# Conference in Memory of Professor Masatake Kuranishi

05/09/2022 9:00 am - 05/12/2022 12:30 pm
Address: 150 Western Ave, Allston, MA 02134

On May 9–12, 2022, the CMSA hosted the conference Deformations of structures and moduli in geometry and analysis: A Memorial in honor of Professor Masatake Kuranishi.

Organizers:  Tristan Collins (MIT) and Shing-Tung Yau (Harvard and Tsinghua)

Videos are available on the conference playlist.

Speakers:

Charles Fefferman (Princeton University)

Teng Fei (Rutgers University)

Robert Friedman (Columbia University)

Kenji Fukaya (Simons Center, Stony Brook)

Akito Futaki (Tsinghua University)

Victor Guillemin (Massachusetts Institute of Technology)

Nigel Hitchin (Oxford University)

Blaine Lawson (Stony Brook University)

Yu-Shen Lin (Boston University)

Melissa C.C. Liu (Columbia University)

Takeo Ohsawa (Nagoya University)

Duong H. Phong (Columbia University)

Sebastien Picard (University of British Columbia)

Paul Seidel (Massachusetts Institute of Technology)

Gabor Szekelyhidi (University of Notre Dame)

Claire Voisin (Institut de Mathematiques, Jussieu, France)

Shing-Tung Yau (Harvard University)

Monday, May 9, 2022

 8:15 am Light breakfast & coffee/tea 8:45–9:00 am Opening Remarks 9:00–10:00 am Kenji Fukaya Title: Gromov Hausdorff convergence of filtered A infinity category Abstract: In mirror symmetry a mirror to a symplectic manifold is actually believed to be a family of complex manifold parametrized by a disk (of radius 0). The coordinate ring of the parameter space is a kind of formal power series ring the Novikov ring. Novikov ring is a coefficient ring of Floer homology. Most of the works on homological Mirror symmetry so far studies A infinity category over Novikov field, which corresponds to the study of generic fiber. The study of A infinity category over Novikov ring is related to several interesting phenomenon of Hamiltonian dynamics. In this talk I will explain a notion which I believe is useful to study mirror symmetry. Video 10:15–11:15 am Nigel Hitchin (Zoom) Title: Deformations: A personal perspective Abstract: The talk, largely historical, will focus on different deformation complexes I have encountered in my work, starting with instantons on 4-manifolds, but also monopoles, Higgs bundles and generalized complex structures. I will also discuss some speculative ideas related to surfaces of negative curvature. Video 11:30–12:30 pm H. Blaine Lawson Title: Projective Hulls, Projective Linking, and Boundaries of Varieties Abstract: In 1958 John Wermer proved that the polynomial hull of a compact real analytic curve γ ⊂ Cn was a 1-dim’l complex subvariety of Cn − γ. This result engendered much subsequent activity, and was related to Gelfand’s spectrum of a Banach algebra. In the early 2000’s Reese Harvey and I found a projective analogue of these concepts and wondered whether Wermer’s Theorem could be generalized to the projective setting. This question turned out to be more subtle and quite intriguing, with unexpected consequences. We now know a great deal, a highpoint of which s a result with Harvey and Wermer. It led to conjectures (for Cω-curves in P2C) which imply several results. One says, roughly, that a (2p − 1)-cycle Γ in Pn bounds a positive holomorphic p-chain of mass ≤ Λ ⇐⇒ its normalized linking number with all positive (n − p)-cycles in Pn − |Γ| is ≥ −Λ. Another says that a class τ ∈ H2p(Pn,|Γ|;Z) with ∂τ = Γ contains a positive holomorphic p-chain ⇐⇒ τ•[Z]≥0 for all positive holomorphic (n−p)-cycles Z in Pn−|Γ| Video 12:30–2:30 pm Lunch Break 2:30–3:30 pm Gabor Szekelyhidi Title: Singularities along the Lagrangian mean curvature flow. Abstract: We study singularity formation along the Lagrangian mean curvature flow of surfaces. On the one hand we show that if a tangent flow at a singularity is the special Lagrangian union of two transverse planes, then the flow undergoes a “neck pinch”, and can be continued past the flow. This can be related to the Thomas-Yau conjecture on stability conditions along the Lagrangian mean curvature flow. In a different direction we show that ancient solutions of the flow, whose blow-down is given by two planes meeting along a line, must be translators. These are joint works with Jason Lotay and Felix Schulze. Video 3:30–4:00 pm Coffee Break 4:00–5:00 pm Takeo Ohsawa Title: Glimpses of embeddings and deformations of CR manifolds Abstract: Basic results on the embeddings and the deformations of CR manifolds will be reviewed with emphasis on the reminiscences of impressive moments with Kuranishi since his visit to Kyoto in 1975. Video

