Title: Symplectic Invariants on Calabi-Yau 3 folds, Modularity and Stability
Speaker: Albrecht Klemm
Abstract: We discuss techniques to calculate symplectic invariants on CY 3-folds $M$, namely Gromov-Witten (GW) invariants, Pandharipande-Thomas (PT) invariants, and Donaldson-Thomas (DT) invariants. Physically the latter are closely related to BPS brane bound states in type IIA string compactifications on $M$. We focus on the rank $r_{\bar 6}=1$ DT invariants that count $\bar D6-D2-D0$ brane bound states related to PT- and high genus GW invariants, which are calculable by mirror symmetry and topological string B-model methods modulo certain boundary conditions, and the rank zero DT invariants that count rank $r_4=1$ $D4-D2-D0$ brane
bound states. It has been conjectured by Maldacena, Strominger, Witten and Yin that the latter are governed by an index that has modularity properties to due $S-$ duality in string theory and extends to a mock modularity index of higher depth for $r_4>1$. Again the modularity allows to fix the at least the $r_4=1$ index up to boundary conditions fixing their polar terms. Using Wall crossing formulas obtained by Feyzbakhsh certain PT invariants close to the Castelnouvo bound can be related to the $r_4=1,2$ $D4-D2-D0$ invariants. This provides further boundary conditions for topological string B-model approach as well as for the $D4-D2-D0$ brane indices.
The approach allows to prove the Castenouvo bound and calculate the $r_{\bar 6}=1$ DT- invariants or the GW invariants to higher genus than hitherto possible.
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Title: From Gromov-Witten to Donaldson-Thomas invariants via Resurgence
Speaker: Murad Alim
Abstract: Enumerative invariants of Calabi-Yau manifolds are most naturally organized in terms of partition functions of physical theories. Higher genus Gromov-Witten invariants of CY threefolds correspond to the expansion coefficients of a series in a formal parameter which corresponds to the topological string coupling. This series is however only asymptotic. I will show how the analysis of finite difference equations and Borel summation reveals the piecewise analytic structure behind the asymptotic expansion. The resulting Stokes jumps of the piecewise analytic structure encode another set of enumerative invariants of the threefold, namely Donaldson-Thomas invariants.
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Title: Bounding Lagrangian intersections using Floer homotopy theory
Speaker: Kenneth Blakey [MIT]
Abstract: I will describe a new lower bound on the number of intersection points of a Lagrangian pair, in the exact setting, using Steenrod squares on Lagrangian Floer cohomology which are defined via a Floer homotopy type.
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Title: Topological strings on nodal Calabi-Yau with topologically non-trivial B-fields
Speaker: Thorsten Schimannek
Abstract: "I will review a recent proposal for the interpretation of the A-model topological string partition function on nodal Calabi-Yau threefolds that carry a flat but topologically non-trivial B-field.
From an enumerative perspective, this is conjectured to encode a refinement of the usual Gopakumar-Vafa invariants with respect to a torsion curve class that only exists in small resolutions that have a trivial canonical class but are not Kaehler. After illustrating the phenomenon in some examples, my focus will be on the relation under mirror symmetry to variations of Hodge structure that have an atypical integral structure.
I will explain how this integral structure can be interpreted in terms of the singular geometries and the B-field and discuss interesting open question"
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Title: Enumerative invariants via Wilson loops
Speaker: Sheldon Katz
Abstract: A compact Calabi-Yau threefold X can be considered to approach a local Calabi-Yau threefold Y in certain limits of Kahler moduli. The enumerative geometry of Y is simpler and physical tools of 5-dimensional gauge theory apply, particularly the use of Wilson loops. Roughly speaking, refined Wilson loop amplitudes are refined BPS numbers of blowups of Y at generic points of the compact surface contained in Y. A general proposal is made for the structure of the refined BPS numbers of X in terms of refined Wilson loops. Using low degree geometric computation and a refined holomorphic anomaly equation for the Wilson loop partition function, refined BPS numbers of elliptically fibered X are computed for many examples, with h^{1,1}(X) as large as 5. In particular, the unrefined limit produces higher genus BPS invariants of X. This talk is based on joint work with Minxin Huang, Albrecht Klemm, and Xin Wang.
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Title: Logarithmic tautological rings
Speaker: Johannes Schmitt
Abstract: "Many moduli spaces arising in enumerative geometry (of smooth curves, abelian varieties, ...) are a priori non-compact. To define enumerative invariants, it is natural to construct a compactification with normal crossings boundary and to study its intersection theory. However, often there are different choices for such compactifications, with some more convenient than others depending on the geometric problem at hand.
