
THE FIELDS
INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES 
Thematic
Program on CalabiYau Varieties: Arithmetic, Geometry
and Physics
October
1519, 2013
Workshop
2 on Enumerative geometry and Calabi–Yau varieties
Principal
Organizers: Mark Gross, Radu Laza, Jaume Gomis, ShingTung
Yau
Note:
Workshops 2 & 3 will be organized jointly with the
Perimeter Institute.
In order to ease participation in both workshops, they
will be held backtoback, one at the Fields Institute
and the other at the Perimeter Institute.
October 2125, 2013 to be held at the Perimeter Institute
Workshop
3 on Physics around Mirror Symmetry
Principal Organizers: Vincent Bouchard, Jaume Gomis,
Sergei Gukov, Johannes Walcher, ShingTung Yau.



Preliminary Schedule as of October 2

October 15
Tuesday

October 16
Wednesday

October 17
Thursday

October 18
Friday

October 19
Saturday

9:3010:30






11:00–12:00






14:00–15:00






15:30–16:30






17:00–19:00

DLS Reception





Speaker 
Title and Abstract 
Abramovich, Dan
Brown University
Lecture Notes

The decomposition formula for logarithmic GromovWitten
invariants
This is joint work of Qile Chen, Mark Gross, Bernd
Siebert and me.
A central aim of logarithmic Gromov–Witen theory is to
find general formulas relating usual Gromov–Witten invariants
of a smooth variety X with appropriate invariants of simpler
varieties which appear as components of a degeneration of
X. The first step is the decomposition formula, which breaks
apart invariants of the singular fiber in combinatorial terms
determined by tropical curves or graphs. I will describe our
work on the decomposition formula, with examples (at least
one example).

Abouzaid, Mohammed
Columbia University 
Formality of Fukaya categories from disc counts
One of the main problems in computing Fukaya categories
is to understand the underlying Ainfinity structure even
for simple and explicitly constructed Lagrangians. I will
explain how one can use counts of holomorphic discs to prove
the formality of certain (subcategories of) Fukaya categories,
focusing on the example of the Milnor surfaces of type A.
This is joint work with I. Smith.

Brav, Chris
IAS 
Hamiltonian local models for symplectic derived stacks
We show that a derived stack with symplectic form
of negative degree can be locally described in terms of generalised
Darboux coordinates and a Hamiltonian cohomological vector
field. As a consequence we see that the classical moduli stack
of vector bundles on a Calabi–Yau threefold admits an
atlas consisting of critical loci of regular functions on
smooth varieties. If time permits, we discuss applications
to the categorification of Donaldson–Thomas theory.
This is joint work with subsets of BenBassat, Bussi, Dupont,
Joyce, and Szendroi.

Bryan, Jim
University of British Columbia 
${\pi}$stable pairs and the crepant resolution conjecture
in DonaldsonThomas theory
We construct curve counting invariants for a Calabi–Yau
threefold Y equipped with a dominant birational morphism ${\pi:
Y \rightarrow X}$. Our invariants generalize the stable pair
invariants of Pandharipande and Thomas which occur for the
case when ${\pi: Y \rightarrow X}$ is the identity. In the
case where ${\pi: Y \rightarrow X}$ is a semismall crepant
resolution, we prove a PT/DT  type formula relating the partition
function of our invariants to the DonaldsonThomas partition
function. In the case where X is the coarse space of a Calabi–Yau
orbifold, our partition function is equal to the Pandharipande–Thomas
partition function of the orbifold.
Our methods include defining a new notion of stability for
sheaves which depends on the morphism ${\pi}$. Our notion
generalizes slope stability which is recovered in the case
where ${\pi}$ is the identity on Y .

Chen, Qile
Columbia University 
Very free curves on Fano complete intersection
The theory of stable log maps are developed for
studying the degeneration of Gromov–Witten invariants.
In this talk, I will introduce another interesting application
of stable log maps to classical birational geometry —
we construct very free curves on Fano complete intersections
in projective spaces over an algebraically closed field of
arbitrary characteristics.
This is a joint work with Yi Zhu.