Tuesday, May 10, 2022

 8:15 am Light breakfast & coffee/tea 9:00–10:00 am Charles Fefferman (Zoom) Title: Interpolation of Data by Smooth Functions Abstract: Let X be your favorite Banach space of continuous functions on R^n. Given an (arbitrary) set E in R^n and an arbitrary function f:E->R, we ask: How can we tell whether f extends to a function F \in X? If such an F exists, then how small can we take its norm? What can we say about its derivatives (assuming functions in X have derivatives)? Can we take F to depend linearly on f? Suppose E is finite. Can we compute an F as above with norm nearly as small as possible? How many computer operations does it take? What if F is required to agree only approximately with f on E? What if we are allowed to discard a few data points (x, f(x)) as “outliers”? Which points should we discard? The results were obtained jointly with A. Israel, B. Klartag, G.K. Luli and P. Shvartsman over many years. Video 10:15–11:15 am Claire Voisin Title: Deformations of K-trivial manifolds and applications to hyper-Kähler geometry Summary: I will explain the Ran approach via the T^1-lifting principle to the BTT theorem stating that deformations of K-trivial compact Kähler manifolds are unobstructed. I will explain a similar unobstructedness result for Lagrangian submanifolds of hyper-Kähler manifolds and I will describe important consequences on the topology and geometry of hyper-Kähler manifolds. Video 11:30– 2:30 pm Victor Guillemin Title: Semi-Classical Functions of Isotropic Type Abstract: The world of semiclassical analysis is populated by objects of “Lagrangian type.” The topic of this talk however will be objects in semi-classical analysis that live instead on isotropic submanifolds. I will describe in my talk a lot of interesting examples of such objects. Video 12:30–2:30 pm Lunch Break 2:30–3:30 pm Teng Fei Title: Symplectic deformations and the Type IIA flow Abstract: The equations of flux compactification of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow known as the Type IIA flow on symplectic Calabi-Yau 6-manifolds. We prove the wellposedness of this flow and establish the basic estimates. We show that the Type IIA flow can be applied to find optimal almost complex structures on certain symplectic manifolds. We prove the dynamical stability of the Type IIA flow, which leads to a proof of stability of Kahler property for Calabi-Yau 3-folds under symplectic deformations. This is based on joint work with Phong, Picard and Zhang. Video Speakers Banquet

