Week 2: International Conference

The objective of the Bridges International Conference is to gather established specialists and young researchers around special geometries and moduli spaces.

Some of the topics will be manifolds with special holonomy (Calabi-Yau, G_2, Spin(7), Quaternionic Kahler, hyperKahler), instantons, various notions of stability (slope-stability, K-stability etc), together with the relevant moduli spaces.

The scientific range of the conference will extend from Algebraic and Differential Geometry to Geometric Analysis and applications to Physics.

Scientific Committee

Speakers

H. Abban (Nottingham, UK)
I. Agricola (Marburg, Ger.)
V. Benedetti (Nice, Fr)
O. Biquard (Sorbonne, Fr)
U. Bruzzo (SISSA, It)
G. Comaschi (Univ. Pau, Fr)
M. Garcia Fernandez (ICMAT, Sp)
O. Garcia-Prada (CSIC, Sp)
M. Haskins (Duke, USA) TBC
L. Ioos (Cergy, Fr)
J. Lauret (Cordoba, Arg)
J. Lotay (Oxford, UK)
A. Napame (Unicamp, Br)
A. Ortu (Gotheburg, Se)
X. de la Ossa (Oxford, UK)
R. Reboulet (Gotheburg, Se)
P. Schwahn (Saclay, Fr)
R. Sena-Dias (IST, Pt)
J. Stoppa (SISSA, It)
R. Terpereau (Lille, Fr)
C. Tonnesen-Friedman (Union College Schenectady, USA)
F. Trinca (London Imp. Coll., UK)

I. Agricola

(Marburg, Ger.)

Invariant Spinors on Homogeneous Spheres

The classification of transitive sphere actions induces nine different homogeneous realizations of the sphere $S^n$ that are strongly linked to Berger's holonomy theorem. We explain what a homogeneous spin structure is and describe which homogeneous spheres admit one (which is then unique), and what geometry is encoded in it. In each of the cases we determine the dimension of the space of invariant spinor fields, give their explicit description, and study the underlying related geometric structures depending on the metric. We will sketch how the highly non-trivial representation theoretic computations are based on a rather unusual description of the spin representation in terms of exterior forms. This is joint work with Jordan Hofmann and Marie-Amelie Lawn, London.

V. Benedetti

(Nice, Fr)

Logarithmic derivations of adjoint discriminants

Given a hypersurface X in the projective space, its sheaf of logarithmic derivations E encodes information about the singularities of X itself. In some cases - for instance when X is a generic determinant - the moduli space of E can be related to the moduli space of deformations of X; in some sense, this amounts to the fact that (deformations of) X can be reconstructed from (deformations of) its singularities. In doing so, it is crucial to obtain a “simple” (alias locally free) presentation of E. In this talk, we exhibit a relationship between projective duality and the sheaf of logarithmic derivations, and show that the latter can be obtained as a push forward from the conormal variety of X. This, together with the so-called Geometric Technique, easily allows to obtain a locally free resolution of E. In order to give an application, we will restrict to the case when X is the adjoint discriminant of a simple Lie group; we will show that the locally free resolution of E in this case splits in two parts: the invariant part, which is a direct sum of line bundles, and the “variant” part, which contains the essential information about X. This is a joint work with Daniele Faenzi and Simone Marchesi.

G. Comaschi

(Univ. Pau, Fr)

Instanton bundles of high rank on Fano threefolds

The notion of mathematical instanton was first introduced by Atiyah, Drinfeld, Hitchin, and Manin to define stable rank 2 vector bundles E on the projective space P3 with c1(E) = 0 and such that H1(E(−2)) = 0. This notion can be generalized to vector bundles of rank greater than 2 and further extended to other Fano threefolds besides the projective space. In this talk, I will present some results about instanton bundles of high rank on Fano threefolds of Picard rank one.

