Mohammed Abouzaid

Mohammed Abouzaid (* 1981 ) is a Moroccan-American mathematician who studies symplectic topology and connections to algebraic geometry and differential topology.

Abouzaid received his bachelor's degree from the University of Richmond in 2002 and his master's degree from the University of Chicago in 2004, completing his PhD in 2007 with Paul Seidel . In his dissertation he applied tropical geometry for a new approach to the conjecture of homological mirror symmetry for toric varieties. As a post-doctoral student , he was at the Massachusetts Institute of Technology from 2007 to 2011 , where he was a visiting scholar in 2011/12, and at the same time as a Research Fellow at the Clay Mathematics Institute from 2007 to 2012. From 2012 to 2013 he was Visiting Associate Professor at the Simons Center for Geometry and Physics. He has been an Associate Professor at Columbia University since 2012 .

He deals with symplectic topology . Symplectic manifolds do not have local invariants, but global invariants, which result from studying pseudoholomorphic curves in the manifold. In this context, Abouzaid particularly examined the Fukaya category , a categorization of the number of intersections of Lagrange submanifolds of the symplectic manifold with half dimension (after Kenji Fukaya , see Floer homology ). He also examined their applications to embedding Lagrangian submanifolds and mirror symmetry .

With Ivan Smith, he proved in 2015 the equivalence of Khovanov homology (after Mikhail Khovanov ) with a link invariant from symplectic geometry, which Paul Seidel and Ivan Smith found in 2004.

In 2007 he became a Clay Research Fellow. In 2015 he and others received a Simons Collaboration Grant for research on homological mirror symmetry. For 2017 he received the New Horizon Prize in Mathematics . In 2014 he was invited speaker at the International Congress of Mathematicians in Seoul ( Family Floer cohomology and mirror symmetry ). He is a fellow of the American Mathematical Society .

He is a citizen of Morocco and the United States.

Fonts

• with Ivan Smith : Khovanov Homology from Floer cohomology , Arxiv, 2015
• with P. Seidel: Altering symplectic manifolds by homologous recombination , Arxiv, 2010
• with Denis Auroux , Ludmil Katzarkov : Lagrangian fibrations on blow ups of toric manifolds and mirror symmetry for hypersurfaces, Pub. Math. IHES, Volume 123, 2016, pp. 199-282, Arxiv
• with Denis Auroux, Alexander I. Efimov, Ludmil Katzarkov, Dmitri Orlov: Homological mirror symmetry for punctured spheres , J. AMS, Volume 26, 2013, pp. 1051-1083, Arxiv
• with Ivan Smith: Exact Lagrangians in plumbings , Geometric and Functional Analysis, Volume 22, 2012, pp. 785-831, Arxiv
• Nearby Lagrangians with vanishing Maslov class are homotopy equivalent , Inventiones Mathematicae, Volume 189, 2012, pp. 251-313, Arxiv
• Framed bordism and Lagrangian embeddings of exotic spheres. Ann. of Math. (2) 175, 2012, no. 1, pp. 71-185. Arxiv
• A cotangent fiber generates the Fukaya category , Advances in Mathematics, Volume 228, 2011, pp. 894–939, Arxiv
• A geometric criterion for generating the Fukaya category , Pub. Math. IHES, Volume 112, 2010, pp. 191-240, Arxiv
• with P. Seidel: An open string analogue of Viterbo functoriality. Geom. Topol. 14 (2010), no. 2, 627-718.
• Morse Homology, Tropical Geometry, and Homological Mirror Symmetry for Toric Varieties , Selecta Math. (NS) 15, 2009, no. 2, pp. 189-270, Arxiv
• Homogeneous coordinate rings and mirror symmetry for toric varieties , Geom. Topol., Volume 10, 2006, 1097-1156, Arxiv
• Family Floer cohomology and mirror symmetry, ICM 2014, Arxiv

Web links

• Homepage
• Website at Claymath
• Videos by Mohammed Abouzaid in the AV portal of the Technical Information Library

Individual evidence

1. ^ Mathematics People ams.org, accessed March 9, 2019
2. ↑ Mohammed Abouzaid in the Mathematics Genealogy Project (English)
3. ↑ Seidel, Smith, A link invariant from the symplectic geometry of nilpotent slices, Arxiv 2004
4. ↑ Communication from Columbia University 2015