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Part B2: The scientific proposal
The Gromov–Witten invariants of a space X record the number of holomorphic curves in X of given genus g
and degree d which meet a given collection of cycles in X. They have important applications in algebraic
geometry, symplectic topology, and theoretical physics:
• in algebraic geometry, they carry information about the enumerative geometry of X;
• in symplectic topology, they give invariants of symplectic manifolds;
• in theoretical physics, they are correlation functions in a certain model of string theory.
Gromov–Witten theory is one of the most active areas of research in geometry, and mathematicians have made
a lot of progress here over the last 20 years. Almost all of this progress, however, has been on questions within
Gromov–Witten theory. It is now time for a fundamental change of perspective, and to apply Gromov–Witten
theory to questions from the rest of mathematics. The key new ingredient here is the recent spectacular advance
in our understanding of Mirror Symmetry, which is due in large part to the PI and his ERC-funded research
team [20, 21, 23, 24].
Section a. State-of-the-art and objectives.
Computing Gromov–Witten invariants is hard. There are only a handful of spaces for which we completely
understand their Gromov–Witten theory. By this I mean the following. One can encode Gromov–Witten
invariants as Taylor coefficients of a generating function – in string theory this function Z X is the partition
function of the topological A-model with target space X. Knowing Z X is the same as knowing all Gromov–
Witten invariants of X.
• For X = point, X = P1 , or X = an orbifold P1 , Z X is known explicitly [44, 49, 64].
• For X a curve, or X a three-dimensional toric Calabi–Yau manifold, there are effective algorithms to
compute Z X [2, 55, 65].
• For X a compact toric manifold, Givental gave a formula [36] for Z X in terms of Zpoint . The PI has
shown, jointly with Iritani and Jiang, that in this case Z X is a generalized modular form [27, 28].
Beyond this, we know very little. The obstruction here is our poor understanding of higher-genus Gromov–
Witten invariants: those with g > 0. If we restrict attention to genus-zero invariants (g = 0) then the situation
is much better. As I will explain below, Mirror Symmetry allows us to compute genus-zero Gromov–Witten
invariants of a broad class of target spaces X; the strongest results here are due to the PI and coauthors. Another defect in our current understanding, which is a major obstruction to applying Gromov–Witten theory
to questions from the rest of mathematics, is that the existing tools for computing Gromov–Witten invariants
align poorly with the basic operations in geometry: passing to a subvariety, passing to a fibration, blow-up, and
degeneration.
I propose here a program of research which will resolve each of these problems. This has three interlinked
strands. In the first strand, the PI and his team will develop powerful new tools for computing Gromov–
Witten invariants. This will drive, and be driven by, progress in the other two strands. Here we will apply
Gromov–Witten theory to questions from other areas of mathematics: birational geometry (in strand two) and
the classification of Fano manifolds (in strand three).
(1) We will fundamentally rework the tools for computing Gromov–Witten invariants.
We will re-engineer the computational tools in Gromov–Witten theory so that they align with basic operations
in geometry: passing to a subvariety, passing to a fibration, blow-up, and degeneration. These new tools will
allow the PI and his team, alongside other mathematicians, to solve problems in enumerative geometry, to test
geometric predictions from string theory, and to uncover connections between Gromov–Witten invariants and
other areas of mathematics (such as representation theory, integrable systems, and the theory of modular forms).
(2) We will determine how Gromov–Witten invariants change under birational transformations.
Birational transformations – such as blow-ups, flips, and flops – are small geometric modifications, analogous
to surgeries in topology. Success here will greatly increase the number of spaces for which we can compute
Gromov–Witten invariants, and will lead to new birational invariants.
(3) We will find and classify Fano manifolds.
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Fano manifolds are ‘atomic pieces’ of mathematical shapes. They are the basic building blocks of algebraic
geometry, and their classification is a long-standing and important open problem. One of the most striking
results to come out of the PI’s ERC Starting Investigator program was the discovery of a totally unexpected
connection between Mirror Symmetry and the classification of Fano manifolds [20, 21]. We will build on
this advance, using it to find and classify higher-dimensional Fano manifolds. This strand of research has
two distinct parts. First we will develop new techniques in combinatorics and massively-parallel computer
algebra that will allow us to discover the classification of higher-dimensional Fano manifolds. Then we will
establish new results in algebraic geometry and mirror symmetry which allow us to prove that this classification
is complete.
Compelling New Questions and Transformative New Methods. This proposal builds directly on the success
of the PI’s ERC Starting Investigator Grant (2009–2015). But it goes far beyond just ‘more of the same’. Indeed
much of the context for this proposal was created by theorems proved by the PI and his coauthors as part of the
Starting Investigator Grant:
• Mirror Theorems for toric stacks and toric complete intersections [23, 24];
• the Crepant Transformation Conjecture in a broad range of cases [28];
• results that link Mirror Symmetry to Fano classification [3, 21];
• ‘genus-zero controls higher genus’ results [27].
Furthermore the PI’s Starting Investigator Grant revealed new and unexpected research directions, for which
techniques and systematization were then in their infancy or unavailable. (A compelling example of this: the
connection between Mirror Symmetry and Fano manifolds.) Now is the time to build the tools that will allow
us to address these questions – questions that did not even exist before the PI’s Starting Investigator Grant.
The State of the Art. We begin by reviewing the methods currently available to compute Gromov–Witten
invariants. An important ingredient here is the quantization formalism. This is a geometric language for
working with Gromov–Witten invariants and quantum cohomology, discovered in joint work by the PI and
Givental [26]. The quantization formalism plays a key methodological role in this program, and we discuss it
in detail on page 3 below. Many of the most effective methods for computing Gromov–Witten invariants are
phrased in terms of the quantization formalism. These include:
A. Mirror Symmetry. The PI and his coauthors have proved Mirror Theorems that determine all genuszero Gromov–Witten invariants of a broad class of toric orbifolds [23] and most genus-zero Gromov–
Witten invariants of a broad class of toric complete intersections [24].
B. The Abelian/non-Abelian Correspondence. This expresses genus-zero Gromov–Witten invariants of
a GIT quotient X//G, where G is a Lie group, in terms of those of X//T, where T ⊂ G is a maximal
torus [12, 17]. Since Mirror Theorems often allow us to compute the invariants of X//T, methods A
and B together allow us to compute genus-zero Gromov–Witten invariants of a wide range of spaces.
C. The Givental–Teleman formula. Givental conjectured, and Teleman proved, a formula for Z X in
terms of Zpoint which applies whenever the quantum cohomology of X is semisimple [36,69]. Semisimplicity is a very strong constraint on X, but nonetheless the Givental–Teleman formula is an important
structural tool. It is one of the few current methods that applies to higher-genus invariants.
The PI and his team will exploit methods A–C throughout the project. We will also generalize methods A and B,
proving a Mirror Theorem for relative Gromov–Witten invariants, and extending the Abelian/non-Abelian Correspondence so that it applies to bundles of non-Abelian quotients X//G over a base. For a high-risk possible
generalization of method C, see the section ‘Virasoro constraints’ on page 7.
The other methods to compute Gromov–Witten invariants currently available are:
D. The GW/DT and GW/PT Correspondences. These express, for a 3-dimensional target space X, the
Gromov–Witten invariants of X in terms of certain counts of stable sheaves on X.
