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Shigeru Mukai: Lectures

January 9, 2019
TITLE: Prime Fano 3-folds and BN-general K3s

ABSTRACT: Fano 3-folds with 2nd Betti number one are classified into 17 deformation types. The anti-canonical degree 2g-2 and the 3rd Betti number 2p are their basic numerical invariants. The sum g+p varies from 12 to 54, and the minimum 12 is attained in 3 cases. In this talk I will explain the linear section theorem in the case (g,p)=(10,2): a prime Fano 3-fold of g=10 is obtained from the (5-dimensional) G2-adjoint variety by taking hyperplane section p=2 times. The basic tool is a rigid, or spherical, vector bundle on a K3 surface S in the anti-canonical linear system. The key property of S used in the proof is the BN-genericity.

Diego Matessi: Lectures

January 7, 2019
TITLE: Polarized tropical manifolds and Lagrangian torus fibrations

ABSTRACT: I will review the notion of polarized tropical manifolds which are the basic combinatorial objects in the Gross-Siebert program. These can be viewed as the basis of a Lagrangian torus fibration on a symplectic Calabi-Yau manifold, but via the Legendre tranfsform they also provide the starting data for the reconstruction of the mirror family using the Gross-Siebert algorithm.

January 9, 2019
TITLE: Conifold transitions and deformations of polarized tropical manifolds

ABSTRACT: Conifold singularities have a nice description in terms of polarized tropical manifolds. I will describe a result where the obstructions to the existence of symplectic resolutions (Smith-Thomas-Yau) and of the complex smoothings on the mirror (Friedman-Tian) can be both read in terms of certain tropical cycles. This suggests an approach, via the Gross-Siebert program, to Morrison’s Conjecture stating that the mirror of a resolution is a smoothing of the mirror. In a joint work with Helge Ruddat this idea leads to the notion of a deformation of a polarized tropical manifold induced by a tropical cycle.

Michele Del Zotto: Lectures

January 11, 2019
TITLE: Aspects of 5d SCFTs and their gauge theory phases

ABSTRACT: In this talk I will revist the geometric engineering of five-dimensional supersymmetric conformal field theories (SCFTs) in M-theory after Intrilligator, Morrison and Seiberg. This esablishes a conjectural bijection assigning to each local isolated Calabi-Yau three-fold singularity a five-dimensional superconformal field theory and viceversa. Focusing on the toric case, I will discuss applications of IIA/M-theory fiberwise duality (i.e. a peculiar instance of collapse) to characterizing the possible gauge theory phases of these systems. This geometric setup clarifies the notion of “UV duality” for such theories. Along the way, I will provide a novel gauge theoretical expression for the 5d prepotential, accounting correctly for the 5d parity anomaly. Based on the preprint arXiv:1812.10451, with Cyril Closset and Vivek Saxena.

Sébastien Boucksom: Lectures

January 8, 2019 and January 9, 2019
TITLE: The essential skeleton of a Calabi-Yau degeneration

ABSTRACT: To any meromorphic degeneration of complex projective varieties corresponds a projective variety over the field of Laurent series, and hence a non-Archimedean analytic space in the sense of Berkovich. This applies in particular to a degeneration of polarized Calabi-Yau manifolds, and has been used in recent years by Nicaise, Xu and their collaborators to approach a version of the Ströminger-Yau-Zaslow conjecture due to Kontsevich-Soibelman. I will provide a gentle introduction to this circle of ideas, mostly based on a joint work with Jonsson, in which the limit of Calabi-Yau volume forms in the associated Berkovich space is analyzed.

Kael Dixon: Lectures

September 14, 2018
TITLE: Asymptotic properties of toric G2 manifolds

A toric G_2 manifold is a 7-manifold M equipped with a torsion-free G_2 structure, which is invariant under the action of a 3-torus T in such a way that there exist multi-moment maps associated to the G_2 3-form and its Hodge dual. These are introduced and studied in a recent paper by Madsen and Swann, where they show that these multi-moment maps induce a local homeomorphism from the space of orbits M/T into R4. In other words, the multi-moment maps provide geometrically motivated local coordinates for M/T. In all of the known examples, this local homeomorphism is a global homeomorphism onto R4. I will describe some partial results toward showing that this is true in general.

