# GBMS Theme 4: Geometry and Visualization

Date: 26 - 28 Aug 2015

Venue: Western Gateway Building, UCC

### Organizers

• Martin Kilian (UCC)
• Ben McKay (UCC)

### Speakers Include(*=TBC)

• Bechtluft-Sachs (NUIM)
• David Brander (TU Denmark)
• Maciej Dunajski (Cambridge)
• Sorin Dumitrescu (Nice)
• Laurent Hauswirth (Paris XII)
• Lynn Heller (Tuebingen)
• Tim Hoffmann (TU Munich)
• Sebastian Klein (Mannheim)
• Katrin Leschke (Leicester)
• Alan McCarthy (Univ New South Wales)
• Ian McIntosh (York)
• Franz Pedit (Amherst)
• Martin Schmidt (Mannheim)

### Talks

#### Wednesday 26 August

• 9am-10am, room 107

Maciej Dunajski, Quartics, Sextics and Beyond

I will review the 19th century classical invariant theory of Cayley and Sylvester, and present a solution of an outstanding open problem concerning binary sextics. This is joint work with Roger Penrose.

• 10am-10:30am, atrium

Coffee and Tea

• 10:30am-11:30pm, room G04

Aaron Bertram, Birational Geometry of Moduli Spaces of Sheaves on the Projective Plane via Stability Conditions

When the minimal model program is applied to a moduli space, it is natural to ask whether the resulting birational models are themselves moduli spaces. In the case of moduli spaces of stable coherent sheaves on some algebraic surfaces, the answer is an emphatic yes, with moduli of Bridgeland-stable objects in the derived category arising as the birational models. In this talk I want to survey some of the known results, particularly focusing on the case of the projective plane.

• 11:30am-12:30pm, room G02

Tim Hoffmann, On a surface theory for quadrilateral nets

I will report on recent joint work with Andrew O. Sageman-Furnas (Furnas) and Max Wardetzky on a discrete version of surface theory for quadrilateral nets. Our approach generalizes the known integrable cases of discrete surfaces (in particular surfaces of constant curvature, which are well understood) into a more general framework.

There are many well working examples of integrable discretizations of special surface classes as well as well working discrete definitions of fundamental forms, curvatures, shape operator, and similar fundamental objects of surface theory but so far little effort has been made to formulate a general framework that covers the integrable cases with their fundamental properties and still works on a broader class of nets.

One of the roots of discrete differential geometry is the theory of discrete integrable surfaces. While in the smooth case these surfaces are often defined by their curvature properties the discretizations originally where done algebraically and only afterwards ad-hoc definitions for curvatures where found. The discrete Steiner formula changed that to some extend unifying a notion of curvature for a wider class of nets but it still relies on analogues of special parametrizations. I will give an overview over these developments and show our new and generic approach to generalize them and unify even more discrete surfaces and a broader class of quadrilateral nets.

• 12:30pm-2pm

Lunch

• 2pm-3pm, room G02

Alan McCarthy, Discrete Projective Minimal (and Q) Surfaces

Surfaces in $\mathbb{RP}^3$ that admit a parametrization by asymptotic coordinates have an associated projective metric of the form $pqdxdy$. A surface $\Sigma\in\mathbb{RP}^3$ is said to be projective minimal if $\Sigma$ is critical for the functional $\iint pqdxdy$. Associated to a point on a projective surface $M$ is the Lie quadric which in turn generically generates four transformation surfaces known as the Demoulin transforms of $M$. These transforms are in asymptotic correspondence with $M$ if $M$ is projective minimal. Conversely if a Demoulin transform is in asymptotic correspondence with $M$, then $M$ is either projective minimal or a special type of surface known as a Q surface. Based on joint work with W.K. Schief, I will provide a summary of some of the continuous theory of projective minimal surfaces and their Demoulin transforms and use this to motivate a definition of discrete projective minimal (and Q) surfaces.

