Mar 19, 2018
Monday
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09:30 AM - 09:45 AM
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Introduction
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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09:45 AM - 10:45 AM
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Riemann-Hilbert problems from Donaldson-Thomas theory
Tom Bridgeland (University of Sheffield)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
I will discuss a class of Riemann-Hilbert problems which arise naturally from the wall-crossing formula in Donaldson-Thomas 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
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10:45 AM - 11:15 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:15 AM - 12:15 PM
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The cohomological Hall algebra and M-theory
Kevin Costello (Perimeter Institute of Theoretical Physics)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
I'll describe a conjectural way to understand the COHA of certain toric Calabi-Yau three-folds in terms of a 5-dimensional gauge theory given by compactifying M-theory on the Calabi-Yau. An analysis of the 5-dimensional 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 Calabi-Yau
- Supplements
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12:15 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Wall-crossing in Gromov-Witten and Landau-Ginzburg theory
Emily Clader (San Francisco State University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
The theory of quasi-maps, developed in recent work of Ciocan-Fontanine and Kim, is a generalization of Gromov-Witten theory that depends on an additional stability parameter varying over positive rational numbers. When that parameter tends to infinity, Gromov-Witten theory is recovered, while when it tends to zero, the resulting theory encodes information related to the "B-model." Ciocan-Fontanine and Kim proved a wall-crossing 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
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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A proof of the Donaldson-Thomas crepant resolution conjecture
Jørgen Rennemo (University of Oslo)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
: Let X be a Calabi-Yau 3-dimensional 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 1-dimensional), the DT crepant resolution conjecture of Bryan-Cadman-Young 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
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Mar 20, 2018
Tuesday
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09:30 AM - 10:30 AM
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Quiver gauge theories and Kac-Moody Lie algebras
Hiraku Nakajima (Kavli Institute for the Physics and Mathematics of the Universe )
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Periods and quasiperiods of modular forms and D-brane masses on the quintic
Albrecht Klemm (Hausdorff Research Institute for Mathematics, University of Bonn)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
We consider one complex structure parameter mirror families $W$ of Calabi-Yau 3-folds with Picard-Fuchs equations of hypergeometric type. By mirror symmetry the even D-brane masses of the orginal Calabi-Yau $M$ 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 D-brane 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 L-values) of $f_4$. These values also determine the behaviour of the Weil-Petersson 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 3-folds and comment on simpler applications to local Calabi-Yau 3-folds and polarized K3 surfaces.
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Enumerative geometry using A1 homotopy theory
Kirsten Wickelgren (Duke University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
There is recent work by Marc Hoyois, Marc Levine, Jesse Kass and myself giving enrichments of enumerative results to the Grothendieck-Witt group of quadratic forms. I will discuss joint work with Jesse Kass along these lines
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Categorified BPS invariants for the category of coherent sheaves on a CY3
Nicholas Davison (University of Edinburgh)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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 cycle-type sheaf defined by Ben-Bassat, 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 above-mentioned 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
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04:30 PM - 06:00 PM
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Reception/ Poster Session
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- Location
- Atrium and Commons area
- Video
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Mar 21, 2018
Wednesday
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09:00 AM - 10:00 AM
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Sheaf quantization of the exact symplectic category
Vivek Shende (University of California, Berkeley)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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
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10:00 AM - 10:30 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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10:30 AM - 11:30 AM
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Lie brackets on the homology of moduli spaces, and wall-crossing formulae
Dominic Joyce (University of Oxford)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Let K be a field, and M be the “projective linear" moduli stack of objects in a suitable K-linear abelian category A (such as the coherent sheaves coh(X) on a smooth projective K-scheme 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 Ringel-Hall algebra. There is also a differential-geometric version of this: if X is a compact manifold with a geometric structure giving instanton-type equations (e.g. oriented Riemannian 4-manifold, 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 wall-crossing formulae under change of stability condition. Several important classes of invariants in algebraic and differential geometry — (higher rank) Donaldson invariants of 4-manifolds (in particular with b^2_+=1), Mochizuki invariants counting semistable coherent sheaves on surfaces, Donaldson-Thomas type invariants for CY 3-folds, Fano 3-folds, and CY 4-folds — 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 wall-crossing formula (from my previous work on motivic invariants) in the Lie algebra (H_*(M), [ , ]).