Cooper, Yaim
Harvard University 
The geometry of stable quotient spaces in genus one
Stable quotient spaces provide an alternative to
stable maps for compactifying spaces of maps. In this talk
I will discuss spaces of stable quotients which compactify
the space of degree d maps of genus 1 curves to ${P^n}$. I
will describe what is known about the geometry of these spaces.
I will also discuss the relationship between these spaces
and the corresponding spaces of stable maps from the perspective
of the minimal model program.

Costello, Kevin
Northwestern University 
Quantization of BCOV theory on CalabiYau manifolds
I’ll discuss some aspects of work in progress with Si
Li on quantization of open and closed BCOV theory on CalabiYau manifolds.
This gives a new formulation of the Bmodel which is local on the CalabiYau.
If time permits, I’ll discuss some calculations in both the open
and closed versions of this theory.

Diaconescu, Duiliu,
E.
University of Alberta 
Parabolic refined invariants and Macdonald polynomials
A string theoretic derivation is given for the
conjecture of Hausel, Letellier and RodriguezVillegas on
the cohomology of character varieties with marked points.
Their formula is identified with a refined BPS expansion in
the stable pair theory of a local root stack. Morever, Haiman’s
geometric con styruction for Macdonald polynomials is shown
to emerge naturally in the context of geometric engineering.

Filippini, Sarah
Fields Institute 
Refined curve counting and wallcrossing
The tropical vertex group of Kontsevich and Soibelman
is generated by formal symplectomorphisms of the 2dimensional
algebraic torus. It plays a role in many problems in algebraic
geometry and mathematical physics. Based on the tropical vertex
group, Gross, Pandharipande and Siebert introduced an interesting
Gromov–Witten theory on weighted projective planes which
admits a very special expansion in terms of tropical counts.
I will describe a refinement or “qdeformation”
of this expansion, motivated by wallcrossing ideas, using
Block–Goettsche invariants. This leads naturally to the
definition of a class of putative qdeformed curve counts.
We prove that this coincides with another natural qdeformation,
provided by a result of Reineke and Weist in the context of
quiver representations, when the latter is well defined.
Joint work with Jacopo Stoppa.

Fukaya, Kenji
Simons Institute 
Perturvation of constant maps, String topology and Perturbative
Chern–Simons Theory
In this talk I will explain the way to obtain a
solution of certain master equation onthe cyclic bar complex
of the deRhamcohomology. This solution is a constant map
part of Lagrangian Floer theory and is related to the two
stories in the title. Something new in this talk, which I
will explain, is the way how to do it without using pseudoholomorphic
curvein the cotangent bundle. So everything works in the story
of finite dimensional spaces. However ‘virtual technique’
is used much.

Gross, Mark
UCSD 
Introduction to Logarithmic Gromov–Witten invariants
Log Gromov–Witten invariants are a generalization
of relative Gromov–Witten invariants.
They can be used to define the notion of curve counts with
specified tangencies along normal crossings divisors, or curve
counts in normal crossings target spaces. I will outline the
basic definitions of these invariants as developed by myself
and Siebert, on the one hand, and Abramovich and Chen, on
the other.

Kontsevich, Maxim
IHES 
Distinguished Lecture 1, What is tropical mathematics?
In tropical mathematics the usual laws of algebra
are changed, the subtraction is forbidden, the division is
always permitted, and 1+1 is equal to 1. Analogs of usual
geometric shapes like lines, circles etc. are replaced by
new figures composed of pieces of lines. I will try to explain
basics of tropical algebra and geometry, its relation with
more traditional domains, and its role in mirror symmetry
which is a remarkable duality originally discovered in string
theory about 20 years ago.
(II) Quivers, cluster varieties and integrable systems
I’ll describe a new approach to cluster varieties
and mutations based on scattering diagrams and wallcrossing
formalism. The central role here is played by certain canonical
transformation (formal change of coordinates) associated with
arbitrary quiver. Also, a complex algebraic integrable system
under some mild conditions produces a quiver, and the associated
canonical transformation is a birational map.
(III) Fukaya category meets Bridgeland stability
Bridgeland’s notion of stability in triangulated
categories is believed to be a mathematical encoding of Dbranes
in string theory. I’ll argue (using physics picture)
that partially degenerating categories with stability should
be described as a mixture between symplectic geometry and
pure algebra. Spectral networks of Gaiotto, Moore and Neitzke
appear as an example.

