# Yorkshire and Durham Geometry Days

A **Yorkshire and Durham Geometry Day** will take place on
**Wednesday December 11, 2019**
in the Department of Mathematics at Durham University. This is to welcome
** Liviana Palmisano** and ** Fernando Galaz-Garcia **
who are joining Durham in this month.

**10:30** Coffee in the Mathematics Department Common Room **CM211**

**11:00** ** Samuel Borza ** (Durham) in ** CM301**

- "CD-condition for negative effective dimension and sub-Riemannian geometry"

Abstract: The celebrated curvature-dimension condition has recently been extended to include negative effective dimensions. This generalisation enlarges the class of metric measure spaces on which a synthetic treatment of curvature can be performed. We will discuss examples satisfying that condition and applications, namely the Brunn-Minkowski and isoperimetric inequalities. We will investigate whether this condition holds for sub-Riemannian manifolds such as the Heisenberg groups.
**12:00** Lunch break.

**1:45** **Mario Micallef
** (Warwick) in **CM301**

- "Instability of covers of minimal surfaces in a PIC manifold"

Abstract:Let E be a hermitian holomorphic vector bundle over a Riemann surface S of nonzero genus. Suppose that the first Chern class of E is non-negative but E admits no holomorphic sections so that the lowest eigenvalue \lambda_1 of the (d-bar)^*(d-bar) operator on E is positive. We show that, by taking suitable high coverings of S, \lambda_1 of the lifted bundle can be made arbitrarily small. This result is then applied to show the instability of a suitable high covering of a minimal surface in a PIC, namely a manifold of positive isotropic curvature. This is work in progress, joint with Rick Schoen.
**3:00** ** Davoud Cheraghi ** (Imperial) in ** CM301 **

- "Topological branched coverings of the sphere and invariant complex structures"

Abstract : Let f be an orientation preserving branched covering of the two dimensional sphere. Is f realised (up to homotopy) by a rational function of the Riemann sphere? If yes, is the corresponding rational function unique up to Mobius transformations (the rigidity)? These questions amount to the existence and uniqueness of a complex structure which is invariant under the action (of the homotopy class) of f.
It turns out that the topological and geometric structure of "the orbits of the branched points” play a key role in these problems. When all the branched points are periodic, a classical result of W. Thurston provides a complete topological characterisation of the branched coverings that are realised by rational functions (and their uniqueness). On the other hand, when the orbits of the branched points form a more complicated subset of the sphere, say a Cantor set, the problem is highly nontrivial and has been extensively studied over the last three decades. In this talk we survey the main results of these studies, and describe a recent advance made on the uniqueness part using a renormalisation technique.
**4:00** Tea break in **CM211**

**4:45** ** Liviana Palmisano** (Durham)
in ** CM301 **

- "Coexistence of attractors and their stability"

Abstract: In unfoldings of rank-one homoclinic tangencies, there exist codimension 2 laminations of maps with infinitely many sinks. The sinks move simultaneously along the leaves. As consequence, in the space of real polynomial maps, there are examples of: Hénon maps, in any dimension, with infinitely many sinks, quadratic Hénon-like maps with infinitely many sinks and a period doubling attractor, quadratic Hénon-like maps with infinitely many sinks and a strange attractor. The coexistence of non-periodic attractors, namely two period doubling attractors or two strange attractors, and their stability is also discussed.
**6:00** Leave from **CM211** for dinner at St Mary's College

**Travel:**
Durham is easy to get to by car and train, and so is the Department of Mathematics, located on the Science Site. Click here for relevant information.

Yorkshire and Durham Geometry Days are jointly organised by the Universities of Durham, Leeds and York, and occur at a
frequency of three meetings per year. Financial support is provided by the London Mathematical Society through a Scheme 3 grant,
currently administered by the University of York. Additional support is provided by the Department of Mathematics,
Durham University.

The
local organizers are:

John
Bolton & Wilhelm Klingenberg,
University of Durham
Derek Harland &
Gerasim Kokarev, University
of Leeds
Ian
McIntosh & Chris
Wood, University of York