In this talk, I introduce the logarithmic Chow ring logCH*, which encodes the intersection theory of all such (suitably chosen) compactifications simultaneously. We will see how natural cycle classes in this ring can be constructed from purely combinatorial/convex geometric data. I discuss applications to double Hurwitz numbers and explicit calculations of logCH* of moduli spaces of rational curves. Finally, I present some recent joint work with Pandharipande, Ranganathan and Spelier, which combines these convex geometric inputs with the intersection theory of strata on our moduli space, and allows to define an additive generating set of logCH* modelled on the tautological classes of the moduli space of stable curves."
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Title: Gromov-Witten theory of Enriques surfaces and quasi-modular forms
Speaker: Georg Oberdieck
Abstract: "It is well-known that the descendent Gromov-Witten potentials of an elliptic curve are quasi-modular forms. In this talk, we present a conjecture that descendent Gromov-Witten potentials of an Enriques surface are quasi-modular forms for the orthogonal group of the Enriques lattice. Orthogonal quasi-modular forms here is a new notion that we introduce. The stated conjectures fit into the larger question regarding the modular interpretation of relative Gromov-Witten potentials of K3 and abelian surface fibrations. The talk is based on joint work with Brandon Williams."
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Title: Double ramification cycles, admissible covers, and the top degree part
Speaker: Aaron Pixton
Abstract: The double ramification (DR) cycle parametrizes curves admitting maps to the projective line with specified ramification profiles over two points. I will begin by discussing joint work with Q. Zhao giving a new formula for the DR cycle in degree one as well as a new conjectural formula in higher degree. This higher degree formula writes the DR cycle as the corresponding admissible covers cycle plus additional error terms coming from contracted components. This can be viewed as motivation for believing an older conjecture, that DR cycles and admissible covers cycles have the same asymptotic growth as the degree and ramification profiles are scaled up. I will conclude by defining the top degree part of the DR cycle and giving some identities that it satisfies, presenting joint work with Y. Bae and S. Molcho.
Title: Castelnuovo bound conjecture and curve-counting invariants
Speaker: Zhiyu Liu
Abstract: The Castelnuovo bound conjecture, which is proposed by physicists, predicts an effective vanishing result for Gopakumar-Vafa invariants of Calabi-Yau 3-folds of Picard number one. In this talk, I will introduce recent advances toward solving this conjecture and discuss relevant results about the bounds of genus of curves in projective threefolds.
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Title: Perverse filtrations and refined Gopakumar-Vafa invariants for local surfaces
Speaker: Junliang Shen
Abstract: I will discuss some roles played by perverse filtrations in enumerative geometry of local surfaces. In physics, this connection was essentially pointed out by Gopakumar-Vafa (via quite different language). It was mathematically formulated by Hosono-Saito-Takahashi, Kiem-Li, Maulik-Toda, etc. I will discuss 3 particular cases (1) local curves T*C (2) K3 surfaces, and (3) del Pezzo surfaces, where the Gopakumar-Vafa theory is tightly connected to geometry and topology of (1) character varieties of surface groups, (2) Lagrangian fibrations of compact hyper-Kaehler manifolds, and (3) Le Potier’s moduli spaces for CP2. Open questions in each case will be discussed.
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Title: Smoothing $L^\infty$ Riemannian metrics with nonnegative scalar curvature outside of a singular set
Speaker: Paula Burkhardt-Guim [Stony Brook University]
Abstract: Abstract: We show that any $L^\infty$ Riemannian metric $g$ on $R^n$ that is smooth with nonnegative scalar curvature away from a singular set of finite $(n-\alpha)$-dimensional Minkowski content, for some $\alpha>2$, admits an approximation by smooth Riemannian metrics with nonnegative scalar curvature, provided that $g$ is sufficiently close in $L^\infty$ to the Euclidean metric. The approximation is given by time slices of the Ricci-DeTurck flow, which converge locally in $C^\infty$ to $g$ away from the singular set. We also identify conditions under which a smooth Ricci-DeTurck flow starting from a $L^\infty$ metric that is uniformly bilipschitz to Euclidean space and smooth with nonnegative scalar curvature away from a finite set of points must have nonnegative scalar curvature for positive times.
