
THE FIELDS
INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES 
Thematic
Program on CalabiYau Varieties: Arithmetic, Geometry
and Physics
October
7–11, 2013
Concentrated Graduate Course
preceding
the
Workshop 2 on Enumerative
Geometry and CalabiYau Varieties
Fields Institute, Room 230


SCHEDULE FOR GRADUATE COURSE
Time

Monday
October 7

Tuesday
October 8

Wednesday
October 9

Thursday
October 10

Friday
October 11

10:00–11:00






11:15–12:15






2:00–3:00






Speaker 
Title and Abstract 
Sara Filippini 
(1) The Tropical Vertex Group, Scattering Diagrams and Quivers
with ManWai Cheung
The tropical vertex group $\mathbb{V}$ introduced by KontsevichSoibelman
plays a role in many problems in algebraic geometry and mathematical
physics. The group itself can be understood in very different ways.
In the approach due to Gross, Pandharipande and Siebert, a central role
is played by tropical curves in $\mathbb{R}^2$ and their enumerative
invariants. This approach leads to a number of applications. On the
one hand, correspondence theorems connect factorizations in $\mathbb{V}$
with GromovWitten theory. On the other hand, these tropical methods
when combined with results of Reineke, allow to relate GromovWitten
theory to the topology of moduli spaces of quiver representations. First
we will describe the tropical vertex group and in particular scattering
diagrams. Then we will sketch the connection with tropical curves and,
if time permits, with moduli spaces of quiver representations.
(2) Introduction to toric degeneration
A toric degeneration is (roughly speaking) a family of varieties
whose central fiber is a union of toric varieties glued pairwise torically
along toric prime divisors. It is possible to encode all information
about the degenerating variety into certain combinatorial data, namely
an affine manifold with singularities together with a compatible piecewiselinear
function. We will introduce singular affine manifolds and the construction
of toric degenerations and discuss the scattering process.

ManWai Cheung 
The Tropical Vertex Group, Scattering Diagrams and Quivers
The tropical vertex group V introduced by KontsevichSoibelman
plays a role in many problems in algebraic geometry and mathematical
physics. The group itself can be understood in very different ways.
In the approach due to Gross, Pandharipande and Siebert, a central role
is played by tropical curves in R2 and their enumerative invariants.
This approach leads to a number of applications. On the one hand, correspondence
theorems connect factorizations in V with GromovWitten theory. On the
other hand, these tropical methods when combined with results of Reineke,
allow to relate GromovWitten theory to the topology of moduli spaces
of quiver representations. First we will describe the tropical vertex
group and in particular scattering diagrams. Then we will sketch the
connection with tropical curves and, if time permits, with moduli spaces
of quiver representations.

Michel van Garrel 
(1) Survey of DonaldsonThomas and PandharipandeThomas theory
This talk is a survey of the definition and properties of
DonaldsonThomas (DT) and PandharipandeThomas (PT) invariants for a
CalabiYau threefold $X$. The focus will be on overviewing some of the
modern developments of the theory. The weighted Euler characteristic
approach will be mentioned. It will be explained how PT invariants yield
a (conjectural) construction of integervalued BPS state counts.
Time permitting, it will be discussed how DT and PT invariants are naturally
realized as counts of objects in the bounded derived category of coherent
sheaves on $X$. In that setting, the wallcrossing formula for going
from DT to PT corresponds to a change of stability condition.
(2)Introduction to Logarithmic Geometry and Log Stable Maps
We give an introduction to logarithmic geometry which will
be fundamental knowledge for the conference talks by Abramovich, Chen
and Gross.
We define log stable maps and explain why a stable curve is a smooth
curve in the logarithmic sense.
(3) Logarithmic GromovWitten Theory
Logarithmic GromovWitten (GW) invariants are a generalization
of GW invariants to a logarithmically smooth situation. One major advantage
is a clarification of the degeneration formula, although its definitive
form is still work in progress. In this talk, we define these invariants
and motivate them from the perspective of the degeneration formula.

Peter Overholser 
(1) Tropical Curves and Disks
I will present a few perspectives on tropical geometry, emphasizing
concrete descriptions and properties of so called "parametrized"
tropical curves, disks, and trees. These objects will play a central
role in the discussion of mirror symmetry for $\mathbb{P}^2$.
(2) Mirror Symmetry for $\mathbb{P}^2$
I will give a sketch of Gross's construction of mirror symmetry
for $\mathbb{CP}^2$. The presentation will rely heavily on the tools
introduced in the week's earlier discussion of tropical geometry.

Nathan Priddis 
(1),(2): Geometric Quantization and its applications to GromovWitten
theory
In the first talk I will try to motivate the methods that
are employed in geometric quantization, such as Feynman diagrams and
Givental's formalism. In the second talk I will introduce the methods
more explicitly and try to give a few examples of how it relates to
GW theory.

Callum Quigley 
(1),(2): Physics of Mirror Symmetry: The Basics
I will review the ideas that lead physicists to Mirror Symmetry:
$N=2$ superconformal field theories, their chiral rings and moduli spaces.
Then I will discuss some simple examples and applications.

Simon Rose 
An introduction to GromovWitten theory
We will go over (quickly!) the motivation and ideas behind
GromovWitten theory, focusing in particular on the case of $\mathbb{P}^2$.
Heavy emphasis will be on examples and concrete computations as much
as possible.

Helge Ruddat 
(1)Introduction to the Fukaya Category
We define Lagrangian Floer homology and the Fukaya category.
We give some examples and explain the idea of the proof of homological
mirror symmetry for the elliptic curve.
(2) Computation of GromovWitten invariants via Tropical Curves
We show that the counting of rational curves on a complete
toric variety which are in general position relative to the toric prime
divisors coincides with the counting of the corresponding tropical curves.
The proof relies on degeneration techniques and log deformation theory
and is a precursor to log GromovWitten theory.

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