E. The Topological Vertex. This is closely related to method D. It gives an algorithm for computing the
Gromov–Witten invariants of a toric 3-dimensional Calabi–Yau manifold X. It gives closed formulas
for the Gromov–Witten invariants of X in any fixed degree d, for all g simultaneously.
F. Localization in equivariant cohomology. If X carries a torus action with isolated fixed points then
one can use the Atiyah–Bott localization formula to compute any Gromov–Witten invariant of X in
finite time.
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G. Virasoro constraints. For some target spaces X the generating function Z X satisfies a family of differential equations called Virasoro constraints. When X is a point the Virasoro constraints completely
determine all Gromov–Witten invariants, but in general this is not the case and often the Virasoro constraints give no information at all.
H. The Degeneration Formula. By degenerating X to a union X+ ∪ X− of two varieties meeting in a
smooth divisor D one can express any Gromov–Witten invariant of X in terms of relative Gromov–
Witten invariants of the pairs (X+ , D) and (X− , D).
Methods D and E here are fundamentally limited to three-dimensional target spaces X, because the underlying
sheaf theory is only well-behaved when dim X = 3. We require techniques that work in all dimensions, so we
will not pursue D and E in this proposal. We will however use an important consequence of D and E, the Katz–
Klemm–Vafa formula [67], to give a new, modular-forms based proof of the classification of Fano 3-folds with
Picard rank 1: see project 4.1 on page 12. We will also pursue a line which is closely related to method D in
philosophy but which applies in all dimensions: this is project 2.3 on page 8, on generalized Gopakumar–Vafa
invariants.
Methods F, G, and H here are in a sense stronger than methods A and B, because they apply to higher-genus
Gromov–Witten invariants as well in genus zero. They share, however, a common defect: rather than computing
the generating function Z X directly, they compute Gromov–Witten invariants one at a time. In their current
form they are fundamentally unsuitable for our goals. This is because of something that is made clear by the
PI’s recent results on Mirror Symmetry and on the Crepant Transformation Conjecture: the structure formed
by all Gromov–Witten invariants transforms in a simple way under natural geometric operations, but the effect
of these operations on individual Gromov–Witten invariants is often very complicated. Thus the structural
approach that we will take throughout this proposal is not only simpler and more powerful than alternative
techniques: it is essential.
The PI and his team will make systematic use of a new method for computing Gromov–Witten invariants,
virtual birational geometry, which is discussed on page 5 below. Furthermore we will re-engineer the existing
methods F, G, and H in a way that fundamentally shifts perspective, emphasizing the structure formed by all
Gromov–Witten invariants at once. The key to this is the quantization formalism.
There is already a body of theory that combines virtual localization (method F) with the quantization formalism
to prove far-reaching structural results: the Mirror Theorem for toric stacks due to the PI and coathors [23], the
Abelian/non-Abelian Correspondence [17], and Brown’s toric bundles theorem [14] are all of this form. We
will build on and extend these methods, proving Mirror Theorems for relative Gromov–Witten invariants and
a higher-genus structure formula for the Gromov–Witten invariants of toric bundles. The latter result implies,
as an immediate consequence of the quantization formalism, that Virasoro constraints hold for the total space
of a toric bundle E → B if and only if they hold for B. (This greatly increases the range of spaces to which
method G applies.) We will develop a quantization formalism for relative Gromov–Witten invariants, thus
greatly extending the power and applicability of method H. This is an archetypal high-risk, high-reward project
– it and its implications are described in detail below (see project 1.5 on page 7).
Section b. Methodology.
Below we describe each strand of the research program in detail, giving precise goals and methods. Before we
do so we discuss two key methological ingredients: the quantization formalism and virtual birational geometry.
The Quantization Formalism. The quantization formalism is a remarkable geometric structure which governs
Gromov–Witten theory and quantum cohomology, discovered in joint work by the PI and Givental [26]. It
has become an essential tool throughout the field – indeed many of the most powerful results for computing
Gromov–Witten invariants need the quantization formalism even to state them correctly (see methods A–C
above). The PI and Givental discovered, roughly speaking, that the Gromov–Witten theory of a target space X
is controlled by linear symplectic geometry in a certain symplectic vector space1
H X = H • (X; C) ⊗ C((z −1 ))
Genus-zero Gromov–Witten invariants of X determine and are determined by a Lagrangian cone L X ⊂ H X
with very special geometric properties2. All Gromov–Witten invariants of X (genus zero or higher genus)
1Morally speaking, the space H X is the localized S 1 -equivariant Floer cohomology of the loop space of X.
2These properties of L X are a geometric reformulation of the universal relations obeyed by genus-zero Gromov–Witten invariants:
the WDVV equations, the Topological Recursion Relations, the String Equation, and the Dilaton Equation.
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are encoded in the generating function Z X . Natural operations in Gromov–Witten theory (such as passing to a
hypersurface in X, or applying a birational transformation to X) correspond to symplectic linear transformations
U of H X : their effect on genus-zero Gromov–Witten invariants is recorded by the effect of U on L X , and
their effect on higher-genus Gromov–Witten invariants is (or is expected to be) recorded by the action of the
D on the generating function Z X .
quantized3 symplectic transformation U
This is an important insight even at the level of genus-zero invariants, as although the effect of natural operations
on the Lagrangian submanifold L X is simple (just a linear symplectic transformation U) the effect on individual
Gromov–Witten invariants can be very complicated. Thus the quantization formalism serves as an important
organizing principle in calculations, allowing progress on computations that before seemed completely intractable. This formalism has allowed the PI and his coauthors to greatly improve the understanding of Mirror
Symmetry for toric varieties and toric stacks [23–25], removing unnecessary positivity and compactness hypotheses. It also allowed us to prove powerful versions of the Quantum Lefschetz Theorem [19,22,24,26]. This
is a ‘quantum cohomology’ version of the Lefschetz Hyperplane Theorem: it determines genus-zero Gromov–
Witten invariants (or quantum cohomology) of a hypersurface or complete intersection in terms of genus-zero
Gromov–Witten invariants (or quantum cohomology) of the ambient space, via a symplectic transformation U
as above. The results [19, 22–26] together are the state of the art for computing Gromov–Witten invariants of
toric complete intersections. Via the Abelian/non-Abelian Correspondence (method B) this also gives access to
genus-zero Gromov–Witten invariants of many subvarieties of Grassmannians and partial flag manifolds.
The symplectic transformations U described above are visible at the level of genus-zero Gromov–Witten invariants, and so the quantization formalism gives a powerful genus-zero controls higher genus principle. In
joint work by the PI and Iritani [27], we gave a precise formulation of this. We reinterpreted the quantization formalism from [26] in purely geometric and co-ordinate-free terms. We defined a Fock sheaf over the
Lagrangian cone L X – recall that L X encodes genus-zero Gromov–Witten invariants – and conjectured that
the higher-genus generating function Z X defines a global section of this Fock sheaf. If Z X is a global section
of the Fock sheaf and L X transforms via a symplectic transformation U then it follows that Z X transforms
D Thus, for global sections of the Fock sheaf, genus zero controls higher genus. The Givental–Teleman
via U.
formula (method C above) has a very simple interpretation in our framework: it implies that, whenever X has
semisimple quantum cohomology, Z X is a global section of the Fock sheaf. Thus, in the semisimple case,
the Givental–Teleman formula proves that genus zero controls higher genus. When combined with our good
understanding of genus-zero Gromov–Witten theory via Mirror Symmetry, this is a powerful tool.