J. Lotay

(Oxford, UK)

Coupled G2-instantons

The equations of motion from heterotic String Theory lead to complex systems of equations in 7 dimensions which couple ambient G2 structures on the manifold with gauge theoretic data. In the spirit of generalized geometry, extra insight into these systems can be understood by looking at structures defined on an extended bundle over the 7-manifold. In this talk, I will explain recent work with A. A. da Silva Jr, M. Garcia Fernandez and H. Sa Earp where we introduce new gauge theoretic objects, which we call "coupled G2-instantons", defined on such an extended bundle, and show that they naturally arise from solutions of the heterotic G2 system, and that they connect to the notion of generalized Ricci-flat metrics.

A. Napame

(Unicamp, Br)

Prescription of singularities on polystable reflexive sheaves

From general theory, if X is a smooth complex variety and E a reflexive sheaf over X, its singular locus is a closed subset of codimension at least three. We will focus on the case of toric varieties. If E is an equivariant sheaf, this locus is a finite union of orbit closures. We will show that it is possible to prescribe singularities on an explicit reflexive sheaf and also to prescribe singularities on a polystable sheaf.

R. Terpereau

(Lille, Fr)

Homogeneous vector Bundles on Fano threefolds

In this talk, we will focus on the problem of constructing homogeneous vector bundles, i.e., vector bundles invariant under the action of an algebraic group, when the algebraic group acts on the basis of the bundle with an open orbit. Producing such bundles is generally challenging, but we will see how instanton bundles, which emerged in the 1970s to explain fundamental forces in physics, offer an approach to construct them when the basis of the bundle is a Fano threefold. This is a joint work with Daniele Faenzi.Laboratoire Paul Painlevé

C. Tonnesen-Friedman

(Union College Schenectady, USA)

Weighted Extremal Twins in the K"ahler setting and their Corresponding Sasakian structures

This talk will describe the interplay between K"ahler and Sasakian geometry. I will focus on K"ahler and Sasaki metrics with special geometric properties and in particular highlight how one can construct explicit examples. The talk will also cover some ongoing joint work with C. P. Boyer, E. Legendre, and H. Huang. I will discuss the existence of what we call {it weighted extremal twins} in which a K"ahler metric is weighted extremal (in the sense of V. Apostolov and D. M. J. Calderbank) with respect to two different (even up to rescale) Killing potentials. This generalizes the twinning phenomenon appearing among certain strongly Hermitian solutions (found by C. LeBrun) to the Einstein-Maxwell equations on the first Hirzebruch surface. Under appropriate conditions, the weighted extremal twins will correspond to two extremal Sasaki structures both compatible with a fixed pseudo-convex CR structure, $(D,J)$, of Sasaki type on a compact smooth manifold $M$. With some very special (Bochner-flat) exceptions, the existence of twins appears to be a discrete phenomenon. This should be contrasted with the well known fact (proven by C. P. Boyer, K. Galicki, and S. Simanca) that the set of extremal Sasakian structures is open when one can deform within the isotopy classes.

BRIDGES meetingin Gauge Theory, extremal structures and stability

June 25th to July 5th, 2024​

June 25th to July 5th, 2024

Save the date!

Week 2: International Conference

Scientific Committee

Speakers

H. Abban

(Nottingham, UK)

I. Agricola

(Marburg, Ger.)

V. Benedetti

(Nice, Fr)

O. Biquard

(Sorbonne, Fr)

U. Bruzzo

(SISSA, It)

G. Comaschi

(Univ. Pau, Fr)

M. Garcia Fernandez

(ICMAT, Sp)

O. Garcia-Prada

(CSIC, Sp)

M. Haskins

(Duke, USA) TBC

L. Ioos

(Cergy, Fr)

J. Lauret

(Cordoba, Arg)

J. Lotay

(Oxford, UK)

A. Napame

(Unicamp, Br)

A. Ortu

(Gotheburg, Se)

X. de la Ossa

(Oxford, UK)

R. Reboulet

(Gotheburg, Se)

P. Schwahn

(Saclay, Fr)

R. Sena-Dias

(IST, Pt)

J. Stoppa

(SISSA, It)

R. Terpereau

(Lille, Fr)

C. Tonnesen-Friedman

(Union College Schenectady, USA)

F. Trinca

(London Imp. Coll., UK)

BRIDGES meeting
in Gauge Theory, extremal structures and stability

June 25th to July 5th, 2024