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Author Archives: Victoria Hain
Shigeru Mukai: Lectures
January 9, 2019
TITLE: Prime Fano 3folds and BNgeneral K3s
ABSTRACT: Fano 3folds with 2nd Betti number one are classified into 17 deformation types. The anticanonical degree 2g2 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 3fold of g=10 is obtained from the (5dimensional) G2adjoint 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 anticanonical linear system. The key property of S used in the proof is the BNgenericity.
Diego Matessi: Lectures
 01/07/2019: Polarized tropical manifolds and Lagrangian torus fibrations
 01/09/2019: Conifold transitions and deformations of polarized tropical manifolds
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 GrossSiebert program. These can be viewed as the basis of a Lagrangian torus fibration on a symplectic CalabiYau manifold, but via the Legendre tranfsform they also provide the starting data for the reconstruction of the mirror family using the GrossSiebert 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 (SmithThomasYau) and of the complex smoothings on the mirror (FriedmanTian) can be both read in terms of certain tropical cycles. This suggests an approach, via the GrossSiebert 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 fivedimensional supersymmetric conformal field theories (SCFTs) in Mtheory after Intrilligator, Morrison and Seiberg. This esablishes a conjectural bijection assigning to each local isolated CalabiYau threefold singularity a fivedimensional superconformal field theory and viceversa. Focusing on the toric case, I will discuss applications of IIA/Mtheory 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 CalabiYau degeneration
ABSTRACT: To any meromorphic degeneration of complex projective varieties corresponds a projective variety over the field of Laurent series, and hence a nonArchimedean analytic space in the sense of Berkovich. This applies in particular to a degeneration of polarized CalabiYau manifolds, and has been used in recent years by Nicaise, Xu and their collaborators to approach a version of the StrömingerYauZaslow conjecture due to KontsevichSoibelman. I will provide a gentle introduction to this circle of ideas, mostly based on a joint work with Jonsson, in which the limit of CalabiYau volume forms in the associated Berkovich space is analyzed.
Kael Dixon: Lectures
September 14, 2018
TITLE: Asymptotic properties of toric G_{2} manifolds
A toric manifold is a 7manifold M equipped with a torsionfree structure, which is invariant under the action of a 3torus T in such a way that there exist multimoment maps associated to the 3form and its Hodge dual. These are introduced and studied in a recent paper by Madsen and Swann, where they show that these multimoment maps induce a local homeomorphism from the space of orbits M/T into R4. In other words, the multimoment 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.
Special Holonomy and Algebraic Geometry, Imperial College, 711 January 2019
Arrival Date: Sunday January 6, 2019. Departure Date: Saturday January 12, 2019.
Talks MonWed will be at Huxley Bldg 140. Talks ThursFri will take be at Huxley Blg 341 [ click for map ]
Registration and coffee/tea are in the Huxley 5th floor common room.
Schedule
MON 7 JAN 
TUE 8 JAN 
WED 9 JAN 
THU 10 JAN 
FRI 11 JAN 

9:00 9:30 
REGISTRATION  
9:3010:30 
Morrison 1  Sun 1  Nordström 2  Li 1  Li 2 
10:3011:00 
Q&A / Discussion  Q&A / Discussion  Q&A / Discussion  Q&A / Discussion  Q&A / Discussion 
11:0011:30 
REFRESHMENTS  REFRESHMENTS  REFRESHMENTS  REFRESHMENTS  REFRESHMENTS 
11:3012:30 
Morrison 2  Nordström 1  Boucksom 2  Sun 2  Del Zotto 
12:3014:00 
LUNCH  LUNCH  LUNCH  LUNCH  LUNCH 
14:0015:00 
Matessi 1  Boucksom 1  Mukai  Braun  Discussion 
15:0015:30 
REFRESHMENTS  REFRESHMENTS  REFRESHMENTS  REFRESHMENTS  REFRESHMENTS 
15:3016:30 
Gross 1  Gross 2  Matessi 2  Discussion  END OF MEETING 
18:00 
MEETING DINNER 
Speakers:
 Sébastien Boucksom (École Polytechnique), The essential skeleton of a CalabiYau degeneration
 Andreas Braun (Oxford), Exceptional Holonomy, String Duality and Vector Bundles
 Michele Del Zotto (Durham, UK), Aspects of 5d SCFTs and their gauge theory phases
 Mark Gross (Cambridge), From SYZ to GS
 Yang Li (Imperial College London), TaubNUT and OoguriVafa
 Yang Li (Imperial College London), TaubNUT (and OoguriVafa) on CalabiYau 3folds
 Diego Matessi (Milan), Polarized tropical manifolds and Lagrangian torus fibrations
 Diego Matessi (Milan), Conifold transitions and deformations of polarized tropical manifolds
 David Morrison (UCSanta Barbara), Degenerations of complex structures on K3 surfaces, following Kulikov, Persson, and Pinkham
 David Morrison (UCSanta Barbara), Supersymmetric torus fibrations of CalabiYau threefolds
 Shigeru Mukai (RIMS, Kyoto), Prime Fano 3folds and BNgeneral K3s
 Johannes Nordström (Bath), Building blocks for twisted connected sums
 Johannes Nordström (Bath), The matching problem for twisted connected sums
 Song Sun (UCBerkeley), Degeneration of CalabiYau metrics under complex structure degenerations
Markus Upmeier: Lectures
September 11, 2018
TITLE: Canonical Orientations for the Moduli Space of G_{2}instantons
ABSTRACT: The moduli space of antiselfdual connections for 4manifolds has been generalized by DonaldsonSegal to special, higherdimensional 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 7manifold 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.
Yuuji Tanaka: Lectures
September 10, 2018
TITLE: On the VafaWitten theory on closed fourmanifolds
ABSTRACT: Vafa and Witten introduced a set of gaugetheoretic equations on closed fourmanifolds around 1994 in the study of Sduality conjecture in N=4 super YangMills 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 algebrogeometric methods; the latter is joint work with Richard Thomas.
HansJoachim Hein: Lectures
September 10, 2018
TITLE: Higherorder estimates for collapsing CalabiYau metrics
ABSTRACT: Consider a compact CalabiYau 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 Ricciflat Kahler forms representing these classes is a basic problem in geometric analysis that has attracted a lot of attention since the celebrated work of GrossWilson (2000) on elliptically fibered K3 surfaces. The limiting behavior of these Ricciflat metrics is still not wellunderstood in general even away from the singular fibers of F. A key difficulty arises from the fact that Yau’s higherorder estimates for the complex MongeAmpere 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 higherorder estimates even though the background curvature blows up.
Donaldson & Sun lectures at 2018 ICM
Regarding the 2018 ICM:
Simon Donaldson presented the opening plenary lecture –
Some recent developments in Kähler geometry and exceptional holonomy. Written version:
https://arxiv.org/abs/1808.03995
Song Sun gave a sectional lecture in Geometry –
Degenerations and moduli spaces in Kähler geometry. Written version: https://math.berkeley.edu/~sosun/ICM.pdf