A Model Example. We now discuss an example which demonstrates the power of this approach: the proof of
the Crepant Transformation Conjecture for higher-genus Gromov–Witten invariants in the compact toric case.
In outline, this goes as follows. Let ϕ : X+ d X− be a crepant birational transformation between toric orbifolds,
such as a toric flop or crepant resolution. The Mirror Theorem [23], by the PI and coauthors, gives a complete
understanding of the genus-zero Gromov–Witten invariants of toric orbifolds. It determines the Lagrangian
submanifolds L X+ and L X− in terms of certain hypergeometric functions I+ and I− . Analytically continuing
the hypergeometric function I+ , using the Mellin–Barnes method [10], and comparing with I− shows that the
Lagrangian submanifolds L X+ , L X− are related by a symplectic transformation U : H X+ → H X− . That is:
U(L X+ ) = L X−
There are distinguished co-ordinate systems on H X+ and H X− , given by Iritani’s integral structure on quantum
cohomology. When we write U in these co-ordinates, we find that it coincides with the Fourier–Mukai transformation on derived categories D b (X+ ) → D b (X− ) induced by the birational transformation ϕ. This is the
genus-zero Crepant Transformation Conjecture for toric orbifolds, proved by the PI with Iritani and Jiang [28].
But the quantum cohomology of compact toric orbifolds is semisimple, so our genus-zero controls higher genus
principle (i.e. Givental–Teleman) now implies that:
D X+ ) = Z X−
U(Z
This is the higher-genus Crepant Transformation Conjecture for compact toric orbifolds, proved by the PI and
Iritani [27].
The Fourier–Mukai transformation here is an isomorphism of derived categories; it comes from analytic continuation of the hypergeometric function I X+ along a path. If we analytically continue around a loop instead
then we get automorphisms of the derived category D b (X+ ), and the corresponding symplectic transformations
3Put differently: the total descendant potential Z X , which is the mathematical counterpart of the partition function in the topological
A-model of string theory, should be thought of as an element of the Fock space arising from the geometric quantization of H X .
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D X+ ) = Z X+ . Thus Z X+ is invariant under a group Γ of autoequivalences of the derived category
U satisfy U(Z
b
D (X+ ). That is, the higher-genus generating function Z X+ of a compact toric orbifold X+ is a generalized
modular form for the group Γ. It is hard to imagine how you would see this result, let alone prove it, without
the structural perspective that the quantization formalism provides.
Homological Mirror Symmetry. One can think of the modularity result just discussed as a shadow of Homological Mirror Symmetry [50]. The Gromov–Witten invariants here should arise from an appropriate derived
Fukaya category; Mirror Symmetry should identify this with the derived category associated to a Landau–
Ginzburg model (which is an object from singularity theory, called ‘the mirror’ to X± ); and modularity should
reflect monodromy-invariance with respect to an appropriate Gauss–Manin connection on the Landau–Ginzburg
side. Homological Mirror Symmetry is still in its infancy, and mathematicians are a very long way from being
able to make or prove these statements precisely. But what is striking here is that not only can we use the
quantization formalism to prove far-reaching consequences of the Homological Mirror Symmetry program, but
we do so using structures that the program suggests should be there. From this point of view, the Lagrangian
submanifold L X which plays such an important role in the quantization formalism is a geometrization of the
notion of variation of semi-infinite Hodge structure [9, 47]. Indeed, the tangent spaces to the cone L X are
precisely an example of a variation of semi-infinite Hodge structure.
Virtual Birational Geometry. Another key methodological ingredient in this proposal is Manolache’s virtual
birational geometry program. Gromov–Witten invariants of X – which, as discussed, give the number or ‘virtual
number’ of curves in X of genus g and degree d that pass through various cycles – are defined as integrals over
Kontsevich’s moduli spaces of stable maps X g, n, d . These moduli spaces are typically singular and have many
components, some of which have unexpectedly large dimension. Thus Gromov–Witten invariants are integrals
not over the usual fundamental cycle of X g, n, d , but against the virtual cycle [11, 57]. This is a distinguished
cycle in X g, n, d , defined in terms of a perfect obstruction theory, which functions as a well-behaved substitute
for the ordinary fundamental cycle of X g, n, d even though this space is singular.
Many of the technical challenges in Gromov–Witten theory stem from the fact that it is hard to work with the
virtual cycle directly. Indeed, one can think of many of the existing methods for computing Gromov–Witten
invariants as methods for avoiding having to deal with the virtual cycle. For example virtual localization,
method F above, replaces a single integral over the moduli space of stable maps to X (where X admits a torus
action) by many integrals over Deligne–Mumford spaces Mg0, n0 . For Deligne–Mumford spaces, the virtual
cycle and the usual fundamental cycle coincide.
We will take a different approach. The model here is the bivariant intersection theory of Fulton and MacPherson [33]. Fulton and MacPherson’s approach to intersection theory – building a functorial theory which combines push-forward and pull-back maps, excess intersection formulas, and so on – greatly extended the range of
problems to which intersection theory could be applied; indeed intersection theory is now part of the standard
toolkit of every algebraic geometer. Manolache is constructing a virtual bivariant intersection theory, which
operates on virtual cycles directly. She has established virtual pull-back and virtual push-forward theorems, as
well as a virtual excess intersection formula [59, 60].
Manolache’s theory is not quite in final form – there should be a stronger virtual excess intersection formula,
which we expect to come in to focus over the course of this program – but nonetheless we are now in a position
to work with virtual cycles directly. This allows for much more direct and geometrically intuitive arguments
than previous approaches. A model example here is the new proof (joint work by the PI and Manolache) of
Givental’s mirror theorem for a toric manifold X. The result here is not new, but the method is completely new
and should be of broad applicability. It is the first proof of the Mirror Theorem which does not rely, at heart,
on localization in equivariant cohomology (method F above). We relate Gromov–Witten invariants, which
are integrals over the moduli space of stable maps X g, n, d , to integrals over the moduli space of quasimaps4
Q g, n, d [15], using virtual birational geometry.
The study of the relationship between stable maps and quasimaps has been pioneered by Ciocan-Fontanine–
Kim, but they use localization methods. Virtual birational geometry allows us to take a much more direct
approach. In the case where X is Fano, we show that the moduli space of stable maps X g, n, d splits into two
±
parts X g,
n, d , each of which is a union of components, such that:
−
• X g,
n, d does not contribute to Gromov–Witten invariants (for dimensional reasons); and
4In genus zero, the moduli space of quasimaps is closely related to Givental’s toric compactification of the moduli space of maps
from P1 to X. The toric compactification played a key role in Givental’s original proof of the Mirror Theorem [34].
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+
• there is a map ϕ : X g,
n, d → Q g, n, d via which the virtual classes agree.
In other words5, moduli spaces of stable maps and quasimaps are ‘virtually birational’ via ϕ : X g, n, d d Q g, n, d .
