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 geometr*y

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.

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