Wednesday, May 11, 2022

 8:15 am Light breakfast & coffee/tea 9:00–10:00 am Shing-Tung Yau (Zoom) Title: Canonical metrics and stability in mirror symmetry Abstract: I will discuss the deformed Hermitian-Yang-Mills equation, its role in mirror symmetry and its connections to notions of stability.  I will review what is known, and pose some questions for the future. Video 10:15–11:15 am Duong H. Phong Title: $L^\infty$ estimates for the Monge-Ampere and other fully non-linear equations in complex geometry Abstract: A priori estimates are essential for the understanding of partial differential equations, and of these, $L^\infty$ estimates are particularly important as they are also needed for other estimates. The key $L^\infty$ estimates were obtained by S.T. Yau in 1976 for the Monge-Ampere equation for the Calabi conjecture, and sharp estimates obtained later in 1998 by Kolodziej using pluripotential theory. It had been a long-standing question whether a PDE proof of these estimates was possible. We provide a positive answer to this question, and derive as a consequence sharp estimates for general classes of fully non-linear equations. This is joint work with B. Guo and F. Tong. Video 11:30–2:30 pm Paul Seidel Title: The quantum connection: familiar yet puzzling Abstract: The small quantum connection on a Fano variety is possibly the most basic piece of enumerative geometry. In spite of being really easy to write down, it is the subject of far-reaching conjectures (Dubrovin, Galkin, Iritani), which challenge our understanding of mirror symmetry. I will give a gentle introduction to the simplest of these questions. Video 12:30–2:30 pm Lunch Break 2:30–3:30 pm Melissa C.C. Liu Title: Higgs-Coulumb correspondence for abelian gauged linear sigma models Abstract: The underlying geometry of a gauged linear sigma model (GLSM) consists of a GIT quotient of a complex vector space by the linear action of a reductive algebraic group G (the gauge group) and a polynomial function (the superpotential) on the GIT quotient. The Higgs-Coulomb correspondence relates (1) GLSM invariants which are virtual counts of curves in the critical locus of the superpotential (Higgs branch), and (2) Mellin-Barnes type integrals on the Lie algebra of G (Coulomb branch). In this talk, I will describe the correspondence when G is an algebraic torus, and explain how to use the correspondence to study dependence of GLSM invariants on the stability condition. This is based on joint work with Konstantin Aleshkin. Video 3:30–4:00 pm Coffee Break 4:00–5:00 pm Sebastien Picard Title: Topological Transitions of Calabi-Yau Threefolds Abstract: Conifold transitions were proposed in the works of Clemens, Reid and Friedman as a way to travel in the parameter space of Calabi-Yau threefolds with different Hodge numbers. This process may deform a Kahler Calabi-Yau threefold into a non-Kahler complex manifold with trivial canonical bundle. We will discuss the propagation of differential geometric structures such as balanced hermitian metrics, Yang-Mills connections, and special submanifolds through conifold transitions. This is joint work with T. Collins, S. Gukov and S.-T. Yau. Video

Thursday, May 12, 2022

 8:15 am Light breakfast & coffee/tea 9:00 am–10:00 am Akito Futaki (Zoom) Title: Transverse coupled Kähler-Einstein metrics and volume minimization Abstract: We show that transverse coupled Kähler-Einstein metrics on toric Sasaki manifolds arise as a critical point of a volume functional. As a preparation for the proof, we re-visit the transverse moment polytopes and contact moment polytopes under the change of Reeb vector fields. Then we apply it to a coupled version of the volume minimization by Martelli-Sparks-Yau. This is done assuming the Calabi-Yau condition of the Kählercone, and the non-coupled case leads to a known existence result of a transverse Kähler-Einstein metric and a Sasaki-Einstein metric, but the coupled case requires an assumption related to Minkowski sum to obtain transverse coupled Kähler-Einstein metrics.Video 10:15 am–11:15 am Yu-Shen Lin Title: SYZ Mirror Symmetry of Log Calabi-Yau Surfaces Abstract: Strominger-Yau-Zaslow conjecture predicts Calabi-Yau manifolds admits special Lagrangian fibrations. The conjecture serves as one of the guiding principles in mirror symmetry. In this talk, I will explain the existence of the special Lagrangian fibrations in some log Calabi-Yau surfaces and their dual fibrations in their expected mirrors. The journey leads us to the study of the moduli space of Ricci-flat metrics with certain asymptotics on these geometries and the discovery of new semi-flat metrics. If time permits, I will explain the application to the Torelli theorem of ALH^* gravitational instantons. The talk is based on joint works with T. Collins and A. Jacob. Video 11:30 am – 12:30 pm Robert Friedman Title: Deformations of singular Fano and Calabi-Yau varieties Abstract: This talk will describe recent joint work with Radu Laza on deformations of generalized Fano and Calabi-Yau varieties, i.e. compact analytic spaces whose dualizing sheaves are either duals of ample line bundles or are trivial. Under the assumption of isolated hypersurface canonical singularities, we extend results of Namikawa and Steenbrink in dimension three and discuss various generalizations to higher dimensions. Video 12:30 pm Concluding Remarks