This implies that Gromov–Witten invariants (which are integrals over X g, n, d ) and quasimap invariants (which
are integrals over Q g, n, d ) coincide, proving the Mirror Theorem. In the case where X is not Fano, the contribu−
tions from X g,
n, d no longer vanish, but instead can be expressed recursively in terms of moduli spaces of stable
maps X g0, n0, d0 with smaller genus and degree. This geometric recursion precisely reproduces the relationship
between genus-zero Gromov–Witten invariants and genus-zero quasimap invariants – which involves Birkhoff
factorization [16] – predicted by the quantization formalism. Thus not only does virtual birational geometry
allow direct and intuitive calculations with virtual cycles, it also fits well with the structural perspective coming
from the quantization formalism. We will exploit this repeatedly when building our new tools.
Strand 1: New Tools in Gromov–Witten Theory.
We now describe in detail the projects that make up strand 1 of this program. The goals here are to re-engineer
the computational tools in Gromov–Witten theory so that they align with basic operations in geometry: passing
to a subvariety, passing to a fibration, blow-up, and degeneration.
1.1. Generalized Quantum Lefschetz. Suppose that E → X is a vector bundle, and that Y ⊂ X is the zero
locus of a regular section of E. If E is split (that is, if E is a direct sum of line bundles) then Y is a complete
intersection. In this case the Quantum Lefschetz theorem, due to the PI and Givental [26], determines (most)
genus-zero Gromov–Witten invariants of Y in terms of genus-zero invariants of X. It is clear how to generalize
Quantum Lefschetz to the non-split case, by replacing E by its resolution by line bundles6. In a different
direction, when E is split and Y is a K3 surface it is clear how to generalize Quantum Lefschetz to compute
so-called reduced Gromov–Witten invariants of Y .
Risk Analysis. These projects are very low risk. But they are also high-reward: the first will allow us to compute
genus-zero Gromov–Witten invariants of a broad class of subvarieties Y ⊂ X. The second will play an important
role in project 4.1 below: leading to a new, modular forms-based proof of the classification of 3-dimensional
Fano manifolds of Picard rank 1.
1.2. The Abelian/non-Abelian Correspondence for Bundles. We will generalize the Abelian/non-Abelian
Correspondence (method B on page 2) to the case of bundles of non-Abelian quotients X//G over a base, by
generalizing the localization techniques used by Brown to prove his toric bundles theorem [14].
Risk Analysis. This is very low-risk. We also have an alternative higher-risk approach, via the Seidel representation and shift operators [43], which we will try first as it would be useful in a wider context. This project is
also high-reward – it will allow us to:
• compute genus-zero Gromov–Witten invariants of a broad range of fibrations, in a way that reflects the
geometry of the fibration. (This aspect combines well with project 1.1 above.)
• compute genus-zero Gromov–Witten invariants of a broad class of blow-ups, in a way that reflects the
geometry of the blow-up.
For the latter result we express the blow-up of X as a subvariety of a Grassmann bundle over X and apply either
the existing Quantum Lefschetz theorem or its generalization in project 1.1. The model here is the result for
codimension-2 blow-ups proved by the PI and coauthors in [21].
1.3. Virtual birational geometry. We will use virtual birational geometry (see page 5) to:
(a) greatly extend the range of spaces to which Brown’s toric bundles theorem and project 1.2 can be
applied. For example, Brown proved his theorem for a projective bundle P(E) → B only when E is
split. But in examples, by combining virtual pull-backs and virtual push-forward theorems with the
splitting principle, we can reduce the proof for a general projective bundle P(E) → B to the case where
B P1 , where Brown’s theorem holds (because every bundle on P1 splits).
(b) determine the relationship between Gromov–Witten invariants and ramified stable map invariants [48].
5Note that the moduli spaces X g, n, d and Q g, n, d are not birational in the traditional sense, as we need to delete whole components
of X g, n, d before the map ϕ becomes well-defined.
6Such resolutions always exist, and in geometric situations – such as when Y is a degeneracy locus – it is often easy to construct
them explicitly.
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Risk Analysis. Project (a) here is high-risk – we do not yet see how to proceed systematically – but the methods
are powerful and the potential payoff is very large. Project (b) is moderate-risk: the geometry here is similar
to that already analyzed by the PI and Manolache in their proof of the Mirror Theorem (see page 5), and we
can always fall back to localization methods. The reward will again be large, with immediate applications to
birational invariants and the generalized Gopakumar–Vafa Conjecture (see project 2.3 below).
1.4. Virasoro constraints. In a joint project with Brown, Givental, and Tseng, we will combine the key idea
that led to Givental’s higher-genus formula [35] with localization methods and Brown’s mirror theorem for
toric bundles [14], to prove that Virasoro constraints hold for the total space of a toric bundle E → B if and
only if they hold for B. A key intermediate result here is a structure theorem for higher-genus Gromov–Witten
invariants of toric bundles which generalizes Givental’s formula for the higher-genus Gromov–Witten invariants
of a toric manifold X. Givental’s formula is:
N
Y
L
−1
D
L
(1)
Z X = SX RX ∆
Zpoint
i=1
where S X and R X are operators constructed from the quantum cohomology of X, ∆ = eU /z with U a diagonal
matrix, and D denotes quantization. Our generalization for the toric bundle E → B has the same form, but with
X = E, Zpoint replaced by Z B , and with ∆ a block-diagonal matrix built from blocks given by the fundamental
solution matrix for the quantum cohomology Frobenius manifold of the base B. (Recall that the fundamental
solution matrix for the quantum cohomology Frobenius manifold of a point is the 1 × 1 matrix (eu/z ); thus our
formula reduces to Givental’s when the base B is a point.)
Risk Analysis. This is low-risk and high reward. As Virasoro constraints are known to hold for curves, K3
surfaces, and Calabi–Yau 3-folds, this will greatly increase the range of spaces for which we have Virasoro
constraints (method G on page 3). A moderate-risk extension would further generalize (1) to the case of
manifolds admitting a torus action with non-isolated fixed locus. There ∆ would be block-diagonal, built from
the fundamental solution matrices of the fixed-point components. The obstruction here is purely technical
(extending the Graber–Pandharipande virtual localization formula to the case of non-isolated 1-dimensional
torus orbits) and should be surmountable using standard techniques. The consequence would be both a further
extension of Virasoro constraints, and a strong suggestion that there is a hidden structure to higher-genus
Gromov–Witten invariants in general, where Z X is obtained from ‘factors’ Z B1 , . . . , Z B k via a standard mixing
procedure:
k
Y
L
−1
D
L
Z X = SX RX ∆
Z Bi
i=1
that involves only the quantum cohomology of X (that is, only genus-zero Gromov–Witten invariants) and
the quantum cohomology of the factors Bi . It would be extremely interesting to understand the geometry or
topology behind this. In the case of Givental’s original formula, this was Mumford’s Conjecture [58, 69] on the
cohomology of the stable mapping class group.
1.5. A Quantization Formalism for Relative Gromov–Witten Invariants. We will develop and exploit a
quantization formalism for relative Gromov–Witten invariants. We will discover this formalism by:
(a) proving a Quantum Riemann–Roch theorem for relative Gromov–Witten invariants.
(b) proving Mirror Theorems for relative Gromov–Witten invariants in the toric setting, using virtual localization on moduli spaces of relative quasimaps (which we will define).