• 3pm-4pm, room G02

Ian McIntosh, Equivariant minimal surfaces in the complex hyperbolic plane and their Higgs bundles

One approach to studying the space of irreducible representations of a surface group (fundamental group of a closed orientable surface) into the isometry group of a real or complex hyperbolic space is to parameterise these by Higgs bundles. It is well-known that the Higgs bundle equations are equivalent to the equations for an equivariant harmonic map into the hyperbolic space. But there are "too many" Higgs bundles: there is one parameterisation for every choice of conformal structure on the surface. One way to try to rectify this is to look for the best conformal structures by focussing on minimal surfaces. I will describe progress in this direction for maps into the complex hyperbolic plane, and how minimal surfaces can tell us more about the properties of a representation.

• 4pm-4:30pm, atrium

Coffee and Tea

• 4:30pm-5:30pm, room G04

Mark Gross, Logarithmic Gromov-Witten Invariants

I will talk about ongoing work defining various new flavours of Gromov-Witten invariants with target spaces being normal crossings or toroidal crossings varieties. These are best done in the category of log schemes, and provide a vast generalization of relative Gromov-Witten invariants. The contents of the talk represent joint with with Abramovich, Chen and Siebert.

• 7pm-, Franciscan Well

Pizza and beer

#### Thursday 27 August

• 9am-10am, room G04

Alexander Schmitt, Semistable quiver sheaves

We will survey the theory of quiver sheaves with special emphasis on the notion of semistability coming from Geometric Invariant Theory. In particular, we will present our recent boundedness results.

• 10am-10:30am, atrium

Coffee and Tea

• 10:30am-11:30am, room G02

David Brander, Geometric Cauchy problems for surfaces associated with harmonic maps

It has been known since the 1990's that harmonic maps from Riemannian or Lorentzian surfaces into Symmetric spaces admit loop group generalizations of the classical Weierstrass representation (Riemannian) or d'Alembert solution of the wave equation (Lorentzian). These allow one to construct solutions to the various geometric problems that are associated (via some version of the Gauss map) to harmonic maps.  The utility of these representations is obstructed by the loss of geometric information in the loop group decomposition that relates the harmonic map to the "Weierstrass" data. Recently, special types of Weierstrass data have been introduced that contain full geometric information along a curve. I will discuss  applications of this technique to the construction of integrable surfaces.

• 12pm-1pm, room 107

Barbara Fantechi, Counting curves on algebraic varieties

Enumerative geometry is one of the oldest parts of mathematics, with entry-level problems that can be explained to the layman (given four lines in space, how many lines meet/intersect all of them?). Yet in recent decades stunning progress has come thanks to a synergy of algebraic and complex geometry, string theory and function theory. In this talk we give a very partial overview, highlighting the interplay between concrete problems and theoretical advances.

• 1pm-2pm

Lunch

• 2pm-3pm, room G02

Martin Schmidt, Deformations of spectral curves

We apply Whitham deformations to the spectral curves of integrable systems. Our main interest concerns the spectral curves of constant mean curvature cylinders of finite type in the round 3-sphere. We prove that for any spectral curve of such a cylinder there starts a continuous path of such spectral curves ending at a spectral curve of genus zero. By using the geometric analysis of mean convex Alexandrov embedded surfaces we can show that almost all continuous deformations of the spectral curves preserves mean convex Alexandrov embeddedness. Along these lines we can show that all mean convex Alexandrov embedded cylinders of finite type are of Delaunay type.

• 3pm-4pm, room G02

Sebastian Klein, A spectral theory for simply periodic solutions of the sinh-Gordon equation

In the talk, I describe a spectral theory for solutions u of the sinh-Gordon equation, which are simply periodic in the sense that their domain of definition is a horizontal strip in the complex plane such that u(z+1)=u(z) holds for all z in N. Real-valued solutions u of the sinh-Gordon equation are of interest in particular, because they give rise to minimal surfaces in the 3-sphere. The case where u is doubly periodic (i.e. has two linear independent periods) corresponds to minimal tori in the 3-sphere; this case has been classified completely by Pinkall/Sterling (1989) and independently by Hitchin (1990). In contrast, the case considered here, of simply periodic solutions u, gives rise to a far larger class of minimal surfaces.