Ruan, Yongbin
Michigan University 
A mathematics theory of gauged linear sigma model
Several years ago, we (Fan, Jarvis and myself)
developed a theory for so called LandauGinzburg model. It
has a variety of applications in integrable hierarchy, LG/CY
correspondence and modularity. LGmodel is a limit of so called
gauged linear sigma model. In the talk, I will discuss a construction
to generalize our ”classical” theory to the general
situation of gauged linear sigma model. Some potential applications
will be discussed.

Ruddat, Helge
Fields Institute 
Speculations on Mirror Symmetry for Riemann surfaces
There has been quite some evidence that some form
of mirror symmetry is valid for curves of higher genus. In
known constructions, the dual geometry is derived from a higherdimensional
Landau–Ginzburg model. We present some ideas of how an
intrinsic form of the mirror construction could be fomulated.

Soibelman, Yan
Kansas State University 
3dimensional CalabiYau manifolds and Hitchin integrable
systems
I am going to discuss the relationship between two
topics mentioned in the title from the point of view of theory
of DonaldsonThomas invariants and wallcrossing formulas
developed by Kontsevich and myself.

Tseng, HsianHua
Ohio State University 
Mirror theorem, Seidel representation, and holomorphic
disks
The quantum cohomology ring QH*(X) of a projective
toric manifold X can be computed in several ways. A presentation
of QH*(X) can be derived from the toric mirror theorem of
Givental, LianLiuYau, and Iritani. McDuffTolman use d Seidel
representations to derive a presentation of QH*(X). More recently,
FukayaOhOhtaOno showed that QH*(X) is isomorphic to the
Jacobian ring of the Lagrangian Floer superpotential of X,
which is defined in terms of counting of holomorphic disks
in X. The purpose of this talk is to explain the geometric
reason underlying the equivalence of these three seemingly
very different approaches, when X is semiFano.

Zinger, Alexsey
Stony Brook 
Mirror Symmetry for Stable Quotients Invariants
I will describe a mirror formula for the direct
analogue of Givental’s Jfunction in the SQ theory. It
is remarkably similar to the mirror formula in the Gromov–Witten
theory, but the former does not involve a change of variables.
This suggests that the mirror map relating the GWinvariants
to the Bmodel of the mirror is more reflective of the choice
of curve counting theory on the A side than of mirror symmetry.
The proof of the mirror formula in the Fano case is as in
the GW–theory. On the other hand, the proof in the Calabi–Yau
case consists of showing that it is a consequence of the Fano
case.
This is joint work with Y. Cooper.

Zaslow, Eric
Northwestern University 
Legendrian knots and constructible sheaves
Given a Legendrian knot, we construct a category,
invariant under Legendrian isotopies up to equivalence. Rankone
objects of our category play a special role. On the one hand,
they define a subcategory which we conjecture to be equivalent
to the bilinearized Legendrian contact homology of the knot.
On the other hand, the moduli of these objects is an interesting
space which, for positive braid closures, enables one to recover
the Khovanov–Rozansky categorified invariant of the topological
type of the knot. I will try to explain all this by working
through simple examples.
This work is joint with Vivek Shende and David Treumann