(c) reinterpreting the recursive relationships in the ‘relative-to-absolute’ correspondence [41, 61] in terms
of quantized operators.
The combination of the Degeneration Formula [42, 54, 56] with the relative-to-absolute correspondence in (c)
is the best available current method for computing with relative Gromov–Witten invariants. In principle this
determines, for example, the all-genus Gromov–Witten invariants of the quintic threefold [61], but in practice
the recursive calculations are so involved that one cannot use them to compute even genus-one invariants. The
reinterpretation proposed here will allow effective calculations.
Risk Analysis. This is one of the highest-reward parts of the proposal, and one of the riskiest. But the approach
here is structured so as to provide multiple lines of attack that reinforce each other, and which will give important advances even if the overall goal is not achieved. It is clear how to do the geometric calculation in (a) – this
is the exact analog of the Grothendieck–Riemann–Roch computation by the PI and Givental [26] which led to
the discovery of the quantization formalism for absolute Gromov–Witten invariants. The combinatorics in (b)
is well under control and, in light of the localization analyses for quasimaps by Ciocan-Fontanine–Kim [16]
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and in the proof of the Mirror Theorem [23] by the PI and coauthors, we should be able to extract from the
localization formulas the statement (and proof) of a Mirror Theorem for relative Gromov–Witten invariants.
Together with (a), this should reveal the quantization formalism for relative invariants that we seek.
It is reasonable to expect that the relative-to-absolute correspondence in (c) will then become the statement that
the generating function Z X for absolute invariants and the corresponding generating function Z X, D for relative
invariants are related by a quantized operator. (This is exactly what happened when the quantization formalism
for absolute Gromov–Witten invariants was defined [26]: the complex recursion relations that relate ‘twisted’
D in the quantization
Gromov–Witten invariants to ordinary invariants were interpreted as a single operator U
formalism that relates the twisted and ordinary Gromov–Witten theories.) This would realise the full power
of the relative-to-absolute correspondence, allowing effective calculations in a much wider setting. Even if we
are unable to find an operator-theoretic formulation, method (b) here (and also its natural generalization to the
setting of toric bundles) will give direct access to the ingredients needed to analyze a wide range of birational
transformations, including many blow-ups, using the Degeneration Formula (cf. [41, 53, 61]).
Strand 2: Mirror Symmetry and Birational Geometry.
We now describe in detail the projects that make up strand 2 of this program. The goals here are to determine how Gromov–Witten invariants change under birational transformations, and to construct new birational
invariants.
2.1. The Toric Case and the Toric Complete Intersection Case. The PI has proved, jointly with Iritani
and Jiang [28], the Crepant Transformation Conjecture for a broad class of crepant transformations between
toric complete intersections. As discussed in the section ‘A Model Example’ on page 4, given a crepant birational transformation ϕ : X+ d X− we established a simple and beautiful formula for the change in genus-zero
Gromov–Witten invariants induced by ϕ, in terms of the quantization formalism and the Fourier–Mukai transformation D b (X+ ) → D b (X− ) induced by ϕ. In the compact toric case, the PI and Iritani proved the Crepant
Transformation Conjecture for higher-genus Gromov–Witten invariants too [27], and showed that the generating function Z X for higher-genus invariants is a generalized modular form.
It is clear how to generalize this to non-crepant birational transformations between toric orbifolds or toric
complete intersections, by combining the Mirror Symmetry methods introduced by the PI and his coauthors
with a recent advance by Acosta in FJRW theory [1]. Here we will see the Fourier–Mukai functor for noncrepant variations of GIT quotient entering [8, 38].
Risk Analysis. This is a perfect post-doc problem: low-risk and high reward. It will allow for the first time
an analysis of how genus-zero Gromov–Witten invariants change under non-crepant birational transformations
such as flips – a major advance. It will also allow us to treat deform-resolve transitions (blow-down-thensmoothings) which generalize the famous conifold transition. There are also higher-risk aspects, with important implications. What is the genus-zero-controls-higher-genus principle here? (In the non-crepant case, the
symplectic transformation U will no longer be an isomorphism.) What are the consequences for modularity?
2.2. Beyond the Toric Complete Intersection Case. We will combine the results of project 2.1 with the
quantization formalism for relative Gromov–Witten invariants from project 1.5 (or, in the worst case, with
the backup outcomes from project 1.5b) to determine how Gromov–Witten invariants change under birational
transformations (crepant or non-crepant) beyond the toric complete intersection case. This would be a vast
generalization of Lee–Lin–Wang’s celebrated work on flops [53].
Risk Analysis. Moderate-to-high risk and very high reward. The experience gained when implementing these
ideas should also help with the development of the quantization formalism in project 1.5.
2.3. Ramified Stable Map Invariants. Building on project 1.3b and project 2.1, we will determine how ramified stable map invariants change under birational transformation. In the three-dimensional Fano case, ramified
stable map invariants are conjectured to coincide with Pandharipande’s generalized Gopakumar–Vafa (GV) invariants [48,66]. Example calculations suggest that GV invariants behave well under birational transformations,
and may lead to new birational invariants.
Risk Analysis. This is risky, because it opens up a completely new direction. But the case of 3-folds seems safe
enough – even just combining project 2.1 with Pandharipande’s definition of GV invariants would be interesting
and new. The 3-fold case will also give a new approach to Pandharipande’s generalized GV Conjecture [66, 70].
I expect that ramified stable map invariants are better-behaved for Fano target spaces (where contributions from
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multiple covers of curves vanish) and so these ideas will give a new approach to many interesting questions in
the birational geometry of Fano manifolds, in and beyond dimension 3. An obvious first question: can we use
this to detect non-rationality?
Strand 3: Mirror Symmetry and the Classification of Fano Manifolds.
Fano manifolds are basic building blocks of algebraic geometry, both in the sense of the Minimal Model Program [13, 68] and as the ultimate source of most explicit examples and constructions. There are finitely many
deformation families of Fano manifolds in each dimension [52]. There is precisely one 1-dimensional Fano
manifold: P1 ; there are 10 deformation families of 2-dimensional Fano manifolds: the del Pezzo surfaces,
known since the 19th century; and there are 105 deformation families of 3-dimensional Fano manifolds. The
3-dimensional classification was completed in the 1990s by Mori and Mukai [63], building on the rank-1 classification by Fano in the 1930s and Iskovskikh in the 1970s; it formed part of the spectacular work on the
geometry of 3-folds for which Mori received the Fields Medal. There is no hope of extending the methods
that worked in dimensions 2 and 3 to higher dimensions, and very little is known about Fano classification in
dimension 4 or more.
In [20], the PI and coauthors announced a program to find and classify Fano manifolds using Mirror Symmetry.
These methods should work in all dimensions. Extensive computational experiments suggest that, under mirror
symmetry, n-dimensional Fano manifolds correspond to certain Laurent polynomials in n variables with very
special properties. A Laurent polynomial f corresponds to a Fano manifold X if the Picard–Fuchs local system
associated to f : (C× )n → C is isomorphic to the quantum cohomology local system7 associated to X; concretely this amounts to demanding that the Picard–Fuchs differential equations associated to f coincide with the
quantum differential equations for X. These quantum differential equations encode (some of the) genus-zero
Gromov–Witten invariants of X.