- Supplements
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11:30 AM - 11:45 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:45 AM - 12:45 PM
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Mirror symmetry and structures of higher genus invariants
Chiu-Chu Melissa Liu (Columbia University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
The Remodeling Conjecture proposed by Bouchard-Klemm-Marino-Pasquetti (BKMP) relates all genus open and closed Gromov-Witten invariants of a symplectic toric Calabi-Yau 3-manifold/3-orbifold to the Eynard-Orantin invariants of the mirror curve of the toric Calabi-Yau 3-fold. It is a version of all genus open-closed mirror symmetry. The goal of this talk is to describe implications of the Remodeling Conjecture on structures of higher genus open-closed Gromov-Witten invariants.
- Supplements
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Mar 22, 2018
Thursday
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09:30 AM - 10:30 AM
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Knot Categorification from Geometry (via String Theory)
Mina Aganagic (University of California, Berkeley)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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 A-branes in the mirror Landau-Ginzburg 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
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Quantum mirrors of log Calabi-Yau surfaces and higher genus curve counting
Pierrick Bousseau (University of Georgia)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Gross-Hacking-Keel have given a construction of mirror families of log Calabi-Yau surfaces in terms of counts of rational curves. I will explain how to deform this construction by counts of higher genus curves to get non-commutative deformations of these mirror families. The proof of the consistency of this deformed construction relies on a recent tropical correspondence theorem
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Matrix factorizations in Gromov-Witten theory
Mark Shoemaker (Colorado State University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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 Gromov-Witten theory, FJRW theory, and so-called hybrid models. This is joint work with Ciocin-Fontanine, Favero, Guere, and Kim
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds
Yukinobu Toda (Kavli Institute for the Physics and Mathematics of the Universe )
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
In this talk, I will discuss birational geometry for Joyce’s d-critical loci, by introducing notions such as ‘d-critical flips’, ‘d-critical flops’, etc. I will show that several wall-crossing phenomena of moduli spaces of stable objects on Calabi-Yau 3-folds are described in terms of d-critical birational geometry, e.g. certain wall-crossing diagrams of Pandharipande-Thomas stable pair moduli spaces forma a d-critical minimal model program. I will also show the existence of semi-orthogonal decompositions of the derived categories under simple d-critical flips satisfying some conditions. This is motivated by a d-critical analogue of Bondal-Orlov, Kawamata’s D/K equivalence conjecture, and also gives a categorification of wall-crossing formula of Donaldson-Thomas invariants
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Mar 23, 2018
Friday
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09:30 AM - 10:30 AM
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On counting curves in Calabi-Yau threefolds
Georg Oberdieck (Massachusetts Institute of Technology)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
The example of K3xE suggests that computing the partition function of a Calabi-Yau threefold requires a good understanding of both its Gromov-Witten and Donaldson-Thomas theory. I will explain two results, one for each theory, related to this discussion.
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Log compactifications of GLSM moduli spaces
Felix Janda (University of Michigan)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
By the work of H.-L. Chang and J. Li, the (arbitrary genus) Gromov-Witten invariants of a quintic threefold can be computed using the relatively simple moduli space of stable maps to P^4 with a p-field.
This is an example of a GLSM moduli space. Another example is the moduli of r-spin curves with a field, which by the work of Chang-Li-Li computes Witten's r-spin 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 non-compact 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
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Tautological integrals on Hilbert schemes of points
Alina Marian (Northeastern University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Enumerative symplectic duality
Andrei Okounkov (Columbia University; University of California, Berkeley)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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 present-day technology.
- Supplements
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