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| == Fall 2010 ==
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| The seminar will be held in room B901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm
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| {| cellpadding="8"
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| !align="left" | date
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| !align="left" | speaker
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| !align="left" | title
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| !align="left" | host(s)
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| |-
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| |September 10
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| |[http://www.math.wisc.edu/~oh/ Yong-Geun Oh] (UW Madison)
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| |[[#Yong-Geun Oh (UW Madison)|
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| ''Counting embedded curves in Calabi-Yau threefolds and Gopakumar-Vafa invariants'']]
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| |local
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| |-
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| |September 17
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| |Leva Buhovsky (U of Chicago)
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| |[[#Leva Buhovsky (U of Chicago)|
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| ''On the uniqueness of Hofer's geometry'']]
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| |[http://www.math.wisc.edu/~oh/ Yong-Geun]
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| |-
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| |September 24
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| |[http://sites.google.com/site/polterov/home/ Leonid Polterovich] (Tel Aviv U and U of Chicago)
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| |[[#Leonid Polterovich (Tel Aviv U and U of Chicago)|
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| ''Poisson brackets and symplectic invariants'']]
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| |[http://www.math.wisc.edu/~oh/ Yong-Geun]
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| |-
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| |October 8
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| |[http://www.math.wisc.edu/~stpaul/ Sean Paul] (UW Madison)
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| |[[#Sean Paul (UW Madison)|
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| ''Canonical Kahler metrics and the stability of projective varieties'']]
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| |local
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| |-
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| |October 15
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| |Conan Leung (Chinese U. of Hong Kong)
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| |[[#Conan Leung (Chinese U. of Hong Kong)|
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| ''SYZ mirror symmetry for toric manifolds'']]
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| |Honorary fellow, local
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| |October 22
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| |[http://www.mathi.uni-heidelberg.de/~banagl/ Markus Banagl] (U. Heidelberg)
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| |[[# Markus Banagl (U. Heidelberg)|
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| ''Intersection Space Methods and Their Application to Equivariant Cohomology, String Theory, and Mirror Symmetry'']]
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| |[http://www.math.wisc.edu/~maxim/ Maxim]
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| |October 29
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| |[http://www.math.umn.edu/~zhux0086/ Ke Zhu] (U of Minnesota)
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| |[[#Ke Zhu (U of Minnesota)|
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| ''Thick-thin decomposition of Floer trajectories and adiabatic gluing'']]
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| |[http://www.math.wisc.edu/~oh/ Yong-Geun]
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| |-
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| |November 5
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| |[http://www.math.psu.edu/tabachni/ Sergei Tabachnikov] (Penn State)
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| |[[#Sergei Tabachnikov (Penn State)|
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| ''Algebra, geometry, and dynamics of the pentagram map'']]
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| |[http://www.math.wisc.edu/~maribeff/ Gloria]
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| |November 19
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| |Ma Chit (Chinese U. of Hong Kong)
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| |[[#Ma Chit (Chinese U. of Hong Kong)|
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| ''A growth estimate of lattice points in Gorenstein cones using toric Einstein metrics'']]
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| |Graduate student, local
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| |December 3
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| |[http://www.math.northwestern.edu/~zaslow/ Eric Zaslow] (Northwestern University)
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| |[[#Eric Zaslow (Northwestern University)|
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| ''Ribbon Graphs and Mirror Symmetry'']]
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| |[http://www.math.wisc.edu/~oh/ Yong-Geun and Conan Leung]
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| |-
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| |December 10
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| |Wenxuan Lu (MIT)
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| |[[#Wenxuan Lu (MIT)|
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| ''Instanton Correction, Wall Crossing And Mirror Symmetry Of Hitchin's Moduli
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| Spaces'']]
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| |[http://www.math.wisc.edu/~oh/ Young-Geun and Conan Leung]
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| |-
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| |-
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| |-
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| |}
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| == Spring 2011 == | | == Spring 2011 == |
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| == Abstracts == | | == Abstracts == |
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| ==Fall 2010==
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|
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| ===Yong-Geun Oh (UW Madison)===
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| ''Counting embedded curves in Calabi-Yau threefolds and Gopakumar-Vafa invariants''
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|
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| Gopakumar-Vafa BPS invariant is some integer counting invariant of the cohomology
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| of D-brane moduli spaces in string theory. In relation to the Gromov-Witten theory,
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| it is expected that the invariant would coincide with the `number' of embedded
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| (pseudo)holomorphic curves (Gopakumar-Vafa conjecture). In this talk, we will explain the speaker's recent
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| result that the latter integer invariants can be defined for a generic choice of
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| compatible almost complex structures. We will also discuss the corresponding
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| wall-crossing phenomena and some open questions towards a complete solution to
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| the Gopakumar-Vafa conjecture.
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|
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| ===Leva Buhovsky (U of Chicago)===
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| ''On the uniqueness of Hofer's geometry''
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|
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| In this talk we address the question whether Hofer's metric is unique among the Finsler-type bi-invariant metrics on the group of Hamiltonian diffeomorphisms. The talk is based on a recent joint work with Yaron Ostrover.
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|
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| ===Leonid Polterovich (Tel Aviv U and U of Chicago)===
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| ''Poisson brackets and symplectic invariants''
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|
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| We discuss new invariants associated to collections of closed subsets
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| of a symplectic manifold. These invariants are defined
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| through an elementary variational problem involving Poisson brackets.
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| The proof of non-triviality of these invariants requires methods of modern
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| symplectic topology (Floer theory). We present applications
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| to approximation theory on symplectic manifolds and to Hamiltonian dynamics.
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| The talk is based on a work in progress with Lev Buhovsky and Michael Entov.