Note here that the correspondence between Fano manifolds X and Laurent polynomials f is not one-to-one: a
given Fano manifold can have (infinitely) many Laurent polynomial mirrors. In [4], the PI and his coauthors
gave a simple and attractive explanation for why this should be so. We introduced the notion of mutations
between Laurent polynomials, which are certain birational changes of variables analogous to cluster transformations, and showed that mutation-equivalent Laurent polynomials have the same Picard–Fuchs local system.
Thus if f is mirror to a Fano manifold X and g is obtained from f by a mutation then g is also mirror to X; in
fact all known mirrors to the same Fano manifold arise this way.
To implement our Fano classification program, we need to solve two problems:
1. what is the class of Laurent polynomials that, under Mirror Symmetry, correspond to Fano manifolds?
2. given such a Laurent polynomial f , how can we construct the corresponding Fano manifold X?
The PI and his team have already made substantial progress here. We believe that we have completely solved
the first problem. We have also almost solved the second problem – see the section on ‘Laurent inversion’
below. For the first problem, we have defined a class of maximally mutable Laurent polynomials [3, 46]: these
are Laurent polynomials f which admit, in a precise sense, as many mutations as possible. The notion of
maximally mutable Laurent polynomial make sense in all dimensions. They typically occur in parametrized
families, and those that do not are referred to as rigid. We conjecture that, under Mirror Symmetry:
Fano manifolds
o
(up to deformation)
are in 1-to-1 correspondence with
/ rigid maximally mutable Laurent polynomials
(up to mutation)
The PI and his coauthors have proved this conjecture in dimensions 2 and 3, showing that the classification
of rigid maximally mutable Laurent polynomials precisely reproduces the known classifications of lowdimensional Fano manifolds [3, 4, 21]. We also extended these methods to give a new classification of a class of
two-dimensional Fano orbifolds, and have begun to analyze higher-dimensional Fano classification from this
perspective [30]; so far we have found 528 new four-dimensional Fano manifolds.
The picture here is in keeping with that suggested by Homological Mirror Symmetry [50]. This suggests that
a Laurent polynomial f mirror to a Fano manifold X records (via its coefficients) counts of holomorphic discs
with boundary on a Special Lagrangian submanifold of L in X. There should be a wall-and-chamber structure
in the moduli space of Special Lagrangians, with f depending only on the chamber containing L; as we move
from one chamber to another, f changes by wall-crossing formulas [7, 51] which are exactly our mutations. We
7The quantum cohomology local system here is the restriction of the quantum connection, which is part of Dubrovin’s quantum
cohomology Frobenius manifold, to the line in H • (X) spanned by c1 (X).
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expect that, if a Fano manifold X is mirror to a Laurent polynomial f , then there is a degeneration from X to
the (singular) toric variety X f defined by the spanning fan of the Newton polytope of f . Thus we make contact
with the work of Gross–Siebert on toric degenerations [37] and with the deformation theory of toric varieties:
we will construct X from its mirror f as a smoothing of the singular toric variety X f .
The work by the PI and coauthors described here has opened the way to the classification of Fano manifolds in
higher dimensions, long thought completely out of reach. First the PI and his team will develop new theory and
algorithms to discover higher-dimensional classifications, and to construct the Fano manifolds involved. Then
we will prove that these classifications are complete. In detail, we will proceed as follows.
3.1. Restricting the Newton polytopes. We will begin by bounding the class of Laurent polynomials f that
we need to consider, by restricting their possible Newton polytopes Newt( f ). We will develop generalizations
to higher dimensions of the notions of singularity content [5] and minimality [45] for polygons. This will lead
to improved algorithms for finding Laurent polynomial mirrors to higher-dimensional Fano manifolds (which
is the current computational bottleneck).
3.2. Laurent inversion. We will develop theory and effective algorithms to construct a Fano manifold X
directly from its mirror Laurent polynomial f , thus solving problem 2 above. A key technique here will be
Laurent inversion, recently discovered by the PI and his team. There are well-understood methods [30], going
back to Givental and Hori–Vafa, for finding Laurent polynomial mirrors to toric complete intersections. We
have discovered how to invert this procedure, reading off complete intersection models for X from a Laurent
polynomial mirror to X. This has broad implications. Given a Laurent polynomial f (such as a rigid maximally
mutable Laurent polynomial) that we expect to be the mirror to a Fano manifold X, in many cases Laurent
inversion allows us to find X directly. Furthermore in examples it allows us to see, by “mutating the ambient
variety”, much or all of the birational geometry of X. Better still, in many cases Laurent inversion constructs,
along with X, an embedded deformation from X to the singular toric variety X f – thus implementing the
smoothing of X f expected from the Gross–Siebert program.
Laurent inversion is one of the most striking and innovative ingredients in this proposal. It is a complete
surprise both from the point of view of Fano classification (where even our existing Mirror Symmetry methods
are revolutionary) and also from the point of view of Mirror Symmetry itself. This idea – that we can ‘invert’
Mirror Symmetry, building a Fano manifold X directly from its Laurent polynomial mirror – has remarkable
ramifications; indeed it is so surprising that the PI and his team are only beginning to come to terms with what
it makes possible. We will systematically work out the implications of this, throughout the program.
We will also generalize the Laurent inversion technique: from toric complete intersections to a broader class
of spaces, tautological subvarieties of quiver Grassmannians8, which includes all Fano manifolds of dimension
≤ 3 [21]. This will allow us to construct the Fano manifolds which correspond to a very broad class of,
and perhaps all, Laurent polynomial mirrors f . One key ingredient here will be the Abelian/non-Abelian
correspondence [17]. An important step in this direction, understanding the interplay between the Abelian/nonAbelian correspondence and the quantum differential equations, has been taken by Galkin–Golyshev–Iritani as
part of their work on the Gamma Conjecture for Fano manifolds.
3.3. Systematic computations. We will systematically find and build databases of maximally mutable Laurent
polynomials in dimensions 4 and 5, and in higher dimensions if this is feasible. We will apply Laurent inversion
(and its generalizations) to these Laurent polynomials, finding the corresponding Fano manifolds. We will
conduct a systematic search for Fano manifolds (in dimensions 4, 5,. . . ) that occur as tautological subvarieties
of quiver Grassmannians, generalizing [30] where we found many new 4-dimensional Fano manifolds as toric
complete intersections. (In this way we would have found all smooth Fano 3-folds.) We will formulate and
prove results that classify9 lattice polytopes up to mutation, generalizing [45].
3.4. Smoothing X f . We will construct the Fano manifold X corresponding to a Laurent polynomial mirror f
by smoothing the singular toric variety X f (discussed above). There are three complementary lines here, all of
which are independently interesting and any of which would solve the problem: the Gross–Siebert smoothing
algorithm, generalized appropriately (this approach has been successfully implemented in dimension 2 by my
PhD student Thomas Prince); the deformation theory of toric varieties, where we need to go well beyond work
of Altmann and Mavlyutov [6, 62], who treat the simplicial case (in examples we have good control of this);
Q
8That is, subvarieties of GIT quotients C N //G, where G = k GL(nk , C), defined by sections of homogeneous vector bundles.
