Spring 06 was the inagurating semester for a new seminar "Geometric Stories". The style of the seminar was designed to be informal with thequestions from the audience encouraged to ensure good understanding of the talks. The scope of the seminar was kept relatively wide (while sticking to a geometric point of view). The goal was to understand sources of different trends in Geometry. The following talks made contributions for achieving this goal:

Apr 11, 2006

Ilia Itenberg (StrasbourgUniversity)
Tropical Welschinger invariants
The Welschinger invariant is designed to bound from below the number of real rational curves which pass through a given generic collection of real points on a real rational surface. In some cases (for example, in the case of toric Del Pezzo surfaces) this invariant can be calculated using Mikhalkin's approach which deals with a corresponding count of tropical curves.
We define a series of relative tropical Welschinger-type invariants of real toric surfaces. In the Del Pezzo case, these invariants can be seen as real tropical analogs of relative Gromov-Witten invariants, and are subject to a recursive formula. As application we obtain new results concerning Welschinger invariants of real toric Del Pezzo surfaces.
(joint work with V. Kharlamov and E. Shustin)

Mar 30, 2006

Julia Viro, (Uppsala University)
Lines and Circles Meeting a Link
We will estimate from below the number of lines meeting given 4 disjoint smooth closed curves in a given cyclic order in the real projective 3-space and in a given linear order in R3. Similarly, we estimate the number of circles meeting in a given cyclic order given 6 disjoint smooth closed curves in R3. The estimations are formulated in terms of linking numbers of the curves and obtained by orienting of the corresponding (0-dimensional) spaces and calculating of their signatures. The calculation is based on a study of a surface swept by projective lines meeting 3 given disjoint smooth closed curves and a surface swept by circles meeting 5 given disjoint smooth closed curves. Higher dimensional generalizations of the results are outlined.

Mar 21, 2006

James Carlson (Clay Mathematical Institute)
New results on the cubic threefold
Clemens and Griffiths showed that the cubic threefold is determined by its Hodge structure. In joint work with Daniel Allcock and Domingo Toledo, we show that a there is a new Hodge theoretic invariant of cubic threefolds which allows one to identify the moduli space with a discrete quotient of the 10-ball, just as the moduli space of cubic curves is identified with a discrete quotient of the unit disk.

Mar 24, 2006

Konstantin Khanin (University of Toronto)
Introduction to Statistical Mechanics (and almost no geometry)
I am planning to explain some of the basic concepts of the equilibrium statistical mechanics: Gibbs states, phase transitions, critical behaviour etc

Mar 16, 2006

Session in common with the Geometry & Models Seminar
Krzysztof Kurdyka (University of Savoie)
An analogue of Lojasiawicz's gradient inequality for maps

Mar 9, 2006

Yan Soibelman (Kansas State University)
Integral affine structures and non-archimedean geometry
I am going to discuss integral affine structures arising in Mirror Symmetry from two points of view. First one is Gromov-Hausdorff theory of collapsing Calabi-Yau manifolds. Second one involves Berkovich theory of non-archimedean analytic spaces. It provides a non-archimedean analog of the Liouville integrability theory. The relationship to tropical geometry will be also discussed.

Mar 3, 2006

Kentaro Hori (University of Toronto)
String theory and mathematics
I plan to show how string theory relates various fields in mathematics, including symplectic geometry, algebraic geometry, commutative algebra, and real algebraic geometry, by taking the example of categories of D-branes in string compactifications.

Feb 16, 2006

Sergey Fomin (University of Michigan)
Catalan numbers and root systems
The Catalan numbers and their generalizations and refinements can be viewed as "type A" versions of more general numbers defined for an arbitrary finite Coxeter group. These numbers come up in a variety of combinatorial, algebraic, and geometric contexts to be surveyed in the talk (hyperplane arrangements, noncrossing partitions, generalized associahedra, and so on), suggesting connections that transcend mere numerology. Joint work with N.Reading.

Feb 10, 2006

Session in common with the *Computability and Complexity in Analysis and Dynamics Seminar *
Alex Nabutovsky (University of Toronto)
Kolmogorov complexity and some of its applications to geometry

Friday, 3 February 2006, 2:00PM -- Fields Institute, Stewart Library,

Speaker: Robert McCann, University of Toronto
Ricci curvature bounds for metric measure spaces.
Abstract: Recently, Lott & Villani (and independently Sturm) used a set of inequalities proved in a Riemannian setting by Cordero-Erausquin, Schmuckenschlaeger and myself to give a definition of what it should mean for a metric space equipped with a Borel reference measure to have a lower bound on its Ricci curvature. This notion survives Gromov-Hausdorff limits, and implies a host of consequences about the local and global geometry of such a metric space.

Thursday, 26 January 2006, 2:00PM -- Fields Institute, Library

Speaker: Robert Lipshitz, Stanford University
Two views of Heegaard-Floer homology
Heegaard-Floer homology, discovered by P. Ozsvath and Z. Szabo, uses symplectic geometry (in particular, Lagrangian intersection Floer homology) to attack problems in 3- and 4- dimensional topology. We will start by recalling the definition of Lagrangian intersection Floer homology in general and introducing the special case of Heegaard-Floer homology. While computing a few examples we will be led to a "cylindrical" reformulation of Heegaard- Floer homology. Time permitting, we will conclude by mentioning a few classical applications of Heegaard-Floer homology, and/or sketching some applications of our reformulation.