular functor for any simple Lie-algebra and any level. In another direction, Andersen and Soren Kold Hansen investigated "Asymptotics of the quantum invariants of surgeries on the figure 8 knot", J. Andersen, S.K.Hansen, MPS-preprint 2003-28.
Dietmar Bisch
In "Singly generated planar algebras of small dimension, Part II", (Adv. Math. 175 (2003), no. 2, 297-318, Dietmar Bisch continued his work with Vaughn Jones on the classification of subactor planar algebras . They consider planar algebras generated by a single "2-box" and succeed in extending their previous results to obtain a complete classification of all standard invariants in this class whose second higher relative commutant has dimension at most 13.
Alexander Braverman and Michael Finkelberg
Alexander Braverman, Dennis Gaitsgory and Michael Finkelberg wrote a paper on "Uhlenbeck spaces via affine Lie algebras" (arXiv:math.AG/0301176) where they study the algebraic geometry of Uhlenbeck compactifications of moduli spaces of bundles on P^2 and the combinatorics of Kashiwara's crystals for affine Kac-Moody Lie algebras.
Michael Finkelberg also wrote papers on "Equivariant K-Homology of affine Grassmannian and Toda lattice", with Roman Bezrukavnikov and Ivan Mirkovic (math.AG/0306413), and "Cherednik algebras and Hilbert schemes in characteristic p", with Roman Bezrukavnikov, Victor Ginzburg, Pavel Etingof and Vadim Vologodsky (arXiv:math.RT/0312474).
Nicolas Monad
Nicolas Monad worked with I. Mineyev and Y. Shalom on "Ideal bicombings for hyperbolic groups and applications" (arXiv:math.GR/0304278).
They show that several Measure Equivalence and Orbit Equivalence rigidity results established earlier by Monod-Shalom hold for all non-elementary hyperbolic groups and their non-elementary subgroups. They also derive superrigidity results for actions of general irreducible lattices on a large class of hyperbolic metric spaces.
He also wrote a paper with Sorin Popa "On co-amenability for groups and von Neumann algebras" (arXiv:math.GR/0301348), where they show that co-amenability does not pass to subgroups, answering a question asked by Eymard in 1972. They also address co-amenability for von Neumann algebras, describing notably how it relates to the former.
Pierre van Moerbeke
Pierre van Moerbeke wrote two papers with Mark Adler, "Virasoro action on Schur function expansions, skew Young tableaux and random walks", arXiv:math.PR/0309202, and "A PDE for the joint distributions of the Airy Process", (arXiv:math.PR/0302329).
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