9One implication of our program is that traditional classification results in polytope theory – classifying reflexive or canonical
polytopes up to change of basis, for example – may be answers to the wrong questions.
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and Laurent inversion, which constructs the required degenerations and deformations in a simple and direct
way (but it is not yet clear if it will apply to all Laurent polynomial mirrors). In any case, Laurent inversion
will serve as a key guide when developing the general theory.
Risk analysis for Strand 3. There is a lot of risk here. But there are so many promising directions and approaches, and the different projects fit together so well, that there is also every chance of success. This is
breakthrough mathematics, with great momentum right now – before our recent work [3, 20, 21], no-one would
have thought the classification of Fano manifolds was even possible. It should be emphasized that this line
of research is fundamentally concrete and data-based, using millions of computational experiments and examples to guide and shape our geometric understanding. This approach was essential to our discovery of the
connections between Fano classification and Mirror Symmetry. It should be similarly effective here – adding
robustness and resilience to the program.
This line of research will also shed light on deep questions with applications in geometry and representation theory. It will suggest a higher-dimensional generalization of cluster algebras, which are key objects in
representation theory and which have a close connection (as made clear by spectacular recent work of Gross–
Hacking–Keel and Kontsevich) to Mirror Symmetry in two dimensions. It involves a tractable case of the
Gross–Siebert program, which is the most promising approach to the Strominger–Yau–Zaslow Conjecture:
one of the landmark conjectures in Mirror Symmetry. Here it will be very interesting to see how much of
the structure revealed by Laurent inversion persists to the general case. Furthermore this work will allow us
to explore the full richness of Homological Mirror Symmetry (the other landmark conjecture in the field) –
derived and Fukaya–Seidel categories, holomorphic disc-counts, motivic Donaldson–Thomas invariants, toric
degenerations – in a broad and interesting class of concrete examples.
3.5. Next-Generation Computational Algebra. The program described here will be driven, as discussed,
by millions of computer experiments using massively-parallel computational algebra and distributed, sharded
databases. Our existing code already outperforms competitor systems, e.g. Mathematica, by orders of magnitude. (The key here is our parallel approach. Existing systems were designed for an outdated computing
paradigm – a single thread on a single CPU – and so are essentially serial in nature.) We will build on this
to form the core of a next-generation, open source Computational Algebra System, which has distributed algorithms, scalability, and database fluency built in from conception. This system could potentially transform
several fields: by bringing new tools to scientists who need symbolic computation applied to huge datasets, and
by treating problem sizes that are far beyond what is currently possible.
We will write the new Computational Algebra System in the modern compiled language Go. Go has been
developed by Google to address their requirements for massively parallel systems and for working with Big
Data. The new system will consist of:
1. A manager layer which will: handle communication with databases and between processes (IPC); run
high-level algorithms using low-level functionality implemented in packages, external modules, and
libraries; and supervise the dependencies between the system’s components. It will also coordinate
appropriate levels of parallelism for the resources available.
2. A series of core modules: type modules that define mathematical primitives (the ring of integers, matrices, polynomial rings, etc.); function modules that implement efficient low-level parallel algorithms
(linear algebra, permutations and sorting, knapsack-style problems, etc.); and call-out modules that
manage calls to other computing resources (C programs, other computational algebra systems, graphics displays, etc.).
3. A hierarchy of high-level modules which focus on key mathematical concepts and algorithms relevant
to the research in this program; examples include rational cones and polyhedra, lattice point enumeration, and sparse linear algebra.
4. A range of database interfaces: connections to data management tools such as SQL and Hadoop,
including protocols for handling local copies and sharing data between processes.
5. A range of user interfaces, so that the system can be used as a plug-in module for existing Computational Algebra Systems such as Mathematica and SAGE. In the long term (beyond this program) we
intend for there to be an interpreted language sitting between the general user and the manager layer.
Go is an ideal language for this project. It has parallelism primitives, based on CSP [39], built in as a core part
of the language, supports modern documentation conventions and code testing practices, and compiles cleanly
and quickly across a range of platforms.
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PI
Post-doc 1
2019
2020
2021
3.2 Laurent inversion and 3.4 smoothings
2.2 beyond toric
3.5 next-gen CAS
1.5 quantization
1.4 Virasoro
1.3 virtual birational geom.
1.1 subvarieties 1.5 quantization
1.2 fibrations
2.1 non-crepant toric c.i.
Post-doc 2
Post-doc 3
2018
GWT
3.1 polytope theory
2.2 beyond toric
2.3 GV invariants
3.5 next-gen CAS
3.2 Laurent inversion 3.3 computations
RA
parallel Hemmecke
parallel Gröbner
Figure 1. Program schedule and the allocation of projects (indicated by number) to team members.
Existing computational algebra systems are vast: it would be completely unrealistic to aim to match their scope
at this stage. But they are also hierarchical, and there are some natural areas to introduce first. Computing-side
structures include hashmap, sets, ordered lists, sorting, permuting, subset selection and knapsack. Mathematicsside areas include: integer and rational arithmetic; finite fields; polynomials; matrices and linear algebra over a
field. These are already enough to tackle significant problems, including the research challenges that will drive
the development of the new system. Once enough of the basic structure is in place, we will add new algorithms
that apply directly to the computations at the heart of strand 3: parallel algorithms for
• computing Hilbert bases for cones (parallelizing Hemmecke’s algorithm);
• computing Gröbner bases (the Faugère–Lacharte algorithm [32]).
There is a very significant technical challenge here but, given our experience and our existing codebase, we are
uniquely well-placed to succeed. Furthermore we will reduce risk by:
• ensuring that the basic development of the core of the system will take place in a small team, with a
single person responsible for the overall technical vision of the project. This will be the PI’s current
post-doc Dr Alexander Kasprzyk, who is one of the strongest mathematician-programmers in Europe.
The PI will assist Kasprzyk with implementation. (This is how we built our existing codebase.)
• making use of the state of the art. This project will build on and complement the existing PolSys project
GBLA, which is working towards multi-core parallel linear algebra for Grobner basis algorithms. (PolSys is a collaboration by INRIA/CNRS/Univ. Pierre et Marie Curie.) Dr C Eder, Dr C Fieker, and the
Singular team at Univ. Kaiserslautern have agreed to share expertise and hard examples.
• ensuring that the new system is tuned against additional challenging problems that are not from the PI’s
research program: from cryptography, and from the Imperial College Bioinformatics Support Service
(which provides computational and statistical analysis and consultancy to bioscientists).
Even the worst-case outcome here would be an exceptionally powerful Computational Algebra System which
is precisely adapted to, but also limited to, the PI’s research agenda. Note that it is likely that incremental
improvements to our existing codebase would allow us to explore up to dimension 5, so this project is not on
the critical path for the remainder of strand 3 (projects 3.1–3.4).
Links Between The Strands.
We close by describing two projects that lie at the intersection of all three strands of the program.
4.1. Fano 3-folds with Picard rank 1. By combining the Katz–Klemm–Vafa formula [67] (which expresses
reduced Gromov–Witten invariants of K3 surfaces in terms of modular forms) with the new Quantum Lefschetz
theorem for reduced Gromov–Witten invariants from project 1.1, we will give a new modular forms-based proof
of the classification of Fano 3-folds with Picard rank 1. This turns an observation by Golyshev – that there is a
correspondence between Fano 3-folds and certain modular forms – from a curiosity into a new proof of one of
the fundamental classification results in modern algebraic geometry.