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|
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| ===Sean Paul (UW Madison)===
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| ''Canonical Kahler metrics and the stability of projective varieties"
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|
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| I will give a survey of my own work on this problem, the basic reference is:
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| http://arxiv.org/pdf/0811.2548v3
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|
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| ===Conan Leung (Chinese U. of Hong Kong)===
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| ''SYZ mirror symmetry for toric manifolds''
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|
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| ===Markus Banagl (U. Heidelberg)===
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| ''Intersection Space Methods and Their Application to Equivariant Cohomology, String Theory, and Mirror Symmetry.''
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|
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| Using homotopy theoretic methods, we shall associate to certain classes of
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| singular spaces generalized geometric Poincaré complexes called intersection
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| spaces. Their cohomology is generally not isomorphic to intersection
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| cohomology.
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| In this talk, we shall concentrate on the applications of the new
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| cohomology theory to the equivariant real cohomology of isometric actions of
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| torsionfree discrete groups, to type II string theory and D-branes, and to
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| the relation of the new theory to classical intersection cohomology under
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| mirror symmetry.
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|
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| ===Ke Zhu (U of Minnesota)===
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| ''Thick-thin decomposition of Floer trajectories and adiabatic gluing''
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|
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| Let f be a generic Morse function on a symplectic manifold M.
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| For Floer trajectories of Hamiltonian \e f, as \e goes to 0 Oh proved that
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| they converge to “pearl complex” consisiting of J-holomorphic spheres
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| and joining gradient segments of f. The J-holomorphic spheres come from the
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| “thick” part of Floer trajectories and the gradient segments come from
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| the “thin” part. Similar “thick-thin” compactification result has
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| also been obtained by Mundet-Tian in twisted holomorphic map setting. In
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| this talk, we prove the reverse gluing result in the simplest setting: we
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| glue from disk-flow-dsik configurations to nearby Floer trajectories of
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| Hamitonians K_{\e} for sufficiently small \e and also show the
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| surjectivity. (Most part of the Hamiltonian K_{\e} is \ef). We will discuss
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| the application to PSS isomorphism. This is a joint work with Yong-Geun Oh.
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|
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| ===Sergei Tabachnikov (Penn State)===
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| ''Algebra, geometry, and dynamics of the pentagram map''
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|
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| Introduced by R. Schwartz almost 20 years ago, the pentagram map acts on plane n-gons, considered up to projective equivalence, by drawing the diagonals that connect second-nearest vertices and taking the new n-gon formed by their intersections. I shall survey recent work on the pentagram map, in particular, I shall demonstrate that the dynamics of the pentagram map is completely integrable. I shall also explain that the pentagram map is a discretization of the Boussinesq equation, a well known completely integrable partial differential equation. A surprising relation between the spaces of polygons and combinatorial objects called the 2-frieze patterns (generalizing the frieze patterns of Coxeter) will be described. Eight new(?) configuration theorems of projective geometry will be demonstrated. The talk is illustrated by computer animation.
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|
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| ===Ma Chit (Chinese U. of Hong Kong)===
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| ''A growth estimate of lattice points in Gorenstein cones using toric Einstein metrics''
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|
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| Using the existence of Einstein metrics on toric Kahler and Sasaki manifolds, a lower bound estimate on the growth of lattice points is obtained for Gorenstein cones. This talk is based on a joint work with Conan Leung.
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|
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| ===Eric Zaslow (Northwestern University)===
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| ''Ribbon Graphs and Mirror Symmetry''
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|
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| I will define, for each ribbon graph, a dg category,
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| and explain the conjectural relation to mirror symmetry.
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| I will being by reviewing how T-duality relates
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| coherent sheaves on toric varieties to constructible sheaves
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| on a vector space, then use this relation to glue
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| toric varieties together. In one-dimension, the
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| category of sheaves on such gluings has a
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| description in terms of ribbon graphs.
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| These categories are conjecturally
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| related to the Fukaya category of a noncompact
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| hypersurface mirror to the variety with toric
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| components.
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|
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| I will use very basic examples.
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| This work is joint with Nicolo' Sibilla
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| and David Treumann.
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|
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|
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| ===Wenxuan Lu (MIT)===
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| ''Instanton Correction, Wall Crossing And Mirror Symmetry Of Hitchin's Moduli
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| Spaces''
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|
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| We study two instanton correction problems of Hitchin's moduli spaces along with
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| their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli space
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| can be put into an instanton-corrected form according to physicists Gaiotto,
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| Moore and Neitzke. The problem boils down to the construction of a set of
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| special coordinates which can be constructed as Fock-Goncharov coordinates
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| associated with foliations of quadratic differentials on a Riemann surface. A
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| wall crossing formula of Kontsevich and Soibelman arises both as a crucial
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| consistency condition and an effective computational tool. On the other hand
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| Gross and Siebert have succeeded in determining instanton corrections of
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| complex structures of Calabi-Yau varieties in the context of mirror symmetry
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| from a singular affine structure with additional data. We will show that the
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| two instanton correction problems are equivalent in an appropriate sense. This
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| is a nontrivial statement of mirror symmetry of Hitchin's moduli spaces which
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| till now has been mostly studied in the framework of geometric Langlands
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| duality. This result provides examples of Calabi-Yau varieties where the
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| instanton correction (in the sense of mirror symmetry) of metrics and complex
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| structures can be determined.
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|
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| ==Spring 2011==
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| ===Mohammed Abouzaid (Clay Institute & MIT)=== | | ===Mohammed Abouzaid (Clay Institute & MIT)=== |
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| ===Alex Suciu (Northeastern)=== | | ===Alex Suciu (Northeastern)=== |
| ''TBA'' | | ''TBA'' |
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| [[Fall-2010-Geometry-Topology]] | | [[Fall-2010-Geometry-Topology]] |