swissmapgeometrytopology
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swissmapgeometrytopology [2015/01/13 10:50] – [FIRST SWISSMAP GEOMETRY&TOPOLOGY CONFERENCE] g.m | swissmapgeometrytopology [2015/01/16 18:00] – [FIRST SWISSMAP GEOMETRY&TOPOLOGY CONFERENCE] g.m | ||
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Ligne 50: | Ligne 50: | ||
Alina Pavlikova (St. Petersburg, RU); | Alina Pavlikova (St. Petersburg, RU); | ||
Maria Podkopaeva (SwissMAP); | Maria Podkopaeva (SwissMAP); | ||
- | Michael Polyak (Haifa, IL); | ||
Arthur Renaudineau (Paris, FR); | Arthur Renaudineau (Paris, FR); | ||
Christoph Schiessl (ETHZ); | Christoph Schiessl (ETHZ); | ||
Ligne 62: | Ligne 61: | ||
| ^ Monday | | ^ Monday | ||
- | ^09: 30 -- 10: 30| Minicourse\\ Lenny NG | Minicourse\\ Lenny NG | Minicourse\\ Lenny NG | Minicourse\\ Lenny NG | Talk\\ | + | ^09: 30 -- 10: 30| Minicourse\\ Lenny NG | Minicourse\\ Lenny NG | Minicourse\\ Lenny NG | Minicourse\\ Lenny NG | Talk\\ |
^11: 00 -- 12: 00| Minicourse\\ Lothar GÖTTSCHE | ^11: 00 -- 12: 00| Minicourse\\ Lothar GÖTTSCHE | ||
^16: 30 -- 17: 30| Talk\\ Conan LEUNG | Talk\\ Sergey GALKIN | ^16: 30 -- 17: 30| Talk\\ Conan LEUNG | Talk\\ Sergey GALKIN | ||
Ligne 72: | Ligne 71: | ||
In this minicourse, we will explore a method for studying topological knots through the symplectic/ | In this minicourse, we will explore a method for studying topological knots through the symplectic/ | ||
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+ | **Lothar GÖTTSCHE** | ||
+ | (1) We first introduce the Severi degrees, which count nodal curves in linear systems on surfaces. | ||
+ | Then, using relative Hilbert schemes of points, we introduce refined invariants. These are are polynomials in a variable y, which specialize | ||
+ | for sufficiently ample linear systems at y=1 to the Severi degrees and are given by nice generating functions. | ||
+ | (2) We review Welschinger invariants, which count real curves. We review plane tropical curves and the tropical definition of Severi degrees and Welschinger invariants. | ||
+ | Then using tropical geometry we introduce refined Severi degrees, polynomials in a variable y that interpolate between Severi degrees | ||
+ | and Welschinger invariants, which are closely related to the refined invariants from before. | ||
+ | (3) We introduce Floor diagrams, which under certain assumptions encode the relevant combinatorial information about plane tropical curves, and can be used | ||
+ | to compute refined Severi degrees. We relate these to Feynman diagrams for an action of a Heisenberg algebra, and use this to show that | ||
+ | the refined Severi degrees can be computed in terms of this Heisenberg algebra action. | ||
+ | (4) Time permitting we will introduce refined Broccoli invariants and discuss some of their properties. | ||
+ | The Welschinger invariants studied above were totally real, i.e. counting real curves | ||
+ | The Broccoli invariants are one way to count real curves | ||
+ | We give a refined version of these invariants, again replacing numbers by polynomials, | ||
+ | The description and properties are considerably nicer than for the nonrefined version. | ||
+ | \\ | ||
+ | |||
**Anton ALEKSEEV** | **Anton ALEKSEEV** | ||
In this talk, we first review the classical moment map theory including symplectic reduction, convexity properties and Duistermaat-Heckman localization. We then pass to more exotic moment map theories with values in solvable and compact Lie groups. | In this talk, we first review the classical moment map theory including symplectic reduction, convexity properties and Duistermaat-Heckman localization. We then pass to more exotic moment map theories with values in solvable and compact Lie groups. | ||
+ | \\ | ||
+ | |||
+ | **Anna BELIAKOVA** //Trace of the categorified quantum groups// \\ | ||
+ | In this talk I will give a gentle introduction to the categorified quantum groups and show that the trace (or 0th Hochschild homology) of the Khovanov-Lauda 2-category is isomorphic to the current algebra. Then I'll discuss some applications of this fact to link homology theories. | ||
+ | (Coauthors: Zaur Guliyev, Kazuo Habiro, Aaron Lauda, and Ben Webster.) | ||
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**Tobias EKHOLM** | **Tobias EKHOLM** | ||
Knot contact homology is based on transporting phenomena in smooth topology (knots in a 3-manifold) to symplectic geometry (Lagrangian conromals in the cotangent bundle). This is a rather general scheme that can be applied also in other situations. We survey some recent results in that direction about cotangent bundles of high-dimensional homotopy spheres and about knot contact homology in other dimensions and codimensions. As will be clear, the 3-dimensional case has many special features. In particular we explain that it is related to topological string theory in a 3-dimensional Calabi-Yau manifold as well as to Chern-Simons gauge theory. | Knot contact homology is based on transporting phenomena in smooth topology (knots in a 3-manifold) to symplectic geometry (Lagrangian conromals in the cotangent bundle). This is a rather general scheme that can be applied also in other situations. We survey some recent results in that direction about cotangent bundles of high-dimensional homotopy spheres and about knot contact homology in other dimensions and codimensions. As will be clear, the 3-dimensional case has many special features. In particular we explain that it is related to topological string theory in a 3-dimensional Calabi-Yau manifold as well as to Chern-Simons gauge theory. | ||
+ | \\ | ||
+ | |||
+ | **Vladimir FOCK** | ||
+ | Fay's trisecant identity is a quadratic relation satisfied by theta | ||
+ | functions on Jacobians of curves. We will present these relation in | ||
+ | different forms and show that they play a key role in solution of | ||
+ | discrete integrable system. An application to abelianization of local | ||
+ | systems on a Riemann surface will be also discussed. | ||
+ | \\ | ||
+ | |||
+ | **Conan LEUNG** | ||
+ | Abstract | ||
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swissmapgeometrytopology.txt · Dernière modification : 2015/01/20 13:46 de g.m