4.2. Gromov–Witten Theory With Terminal Singularities. We have found examples of Laurent polynomials f such that applying Laurent inversion to f yields a Fano variety X (a complete intersection in a toric
orbifold) with terminal singularities. That suggests that one might be able to define Gromov–Witten invariants
of spaces with terminal singularities, and that Mirror Symmetry might extend to this realm. (At the moment
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Gromov–Witten invariants can only be defined for orbifolds.) Most terminal singularities do not admit crepant
resolutions but one might hope that, by looking at different non-crepant resolutions of a terminal variety X and
using the results of Strand 2, we could find the Gromov–Witten theory of X inside that of its resolutions. This
is necessarily speculative, since it involves ideas that are far into each strand of the program, but the outcome
would be striking: we would be able to define Gromov–Witten invariants in precisely the same category, that
of terminal varieties, that is singled out by the Minimal Model Program and birational geometry.
Section c. Resources (including project costs).
Cost Category
Total in Euro
Direct Personnel
PI
Costs
Senior Staff
Postdocs
Students
Other
i. Total Direct Costs for Personnel (in Euro)
Travel
Equipment
Other goods
Consumables – Equipment / Workshops/
and services
Conferences / Artists in residence
Publications (including Open Access fees), etc.
Other Audit.
ii. Total Other Direct Costs (in Euro)
A – Total Direct Costs (i + ii) (in Euro)
B – Indirect Costs (overheads) 25% of Direct Costs (in Euro)
C1 – Subcontracting Costs (no overheads) (in Euro)
C2 – Other Direct Costs with no overheads (in Euro)
Total Estimated Eligible Costs (A + B + C) (in Euro)
Total Requested EU Contribution (in Euro)6
377,335 €
0
670,571 €
274,262 €
0
1,322,168 €
94,048 €
0
152,945 €
15,867 €
14,968 €
277,828 €
1,599,996 €
399,999 €
0
0
1,999,995 €
1,999,995 €
For the above cost table, please indicate the % of working time the PI dedicates to
the project over the period of the grant:
50%
The PI commits to spending at least 50% of his time on the project over the period of the grant, and to spending
at least 80% of his working time in the EU during the period of the grant.
A Note on Consumables. The consumables total of e152,945 is made up of HPC equipment (e36,888),
artist-in-residence costs (e61992), and a research conference plus three workshops (e54,065). These costs are
justified below. The HPC equipment being requested is included under the consumables heading as it falls below the e72500 threshold required to be classified as equipment under Imperial College’s accounting practices.
This Project Requires A Team Of Researchers. This is because of the scale and ambition of the proposal:
there are far too many projects here for the PI to tackle alone. And a great strength of the program – its robustness – comes from making simultaneous progress on many fronts, with the ideas developed in the different
projects reinforcing and complementing each other. This requires a team. The project requires: 50% of the PI’s
time; 3 post-docs, employed for 3 years each; and one Research Assistant, employed for 4 years. The post-docs
should be experienced researchers with appropriate backgrounds and at least 2–3 years of research experience.
Post-doc 1 (in strand 1) should have expertise in Gromov–Witten theory or a related branch of algebraic geometry; post-doc 2 (in strand 2) should have expertise in birational geometry, moduli theory, or FJRW theory;
post-doc 3 (in strand 3) should have expertise in polyhedral combinatorics and scientific computing; and the
Research Assistant (in strand 3) should have skills and experience in scientific computing and algorithm design.
Team Structure and Team Roles. Each post-doc will work in collaboration with, and closely supervised by,
the PI. The RA will be co-supervised by the PI and post-doc 3. The program schedule and the allocation of
projects to team members is shown in the Gantt chart on page 12. Note that the PI will be responsible for the
highest-risk parts of the program.
We will hold two working seminars each week, one with the PI, post-docs 1 and 2, and my colleague Dr
Manolache (for strands 1 and 2), and the other with the PI, post-doc 3, the RA, and my colleagues Dr Kasprzyk
and Prof. Corti (for strand 3).
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High-Performance Computing (HPC) Equipment. The program proposed here will be driven by large-scale
computational experiments, which require significant HPC equipment. My research group currently uses an
HPC setup (200 cores) funded by the UK Engineering and Physical Sciences Research Council, which is likely
to come to the end of its life during the program proposed here. (It is already out of warranty.) Our existing
HPC setup has been almost 100% utilised ever since it was purchased, and I expect this to be true for the HPC
setup requested here too (160 cores with 1Tb of RAM and 16Tb disk space). The HPC equipment will be
housed in the Imperial College High Performance Computing center, with technical support and maintenance
provided at no cost to the ERC. It will be dedicated to the exclusive use of the PI and this project.
Existing Resources That Will Contribute To The Project. The following existing resources at Imperial
College will contribute to this project, at no cost to the ERC. My colleague Dr Manolache (formerly a Marie
Curie post-doc in my group; now a Royal Society Dorothy Hodgkin Fellow at Imperial College) has agreed to
commit 50% of her time to this project, both for research collaboration and co-supervising post-doc 1 and a
PhD student. My Department will provide two studentships. Imperial College will provide HPC resources as
described above, as well as shared access to the Imperial College 16,000 core HPC cluster.
Travel Costs and Visitor Program. The budget includes travel costs for the PI and post-docs (each, per year:
1 US/Asian conference, 2 European conferences/research visits, 3 UK research visits) and the RA (in total: 1
US/Asian conference, 3 European conferences, 3 UK research visits). Total team travel costs: e54K. It also
includes funding (of e8K per year) to invite leading mathematicians in the field from around the world for short
visits, to give intensive mini-courses to the team. These visits will be purely intellectual in nature and none of
the visitors will receive any ERC funding other than travel and subsistence expenses.
Dissemination Activities. The proposal includes funding for Open Access publication costs (3 papers per year
over the life of the grant), 3 research workshops, and a high-profile international conference. Workshop costs
are based on the actual costs of workshops that I organized this year: for each 2-day workshop, travel and
accommodation costs for 10 participants from the UK (outside London), and 5 participants from elsewhere in
Europe, for a total of e10.5K per workshop. The conference costs are based on the actual costs of a conference
organised this year by one of my colleagues: travel and accommodation costs for 5 participants from the US or
Asia, 10 participants from the UK (outside London), and 10 participants from elsewhere in Europe, plus venue
hire and refreshments, for a total of e23K.
Public Engagement. The proposal includes funding for two residencies by visual artists at Imperial College (3
days/week for 9 months each), and associated costs for materials, exhibitions, and outreach events. Costs here
are based on the actual costs of the residency at Imperial by my artist collaborator Gemma Anderson in 2012.
A World-Leading Team. A project of this scale and ambition requires an ERC grant to succeed. Supported by
my existing ERC Starting Investigator Grant, I have built a world-leading research group at Imperial College
London, with a distinctive perspective and the prestige to attract the strongest post-docs and PhD students from
around the world. This proposal will consolidate and build on that success, creating and maintaining a European
centre of excellence in one of the most active and internationally-competitive fields of mathematics.
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14
COATES
Part B2
GWT
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