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Title: Modular forms from Betti numbers
Speaker: Pierrick Bousseau
Abstract: Modular forms are complex analytic functions with striking symmetries, which play a fundamental role in number theory. In the last few decades, there have been a series of astonishing predictions from theoretical physics that various basic mathematical numbers when put in a generating series, end up being modular forms when there is no known mathematical reason for such hidden structure. In this talk, we will first provide a gentle introduction to modular forms. We will then focus on spaces parametrizing complex plane algebraic curves with line bundles, and prove that generating series of their Betti numbers are modular forms. This verifies physical predictions, using various tools of modern enumerative algebraic geometry. Part of this is joint work with Honglu Fan, Shuai Guo, and Longting Wu.
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Title: Global Kuranishi charts for symplectic Gromov-Witten theory
Speaker: Amanda Hirschi
Abstract: In 2021, Abouzaid, McLean and Smith constructed a presentation of the moduli space of pseudo-holomorphic stable rational curves in symplectic manifold that dramatically simplifies the construction of a virtual fundamental class. I will describe a generalisation of their construction to higher genus and some applications to symplectic Gromov-Witten theory. This is partially joint work with Mohan Swaminathan.
Title: Tropical psi classes and tropicalizations of psi classes
Speaker: Renzo Cavalieri
Abstract: The general rule for the interactions between tropical geometry and moduli spaces of course is the following: everything you may wish is going to work like a charm in genus zero, and break down horribly in higher genus. This is the case for the tautological intersection theory of psi classes, a class of fundamental objects in the geometry of moduli spaces of curves: the generating function of their intersection numbers has made waves, pun intended, when it was noticed that it is a tau function for the KdV hierarchy. Back to tropical geometry: in genus zero tropical psi classes have been first defined by Mikhalkin in the early 2000's, then through the work of Kerber-Markwig and Katz it was shown that intersection numbers of tropical psi classes agree with their algebraic counterparts.
In work with A.Gross and H.Markwig (2021), we were able to make sense of tropical psi classes in higher genus, by making the tropical moduli space of curves into a stack for families of tropical curves with an affine structure. This is a combinatorial theory that recovers the algebraic intersection numbers, but can also produce results that do not have a counter part in algebraic geometry. To this end, in recent work with A.Gross we answer the question of when we can show that tropical psi classes are tropicalizations. In order to even make sense of the statement, we had to introduce a notion of tropicalization for families of curves based on the Picard theory of the base. "
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Title: Lorentz-covariant description of relativistic guiding-center motion
Abstract: When a charged particle moves through a magnetic field, it undergoes rapid cyclotron motion along with a slower movement of the guiding center. The guiding center primarily follows the magnetic field lines but gradually drifts away from them. Developing a manifestly Lorentz-covariant description of this motion has been a long-standing challenge in plasma physics and astrophysics. Although equations of motion formulated in the 1960s were implicitly Lorentz invariant, they were written in a cumbersome component form, which hindered any straightforward extension to general relativity until very recently. We present a relativistically invariant action for the guiding center that reproduces all known results. We further generalize this framework to curved spacetime, demonstrating that the so-called "curvature drift" and "gravitational drift" of the guiding center are connected by Einstein's principle of equivalence. This talk is based on arXiv:2405.08073.
Title: Gromov-Witten theory of complete intersections
Speaker: Hülya Argüz
Abstract: I will explain an algorithm for calculating all genus Gromov-Witten invariants of complete intersections in projective spaces. While the focus of previous work has been primarily on invariants with insertions pulled back from the ambient projective space, we study invariants with arbitrary insertions, in particular with ``primitive insertions'' that are not pulled back from the ambient space. I will describe several techniques we developed to achieve this, utilizing monodromy, degeneration, and nodal relative geometry. This is joint work with Pierrick Bousseau, Rahul Pandharipande, and Dimitri Zvonkine.
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The Simons Center is pleased to present pianist Liya Nigmati,
Mirrors of Sound: A Musical Odyssey
Solo recital performance at the Simons Center for Geometry and Physics
Wednesday February 12, 2025
5:00pm, Della Pietra Family Auditorium, room 103
Program
G. Mahler ‘Adagietto’ from Symphony No. 5 (arr. by O. Singer) (11 minutes)
J. S. Bach ‘Italian Concerto’ (13 minutes)
C. Debussy ‘Reflets dans l’eau’, ‘Hommage à Rameau’, ‘Mouvement in C’ from Images (Book 1) (15 minutes)
A. Piazzolla Milonga Del Angel (4 minutes) and Libertango (3 minutes)
W. A. Mozart – A. Volodos Turkish March (3 minutes)
For more information please visit: https://scgp.stonybrook.edu/archives/45308
Title: Genus 1 Gromov-Witten invariants of Hilbert scheme of points on the affine plane
Speaker: Hsian-Hua Tseng
Abstract: We discuss some results on genus 1 Gromov-Witten invariants of Hilbert scheme of points on the affine plane, including a determination of multi-point series in terms of one-point series and a close formula for an one-point series.
