Mar 19, 2018
Monday

09:30 AM  09:45 AM


Introduction

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
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09:45 AM  10:45 AM


RiemannHilbert problems from DonaldsonThomas theory
Tom Bridgeland (University of Sheffield)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will discuss a class of RiemannHilbert problems which arise naturally from the wallcrossing formula in DonaldsonThomas theory. As well as giving an explicit description of the simplest example, I will try to explain the motivating analogy with the deformed flat connection in the theory of Frobenius manifolds.
 Supplements



10:45 AM  11:15 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
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11:15 AM  12:15 PM


The cohomological Hall algebra and Mtheory
Kevin Costello (Perimeter Institute of Theoretical Physics)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I'll describe a conjectural way to understand the COHA of certain toric CalabiYau threefolds in terms of a 5dimensional gauge theory given by compactifying Mtheory on the CalabiYau. An analysis of the 5dimensional gauge theory leads to a conjectural expression for the COHA as the positive part of the double current algebra built from a supergroup. This expression reproduces known formulae for the refined DT invariants of the toric CalabiYau
 Supplements



12:15 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
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02:00 PM  03:00 PM


Wallcrossing in GromovWitten and LandauGinzburg theory
Emily Clader (San Francisco State University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The theory of quasimaps, developed in recent work of CiocanFontanine and Kim, is a generalization of GromovWitten theory that depends on an additional stability parameter varying over positive rational numbers. When that parameter tends to infinity, GromovWitten theory is recovered, while when it tends to zero, the resulting theory encodes information related to the "Bmodel." CiocanFontanine and Kim proved a wallcrossing formula exhibiting how the theory changes with the stability parameter, and in this talk, we discuss an alternative proof of their result as well as a generalization to other gauged linear sigma models. This is joint work with Felix Janda and Yongbin Ruan.
 Supplements



03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
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03:30 PM  04:30 PM


A proof of the DonaldsonThomas crepant resolution conjecture
Jørgen Rennemo (University of Oslo)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
: Let X be a CalabiYau 3dimensional orbifold, and let Y be a crepant resolution of the coarse moduli space of X. When X satisfies the "hard Lefschetz condition" (that is, when the fibres of the resolution are at most 1dimensional), the DT crepant resolution conjecture of BryanCadmanYoung gives a precise relation between the DT curve counts of X and Y. I will explain a proof of this conjecture via wall crossing and the motivic Hall algebra. This is joint work with Sjoerd Beentjes and John Calabrese.
 Supplements




Mar 20, 2018
Tuesday

09:30 AM  10:30 AM


Quiver gauge theories and KacMoody Lie algebras
Hiraku Nakajima (Kavli Institute for the Physics and Mathematics of the Universe )

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Quiver gauge theories give two types of algebraic symplectic varieties, which are called quiver varieties and Coulomb branches respectively. The first ones were introduced by the speaker in 1994, and their homology groups are representations of Kac–Moody Lie algebras. The second ones were introduced by the speaker and Braverman, Finkelberg in 2016. The two types of varieties are very different (e.g., dimensions are different), but are expected to be related in rather mysterious ways. As an example of mysterious links, I would like to explain a conjectural realization of Kac–Moody Lie algebra representations on homology groups of Coulomb branches, which the speaker proves in affine type A. It nicely matches with geometric Satake correspondence for usual finite dimensional complex simple groups and loop groups
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
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11:00 AM  12:00 PM


Periods and quasiperiods of modular forms and Dbrane masses on the quintic
Albrecht Klemm (Hausdorff Research Institute for Mathematics, University of Bonn)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We consider one complex structure parameter mirror families $W$ of CalabiYau 3folds with PicardFuchs equations of hypergeometric type. By mirror symmetry the even Dbrane masses of the orginal CalabiYau $M$ can be identified with four periods w.r.t. to an integral symplectic basis of $H_3(W,Z)$ at the point of maximal unipotent monodromy. It was discovered by Chad Schoen in 1986 that the singular fibre of the quintic at the conifold point gives rise to a Hecke eigen form of weight four $f_4$ on $\Gamma_0(25)$ whose Fourier coefficients $a_p$ are determined by counting solutions in that fibre over the finite field $\mathbb{F}_{p^k}$. The Dbrane masses at the conifold are given by the transition matrix $T_{mc}$ between the integral symplectic basis and a Frobenius basis at the conifold. We predict and verify to very high precision that the entries of $T_{mc}$ relevant for the D2 and D4 brane masses are given by the two periods (or Lvalues) of $f_4$. These values also determine the behaviour of the WeilPetersson metric and its curvature at the conifold. Moreover we describe a notion of quasi periods and find that the two quasi period of $f_4$ appear in $T_{mc}$. We extend the analysis to the other hypergeometric one parameter 3folds and comment on simpler applications to local CalabiYau 3folds and polarized K3 surfaces.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
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02:00 PM  03:00 PM