Confirmed Participants to the Workshop as of October 9,
2013
Full Name 
University/Affiliation 
Arrival Date 
Departure Date 
Abouzaid, Mohammed 
Columbia University 
14Oct13 
16Oct13 
Abramovich, Daniel 
Brown University 
14Oct13 
20Oct13 
AmirKhosravi, Zavosh 
University of Toronto 
01Jul13 
30Dec13 
Bertolini, Marco 
Duke University 
14Oct13 
26Oct13 
Brav, Christopher 
Institute for Advanced Study 
14Oct13 
19Oct13 
Bryan, Jim 
University of British Columbia 
14Oct13 
19Oct13 
Ceballos, Cesar 
York University 
13Aug13 
20Dec13 
Chan, Kwokwai 
The Chinese University of Hong Kong 
13Oct13 
20Oct13 
Chen, Qile 
Columbia University 
14Oct13 
20Oct13 
Cheung, ManWai 
UC San Diego 
14Oct13 
20Oct13 
Costello, Kevin J 
Northwestern University 
14Oct13 
16Oct13 
Dedieu, Thomas 
Université Paul Sabatier 
06Oct13 
20Oct13 
Diaconescu, DuiliuEmanuel 
University of Alberta 
16Oct13 
22Oct13 
Fang, Bohan 
Columbia University 
15Oct13 
19Oct13 
Filippini, Sara Angela 
Fields Institute 
01Jul13 
31Dec13 
Fisher, Jonathan 
University of Toronto 
01Jul13 
31Dec13 
Fukaya, Kenji 
Stony Brook University 
16Oct13 
20Oct13 
GarciaRaboso, Alberto 
University of Toronto 
01Aug13 
31Dec13 
GonzalezDorrego, Maria R. 
University of Toronto 
16Sep13 
19Oct13 
Gross, Mark 
UC San Diego 
14Oct13 
20Oct13 
Gualtieri, Marco 
University of Toronto 
05Sep13 
05Dec13 
Jinzenji, Masao 
Hokkaido University 
14Oct13 
18Oct13 
Karigiannis, Spiro 
University of Waterloo 
15Oct13 
19Oct13 
Kasa, Michael 
UC San Diego 
14Oct13 
20Oct13 
Kelly, Tyler 
University of Pennsylvania 
15Oct13 
19Oct13 
Lau, SiuCheong 
Harvard University 
15Oct13 
19Oct13 
Laza, Radu 
Stony Brook University 
15Oct13 
19Oct13 
LiBland, David 
Berkeley 
24Sep13 
16Oct13 
Marcus, Steffen 
University of Utah 
11Oct13 
18Oct13 
Molnar, Alexander 
Queen's University 
01Jul13 
31Dec13 
Moraru, Ruxandra 
University of Waterloo 
15Oct13 
19Oct13 
Movasati, Hossein 
Instituto Nacional de Matemática Pura e Aplicada 
15Oct13 
22Nov13 
Odaka, Yuji 
Imperial college (/Kyoto university) 
14Oct13 
18Oct13 
Overholser, Douglas 
University of California, San Diego 
01Jul13 
31Dec13 
Park, B. Doug 
University of Waterloo 
01Sep13 
20Dec13 
Perunicic, Andrija 
Fields Institute 
02Jul13 
31Dec13 
Quigley, Callum 
University of Alberta 
10Oct13 
25Oct13 
Rayan, Steven 
University of Toronto 
15Jun13 
30Jan14 
Rose, Simon 
Fields Institute 
01Jul13 
31Dec13 
Ross, Dustin 
University of Michigan 
14Oct13 
19Oct13 
Ruan, Yongbin 
University of Michigan 
15Oct13 
19Oct13 
Ruddat, Helge 
Universität Mainz 
25Jun13 
31Dec13 
Schaug, Andrew 
University of Michigan 
15Sep13 
20Oct13 
Selmani, Sam 
McGill University 
15Sep13 
20Nov13 
Silversmith, Robert 
University of Michigan 
14Oct13 
20Oct13 
Soibelman, Yan 
Kansas State University 
14Oct13 
26Oct13 
Soloviev, Fedor 
University of Toronto 
01Jul13 
30Dec13 
Thompson, Alan 
Fields Institute 
01Jul13 
30Dec13 
Tseng, HsianHua 
Ohio State University 
14Oct13 
20Oct13 
van Garrel, Michel 
Fields Institute 
01Jul13 
31Dec13 
Yui, Noriko 
Queen's University 
02Jul13 
20Dec13 
Zaslow, Eric 
Northwestern University 
17Oct13 
20Oct13 
Zhang, Zheng 
Stony Brook University 
15Oct13 
19Oct13 
Zhu, Yuecheng 
University of Texas at Austin 
01Jul13 
23Nov13 
Zinger, Aleksey 
Stony Brook University 
17Oct13 
20Oct13 
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