Slides of Lecture

Markus Upmeier: Lectures

September 11, 2018
TITLE: Canonical Orientations for the Moduli Space of G2-instantons

ABSTRACT: The moduli space of anti-self-dual connections for 4-manifolds has been generalized by Donaldson-Segal to special, higher-dimensional geometries. I will discuss a technique for fixing canonical orientations on these moduli spaces in dimension 7 and for the gauge group SU(n). These orientations depend on the choice of a flag structure, an additional piece of data on the underlying 7-manifold introduced by Joyce. After discussing the reconstruction of an SU(n)-bundle from its ‘Poincaré dual’ submanifold, the definition of canonical orientations will be presented, based on the excision principle from index theory.

Slides of lecture

Yuuji Tanaka: Lectures

September 10, 2018
TITLE: On the Vafa-Witten theory on closed four-manifolds

ABSTRACT: Vafa and Witten introduced a set of gauge-theoretic equations on closed four-manifolds around 1994 in the study of S-duality conjecture in N=4 super Yang-Mills theory in four dimensions. They predicted from supersymmetric reasoning that the partition function of the invariants defined through the moduli spaces of solutions to these equations would have modular properties. But little progress has been made other than their original work using results by Goettsche, Nakajima and Yoshioka.

However, it now looks worth trying to figure out some of their foreknowledge with more advanced technologies in analysis and algebraic geometry fascinatingly developed in these two decades. This talk discusses issues to construct the invariants out of the moduli spaces, and presents possible ways to sort them out by analytic and algebro-geometric methods; the latter is joint work with Richard Thomas.

Hans-Joachim Hein: Lectures

September 10, 2018
TITLE: Higher-order estimates for collapsing Calabi-Yau metrics

ABSTRACT: Consider a compact Calabi-Yau manifold X with a holomorphic fibration F: X to B over some base B, together with a “collapsing” path of Kahler classes of the form [F*(omega_B)] + t * [omega_X] for t in (0,1]. Understanding the limiting behavior as t to 0 of the Ricci-flat Kahler forms representing these classes is a basic problem in geometric analysis that has attracted a lot of attention since the celebrated work of Gross-Wilson (2000) on elliptically fibered K3 surfaces. The limiting behavior of these Ricci-flat metrics is still not well-understood in general even away from the singular fibers of F. A key difficulty arises from the fact that Yau’s higher-order estimates for the complex Monge-Ampere equation depend on bounds on the curvature tensor of a suitable background metric that are not available in this collapsing situation. I will explain recent joint work with Valentino Tosatti where we manage to bypass Yau’s method in some cases, proving higher-order estimates even though the background curvature blows up.

Toby Wiseman: Lectures

June 4, 2018
TITLE: Some applications of Ricci flow in physics

ABSTRACT: I will review two areas where Ricci flow makes contact with physics. Firstly I will review how Ricci flow arises from the renormalisation group equations of 2d `sigma models’ (I will try to explain what these words mean!). Secondly I will review a more recent link, where Ricci flow may be thought of as an algorithm to numerically find solutions to Einstein’s gravitational equations in exotic settings. In physics in both cases it is interesting to consider how black holes evolve under Ricci flow. Static black holes may be thought of as Riemannian geometries, while stationary black holes cannot, but still may be evolved using a ‘Lorentzian’ signature Ricci flow. I will also discuss the existence of Ricci solitons which are important to understand in the second context.

Lu Wang: Lectures

June 7, 2018
TITLE: Properties of self-similar solutions of mean curvature flow

ABSTRACT: I will survey some known results as well as some open problems about self-similar solutions of the mean curvature flow.