Enumerative geometry using A1 homotopy theory
Kirsten Wickelgren (Duke University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
There is recent work by Marc Hoyois, Marc Levine, Jesse Kass and myself giving enrichments of enumerative results to the GrothendieckWitt group of quadratic forms. I will discuss joint work with Jesse Kass along these lines
 Supplements



03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
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03:30 PM  04:30 PM


Categorified BPS invariants for the category of coherent sheaves on a CY3
Ben Davison (University of Glasgow)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
This talk is on joint work with Sven Meinhardt. The (refined) BPS invariants for a CY3 category C (equipped with orientation data) are defined as plethystic logarithms of partition functions recording the weights on the total hypercohomology of the vanishing cycletype sheaf defined by BenBassat, Brav, Bussi, Dupont and Joyce. In the case that C=Coh(X), for X a CY3 variety, the integrality conjecture states that these plethystic logarithms produce Laurent polynomials, as opposed to Laurent formal power series. Then, setting q^{1/2}=1 we recover (more down to earth!) BPS/DT invariants.
Heuristically, these polynomials are the weight polynomials for the "space of BPS states"  although there is no definition of such a space in the picture, hence the inverted commas. We have shown that this picture can be categorified, in the sense that the abovementioned hypercohomology is isomorphic to the symmetric algebra generated by an explicit mixed Hodge structure, which we can then define to be the mixed Hodge structure on the cohomology of the space of BPS states. This produces the following strengthening of the integrality conjecture: for a fixed stability condition and slope, there is a Lie algebra g, endowed with a mixed Hodge structure, graded by Chern classes of the same fixed slope, and for which each graded piece is finite dimensional, whose graded components recover the refined BPS invariants as their weight polynomials.
 Supplements



04:30 PM  06:00 PM


Reception/ Poster Session

 Location
 Atrium and Commons area
 Video


 Abstract
 
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Mar 21, 2018
Wednesday

09:00 AM  10:00 AM


Sheaf quantization of the exact symplectic category
Vivek Shende (University of California, Berkeley)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will explain how, using microlocal sheaf theory, one can associate categories to exact symplectic manifolds, objects to exact lagrangians, and functors to exact lagrangian correspondences. Time permitting I will also discuss comparison with the Fukaya category, and applications to homological mirror symmetry.
 Supplements



10:00 AM  10:30 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
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10:30 AM  11:30 AM


Lie brackets on the homology of moduli spaces, and wallcrossing formulae
Dominic Joyce (University of Oxford)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Let K be a field, and M be the “projective linear" moduli stack of objects in a suitable Klinear abelian category A (such as the coherent sheaves coh(X) on a smooth projective Kscheme X) or triangulated category T (such as the derived category D^bcoh(X)). I will explain how to define a Lie bracket [ , ] on the homology H_*(M) (with a nonstandard grading), making H_*(M) into a graded Lie algebra. This is a new variation on the idea of RingelHall algebra. There is also a differentialgeometric version of this: if X is a compact manifold with a geometric structure giving instantontype equations (e.g. oriented Riemannian 4manifold, G2 manifold, Spin(7) manifold) then we can define Lie brackets both on the homology of the moduli spaces of all U(n) or SU(n) connections on X for all n, and on the homology of the moduli spaces of instanton U(n) or SU(n) connections on X for all n. All this is (at least conjecturally) related to enumerative invariants, virtual cycles, and wallcrossing formulae under change of stability condition. Several important classes of invariants in algebraic and differential geometry — (higher rank) Donaldson invariants of 4manifolds (in particular with b^2_+=1), Mochizuki invariants counting semistable coherent sheaves on surfaces, DonaldsonThomas type invariants for CY 3folds, Fano 3folds, and CY 4folds — are defined by forming virtual classes for moduli spaces of “semistable” objects, and integrating some cohomology classes over them. The virtual classes live in the homology of the “projective linear" moduli stack. Yuuji Tanaka and I are working on a way to define virtual classes counting strictly semistables, as well as just stables / stable pairs. I conjecture that in all these theories, the virtual classes transform under change of stability condition by a universal wallcrossing formula (from my previous work on motivic invariants) in the Lie algebra (H_*(M), [ , ]).
 Supplements



11:30 AM  11:45 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:45 AM  12:45 PM


Mirror symmetry and structures of higher genus invariants
ChiuChu Melissa Liu (Columbia University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The Remodeling Conjecture proposed by BouchardKlemmMarinoPasquetti (BKMP) relates all genus open and closed GromovWitten invariants of a symplectic toric CalabiYau 3manifold/3orbifold to the EynardOrantin invariants of the mirror curve of the toric CalabiYau 3fold. It is a version of all genus openclosed mirror symmetry. The goal of this talk is to describe implications of the Remodeling Conjecture on structures of higher genus openclosed GromovWitten invariants.
 Supplements