I will describe how one can associate to a simply periodic solution u so-called spectral data (Sigma,D). Here Sigma is the spectral curve associated to u, which is a non-compact, hyperelliptic Riemann surface above the punctuated complex plane, and D is a divisor on Sigma, corresponding to a holomorphic line bundle. Unlike in the doubly periodic case (studied by Hitchin), Sigma generally has infinitely many branch points, and the set of the branch points has two accumulation points.

The "direct problem" for a given simply periodic solution u is to construct the corresponding spectral data (Sigma,D), and to discuss their behavior. In this context, I will in particular characterise the asymptotic behavior of the spectral divisor D near the accumulation points of the set of branch points of Sigma. It turns out that the divisor asympotically approximates the spectral divisor of the "vacuum solution" u=0 with a certain order.

Finally, the "inverse problem" will be discussed. This is to reconstruct the solution u from its spectral data (Sigma,D).

• 4pm-4:30pm, atrium

Coffee and Tea

• 4:30pm-5:30pm, room G02

Laurent Hauswirth, Alexandrov embedded annuli and integrable systems

We study geometric deformations of CMC and minimal annuli of
finite type in S(2)xR and S(3) when we deform the associated spectral
curve. This deformation induces a topology on the space of annuli which
is of a local nature. We describe the topology on compact sets and how the
control of the spectral curve induces global properties on the deformation.
In particular Alexandrov embeddedness is preserved under the deformation
of the spectral curve.

• 8pm-, ???

Conference dinner

#### Friday 28 August

• 9am-10am, room G02

Sorin Dumitrescu, Quasihomogeneous real and complex geometric structures

This talk deals with rigid geometric structures which are quasihomogeneous, in the sense that they are locally homogeneous on an open dense subset of a manifold, but not on all of the manifold.

Our motivation comes from Gromov's open-dense orbit theorem and its application to prove the differential rigidity of some smooth Anosov systems. More precisely, we will present the classification of quasihomogeneous real analytic connections on surfaces (collaboration with A. Guillot) and the case of real analytic Lorentz metrics on threefolds (collaboration with K. Melnick). We will also present the corresponding classification results on complex manifolds.

• 10am-10:30am, atrium

Coffee and Tea

• 10:30am-11:30am, room G02

Katrin Leschke, Lopez-Ros deformation of minimal surfaces revisited

The Lopez-Ros deformation of a minimal surface in 3-space preserves completeness and finite total curvature. Lopez and Ros used this property to show that any complete, embedded, genus zero minimal surface with finite total curvature is a catenoid or a plane. The deformation since then has been used in various aspects of minimal surface theory, e.g., in the study of properness of complete embedded minimal surfaces or the Calabi-Yau problem. In this talk we will approach this deformation from the point of view of integrable systems: the Gauss map of a minimal surface is harmonic and thus we can define a dressing operation. Since a minimal surface is not uniquely determined by its Gauss map, to obtain a new minimal surface we also have to prescribe the deformation of the support function. The Lopez-Ros deformation is then indeed a special case of the simple factor dressing of the minimal surface. This allows to construct new (non-embedded) examples of doubly-periodic minimal surfaces and singly periodic minimal surfaces with planar ends.

• 11:30am-12:30pm, room G02

Lynn Heller, Navigating the space of symmetric CMC surfaces

We consider conformal immersions from a compact Riemann surface into the round 3-sphere with constant mean curvature (CMC surfaces). For tori integrable systems techniques lead to a deep understanding of the moduli space of all CMC immersions. In this talk I want to introduce an integrable systems approach for higher genus CMC surfaces with symmetries, which has been experimentally exploited to obtain a picture of the moduli space of embedded CMC surfaces of genus 2. This is joint work with S. Heller, N. Schmitt.

• 12:30pm-2pm

Lunch

• 2pm-3pm, room 107

Franz Pedit, Integrable surface geometry for surfaces of non-abelian topology

We give a short historical introduction to the theory of constant mean curvature surfaces and discuss recent advances for surfaces whose fundamental groups are non-abelian. We explain how to use the Riemann-Hilbert correspondence between local systems, representation varieties and holomorphic bundles to give a description of those surfaces in terms of algebro-geometric data. Computer experiments and visualizations support and guide the theoretical investigations making the talk quite accessible to non-experts.

• 4pm-4:30pm, atrium

Coffee and Tea