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Title: Homological mirror symmetry for Batyrev mirror pairs
Speaker: Sheel Ganatra
Abstract: I will give an overview of my recent proof (joint with Hanlon, Hicks, Pomerleano, and Sheridan) of Kontsevich’s homological mirror symmetry conjecture for a large class of mirror pairs of compact Calabi–Yau hypersurfaces in toric varieties. These mirror pairs were first constructed by Batyrev from dual reflexive polytopes, and our result holds in characteristic zero and in all but finitely many positive characteristics. I will also say a few words about how our result (along with conjectural structural expectations in open-closed Floer theory) might imply all-genus mirror symmetry for such pairs.
Title: Integer-valued Gromov-Witten invariants
Speaker: Guangbo Xu
Abstract: (joint work with Shaoyun Bai) Moduli spaces in Gromov-Witten theory have two somewhat unsatisfying features: 1. it is hard to separate the contributions of simple curves and those of multiple covers; 2. the symmetry of curves leads to rational but not integral invariants. Following a proposal of Fukaya-Ono, we developed a new method to define counts of pseudoholomorphic curves with prescribed symmetry types, by turning on a specific kind of single-valued perturbations. In particular, this technology (which is purely topological) leads to integer-valued curve counting invariants (for all symplectic manifolds in all genera), which can be interpretated as the counts of curves with trivial automorphism group. Additionally, there are evidences suggesting that these integers agree with Gopakumar-Vafa invariants for Calabi-Yau threefolds. In this talk I will explain the idea underlying this new technology and the structures of these new invariants.
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Title: Local and global in enumerative geometry
Speaker: Local and global in enumerative geometry
Abstract: For a restricted class of targets X, I will talk about some examples and general expectations connecting local computations (equivariant counts of maps to X from P^1 or a formal disk) to counts of maps to X from a complete curve, that may be fixed or move in moduli.
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Title: Bordism valued GW invariants
Speaker: Mohammed Abouzaid
Abstract: The geometric input of Gromov-Witten theory are moduli spaces of (pseudo)-holomorphic curves with target a (closed) symplectic manifold. It has long been known that these are not in general manifolds, because of the presence of symmetries, and that they are not in general orbifolds either, since symmetries can obstruct transversality. One model for the structure they carry is that of derived orbifolds. This motivates the study of the bordism groups of stably complex derived orbifolds as a universal receptacle for Gromov-Witten invariants in symplectic topology. I will describe joint work with Shaoyun Bai, which uses the functoriality of the resolution of singularities algorithm for complex algebraic varieties, together with refinements of Fukaya and Ono's old idea of normally complex perturbations, to split the inclusion of the bordism group ofstably complex manifolds (unitary bordism) into this mysterious group, and thus proving the existence of well-defined GW invariants valued in complex cobordism groups.
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Title: Symplectic pushforwards and DT theory
Speaker: Hyeonjun Park
Abstract: I will introduce how to pushforward shifted symplectic fibrations along base changes. This yields an étale local structure theorem for shifted symplectic derived Artin stacks via Hamiltonian reduction. One application is a construction of cohomological Hall algebras for Calabi-Yau 3-folds, which is joint work with Tasuki Kinjo and Pavel Safronov. Another application is deformation invariance of Donaldson-Thomas invariants for Calabi-Yau 4-folds.
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Title: The quantum spectrum and Gamma structure for standard flips
Speaker: Yefeng Shen
Abstract: In this talk, we investigate the quantum spectrum and Gamma structure for projective bundles, blow-ups, and standard flips in a particular setup. By restricting the quantum cohomology to a fiber curve direction, both quantum spectrum and asymptotic behavior become computable. Using a sequence of reductions and asymptotic expansions of Meijer G-functions, we obtain a decomposition of the cohomology of standard flips into asymptotic Gamma classes. This decomposition is compatible with the semi-orthogonal decomposition for standard flips constructed in the work of Bondal-Orlov and Belmans-Fu-Raedschelders. The talk is based on work joint with Mark Shoemaker.
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Title: TBA
Speaker: Eric Bedford [Stony Brook University]
Abstract: TBA
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Public holiday
Title: TBA
Speaker: Jiakai Li [Harvard University]
Abstract: TBA
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