Mar 22, 2018
Thursday

09:30 AM  10:30 AM


Knot Categorification from Geometry (via String Theory)
Mina Aganagic (University of California, Berkeley)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will describe three paths to categorification of RTW invariants of knots, and the relations between them. The first is based on a category of B type branes on resolutions of slices in affine Grassmannians, the second on a category of Abranes in the mirror LandauGinzburg theory. The third is the approach based on counting solutions to five dimensional
equations with gauge theory origin. All three approaches can be deduced starting from a string theory in six dimensions. This is based in joint works with Andrei Okounkov and with Dimitrii Galakhov.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Quantum mirrors of log CalabiYau surfaces and higher genus curve counting
Pierrick Bousseau (Imperial College, London)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
GrossHackingKeel have given a construction of mirror families of log CalabiYau surfaces in terms of counts of rational curves. I will explain how to deform this construction by counts of higher genus curves to get noncommutative deformations of these mirror families. The proof of the consistency of this deformed construction relies on a recent tropical correspondence theorem
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Matrix factorizations in GromovWitten theory
Mark Shoemaker (Colorado State University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Originally introduced by Eisenbud in the context of commutative algebra, matrix factorizations have since earned a prominent role in mathematical physics. In this talk I will describe recent work using matrix factorizations to define a cohomological field theory given the input data of a gauged linear sigma model. This construction gives a new description of the virtual class in GromovWitten theory, FJRW theory, and socalled hybrid models. This is joint work with CiocinFontanine, Favero, Guere, and Kim
 Supplements



03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Birational geometry for dcritical loci and wallcrossing in CalabiYau 3folds
Yukinobu Toda (Kavli Institute for the Physics and Mathematics of the Universe )

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In this talk, I will discuss birational geometry for Joyce’s dcritical loci, by introducing notions such as ‘dcritical flips’, ‘dcritical flops’, etc. I will show that several wallcrossing phenomena of moduli spaces of stable objects on CalabiYau 3folds are described in terms of dcritical birational geometry, e.g. certain wallcrossing diagrams of PandharipandeThomas stable pair moduli spaces forma a dcritical minimal model program. I will also show the existence of semiorthogonal decompositions of the derived categories under simple dcritical flips satisfying some conditions. This is motivated by a dcritical analogue of BondalOrlov, Kawamata’s D/K equivalence conjecture, and also gives a categorification of wallcrossing formula of DonaldsonThomas invariants
 Supplements




Mar 23, 2018
Friday

09:30 AM  10:30 AM


On counting curves in CalabiYau threefolds
Georg Oberdieck (Massachusetts Institute of Technology)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The example of K3xE suggests that computing the partition function of a CalabiYau threefold requires a good understanding of both its GromovWitten and DonaldsonThomas theory. I will explain two results, one for each theory, related to this discussion.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Log compactifications of GLSM moduli spaces
Felix Janda (University of Michigan)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
By the work of H.L. Chang and J. Li, the (arbitrary genus) GromovWitten invariants of a quintic threefold can be computed using the relatively simple moduli space of stable maps to P^4 with a pfield.
This is an example of a GLSM moduli space. Another example is the moduli of rspin curves with a field, which by the work of ChangLiLi computes Witten's rspin class. One subtlety in using these moduli spaces for computations lies in the cosection localized virtual class that is necessary to produce invariants from these noncompact moduli spaces.
In my talk, I will discuss joint work in progress with Q. Chen, Y.
Ruan on how to derive the same invariants from an ordinary virtual class on a compactified moduli space constructed using logarithmic geometry. This virtual class is amenable to torus localization methods.
In the case of a quintic threefold, in joint work in progress with S.
Guo and Y. Ruan, we apply this method to prove the holomorphic anomaly equations. There are further applications in joint work with A.
Sauvaget and D. Zvonkine, and with X. Wang.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Tautological integrals on Hilbert schemes of points
Alina Marian (Northeastern University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Recently there has been progress in evaluating the series of Segre integrals associated with tautological vector bundles on Hilbert schemes of points on surfaces. The talk will explain some of the new ideas in the subject, including the connection between Segre theory and Verlinde theory in the Hilbert scheme context. It will be mostly based on joint work with Dragos Oprea and Rahul Pandharipande.
 Supplements



03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Enumerative symplectic duality
Andrei Okounkov (Columbia University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
This will be a report on a joint work in progress with Mina Aganagic.
Its goal is to prove the equality of curve counts in dual geometries, whenever both counts can be defined using presentday technology.
 Supplements



