PPT-Duality for Immersed Manifolds

Daniel Dreibelbis University of North Florida USA Umbilic Bracelet Outline Define duals and dual generalizations Describe the singularities of duals of hypersurfaces Define dual sphere bundles and connect their singularities. [40]G.AnandaSwarup.Onembeddedspheresin3-manifolds.Math.Ann.,203:89{102,1973.[41]G.AnandaSwarup.Projectiveplanesinirreducible3-manifolds.Math.Z.,132:305{317,1973.[42]G.AnandaSwarup.Pseudo-isotopiesofS3 a a i = L a two a space. a a a a a a = Z a a a finite sketched I a a a (dr) a T(E p LP Duality YiZhang Yi Outline Outline DualofLPs Dual Strong Duality No gap: primal optimal, dual opitmal Nogap:primalinfeasibledualunbounded No infeasible No gap: primal unbounded, dual infeasible Ifi 4SIDDHARTHAGADGILToconstructnon-orientable3-manifolds,onegluesnon-orientablehandlebodiesofthesamegenusalongtheirboundaries.Afundamentaltheoremassertsthattheseconstructionsgiveall3-manifolds.Theorem2.E Lecture 9. N. Harvey. http://www.math.uwaterloo.ca/~harvey/. TexPoint. fonts used in EMF. . Read the . TexPoint. manual before you delete this box. .: . A. A. A. A. A. A. A. A. A. A. Outline. Complementary Slackness. Sparse Approximations. Winter 2013. Lecture . 1. N. Harvey. TexPoint. fonts used in EMF. . Read the . TexPoint. manual before you delete this box. .: . A. A. A. A. A. A. A. A. A. A. Linear Program. Four-Manifolds Constructed via Plumbing ~ . a ~ k + 1 = + 1 ag _ 1 = _ 1 + 1 F 1 1 (in fact one can reduce to the +t cases RI and RII with F1 = • F o a . = S E 2 L~ L 2 E 2 L 2 as required. requ analysis and its applications on Riemannian manifolds. S. . Hosseini. FSDONA 2011, Germany.. Nonsmooth analysis. However, in many aspects of mathematics such as . control theory . and . matrix analysis, . 1. , . Piotr. DACKO. 2. & . Cengizhan. MURATHAN. 1 . . 1 . Uludağ. University, Art and Science Faculty, Department of Mathematics, Bursa-TURKEY. ,. 2. . Wroclaw. , POLAND . 1. . Preliminaries. The same density.. Different densities.. Not enough information to determine density relation.. Two objects, made of the same material have the same mass and density, but are differently shaped. The objects have the same volume.. Siye Wu (KITP Langlands Duality 8-8-08) S-duality in Vafa-Witten theoryPage 2 Siye Wu (KITP Langlands Duality 8-8-08) S-duality in Vafa-Witten theoryPage 3 Siye Wu (KITP Langlands Duality 8-8-08) S-du Siye Wu KITP Langlands Duality 8-8-08 S-duality in Vafa-Witten theoryPage 2Siye Wu KITP Langlands Duality 8-8-08 S-duality in Vafa-Witten theoryPage 3Siye Wu KITP Langlands Duality 8-8-08 S-duality in HANDBOOKOFKNOTTHEORYEditedbyWilliamMenascoandMorwenThistlethwaite2005ElsevierB.V.Allrightsreserved In1926,Artin[3]describedtheconstructionofcertainknotted2-spheresin.Theintersectionofeachoftheseknotsw – D1-branes in 2d black hole and their dual . Yasuaki. . Hikida. (Keio University). Refs.. [1] YH, V. . Schomerus. (DESY), JHEP0710(2007)064.. [2] YH, V. . Schomerus. (DESY), JHEP0903(2009)095..