Topology & Geometry

The Topology & Geometry seminar meets on Tuesdays at 13:00 at room 70 in the Ross Building.
2019 Dec 31

T&G: Dror Bar-Natan (Toronto), Some Feynman Diagrams in Pure Algebra

11:00am to 12:30pm

Location: 

Room 70, Ross Building, Jerusalem, Israel
I will explain how Feynman diagrams arise in purealgebra: how the computation of compositions of maps of a certain naturalclass, from one polynomial ring into another, naturally leads to a certaincomposition operation of quadratics and to Feynman diagrams.

I will also explain, with very little detail, how this is used inthe construction of some very well-behaved poly-time computable knotpolynomials, and then with better detail, why I care about having suchinvariants.

Joint work with Roland van der Veen.

2019 Dec 31

T&G: Albert Schwarz (UC Davis), Geometric approach to quantum theory

1:00pm to 2:30pm

Location: 

Room 209, Manchester Building, Jerusalem
In geometric approach the starting point of the formulation of quantum theory is the cone of states. I'll explain this approach, the algebraic approach , the textbook formulation of quantum mechanics and the relations between these approaches. I'll show how the formulas for probabilities can be obtained from basic principles. I'll talk in more detail about the case when the cone of states is homogeneous and about the relation of homogeneous cones to Jordan algebras and homogeneous complex domains.
2019 Dec 17

T&G: Yanir Rubinstein (UMD/WIS), What is the small angle limit of a teardrop?

1:00pm to 2:30pm

Location: 

Room 209, Manchester Building, Jerusalem
In joint work with K. Zhang we construct some explicit canonical geometries on various classes of complex manifolds, following a general symmetry principle pioneered by Calabi in the 70's. Our focus is to allow edge type singularities (that are the natural higher-dimensional analogues of conical Riemann surfaces studied by Picard and others since the 19th century) and study Gromov-Hausdorff limits as the angle in the cone tends to zero.
2020 Jan 07

T&G: Ben Gammage (Harvard and Miami), Mirror symmetry and Hitchin systems

1:00pm to 2:30pm

Location: 

Room 209, Manchester Building, Jerusalem
Mirror symmetry relates the algebraic and symplectic geometry of spaces which are related by dualizing a Lagrangian torus fibration. From the perspective of representation theory this is particularly interesting, with ties to geometric Langlands duality, in cases where the spaces are hyperkähler, and the Lagrangian tori are actually holomorphic Lagrangian. Such spaces, which arise as moduli spaces of four-dimensional field theories, include character varieties, multiplicative quiver varieties, and the "K-theoretic Coulomb branches" of Braverman-Finkelberg-Nakajima.
2019 Nov 05

T&G: Andrei Caldararu (UW Madison), A survey of categorical enumerative invariants

1:00pm to 2:30pm

Location: 

Room 209, Manchester Building, Jerusalem
I will survey recent progress in defining and computing categorical enumerative invariants, analogues of Gromov-Witten invariants defined directly from a cyclic A_infinity category and a choice of splitting of the Hodge filtration on its periodic cyclic homology. A proposed definition of such invariants appeared in 2005 in work of Costello, but the original approach had technical problems that made computations impossible.
2019 Mar 26

T&G: Vivek Shende (Berkeley), Quantum topology from symplectic geometry

1:00pm to 2:30pm

Location: 

Room 110, Manchester Building, Jerusalem, Israel
The discovery of the Jones polynomial in the early 80's was the beginning of ``quantum topology'': the introduction of various invariants which, in one sense or another, arise from quantum mechanics and quantum field theory. There are many mathematical constructions of these invariants, but they all share the defect of being first defined in terms of a knot diagram, and only subsequently shown by calculation to be independent of the presentation. As a consequence, the geometric meaning has been somewhat opaque.
2019 Mar 12

T&G: John Pardon (Princeton), Structural results in wrapped Floer theory

1:00pm to 2:30pm

Location: 

Room 110, Manchester Building, Jerusalem, Israel
I will discuss results relating different partially wrapped Fukaya categories. These include a K\"unneth formula, a `stop removal' result relating partially wrapped Fukaya categories relative to different stops, and a gluing formula for wrapped Fukaya categories. The techniques also lead to generation results for Weinstein manifolds and for Lefschetz fibrations. The methods are mainly geometric, and the key underlying Floer theoretic fact is an exact triangle in the Fukaya category associated to Lagrangian surgery along a short Reeb chord at infinity.
2019 Mar 19

T&G: Viatcheslav Kharlamov (Strasbourg), Segre indices, Welschinger weights, and an invariant signed count of real lines on real projective hypersurfaces

1:00pm to 2:30pm

Location: 

Room 110, Manchester Building, Jerusalem, Israel
As it was observed a few years ago, there exists a certain signed count of real lines on real projective hypersurfaces of degree 2n+1 and dimension n that, contrary to the honest "cardinal" count, is independent of the choice of a hypersurface, and by this reason provides, as a consequence, a strong lower bound on the honest count. Originally, in this invariant signed count the input of a line was given by its local contribution to the Euler number of an appropriate auxiliary universal vector bundle.
2019 Jan 23

T&G: Sylvain Cappell (NYU), Atiyah-Bott classes and extending representations of fundamental groups of 3-manifolds from part of the boundary

1:00pm to 2:00pm

Location: 

Room 70, Ross Building, Jerusalem, Israel
We consider the problem of extending a representation of the fundamental group of 3-manifolds from part of the boundary surfaces. Applications to links will be discussed. Combining this with some cohomology classes of Atiyah and Bott leads to new multivariable polynomial invariants of 3-manifolds with boundary.
This is joint work with Edward Miller.
No background in 3-dimensional topology will be assumed in this survey and research talk.
2019 Jan 15

T&G: Michael Khanevsky (Technion), Geometry of sets of Hamiltonian isotopic curves in a symplectic surface

2:00pm to 3:30pm

Location: 

Room 209, Manchester Building, Jerusalem, Israel

Given two Hamiltonian isotopic curves in a surface, one would like to tell whether they are "close" or "far apart". A natural way to do that is to consider Hofer's metric which computes mechanical energy needed to deform one curve into the other. However due to lack of tools the large-scale Hofer geometry is only partially understood. On some surfaces (e.g. S^2) literally nothing is known.
2019 Jan 08

T&G: David Treumann (Boston College), The Fargues-Fontaine curve for symplectic geometers -- NOTE special time and location

4:00pm to 5:30pm

Location: 

Room 70, Ross Building, Jerusalem, Israel
I will review homological mirror symmetry for the torus, which describes Lagrangian Floer theory on T^2 in terms of vector bundles on the Tate elliptic curve --- a version of Lekili and Perutz's works "over Z", where t is the Novikov parameter. Then I will describe a modified form of this story, joint with Lekili, where the Floer theory is altered by a locally constant sheaf of rings on T^2 (an "F-field").
2018 Dec 25

T&G: Or Hershkovits (Stanford), Mean Curvature Flow of Surfaces -- NOTE special time and location

1:00pm to 2:00pm

Location: 

Room 70, Ross Building, Jerusalem, Israel
In the last 35 years, geometric flows have proven to be a powerful tool in geometry and topology. The Mean Curvature Flow is, in many ways, the most natural flow for surfaces in Euclidean space. In this talk, which will assume no prior knowledge, I will illustrate how mean curvature flow could be used to address geometric questions.

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