2023 Differential Geometry Seminars

School of Mathematical Sciences – The University of Adelaide.

These seminars are held in hybrid mode.
Time & location in semester 1, 2023: Fridays at 10:10am in Enginieering Nth N132 and on Zoom.
Time & location in semester 2, 2023: Fridays at 10:10am in Enginieering Maths EM213 and on Zoom.
Contact: Thomas Leistner (for Semester 1) and David Baraglia (for Semester 2).

2023 Seminars

• Michael Albanese (University of Waterloo)
  Title: Aspherical 4-Manifolds, Complex Structures, and Einstein Metrics
  Date: 24 November 2023, 10:10am
  
  Abstract: Using results from the theory of harmonic maps, Kotschick proved that a closed hyperbolic four-manifold cannot admit a complex structure. We give a new proof which instead relies on properties of Einstein metrics in dimension four. The benefit of this new approach is that it generalizes to prove that another class of aspherical four-manifolds (graph manifolds with positive Euler characteristic) also fail to admit complex structures. This is joint work with Luca Di Cerbo.
   
• Thomas Fils (University of Sydney)
  Title: Translation surfaces and isoperiodic foliation
  Date: 10 November 2023, 10:10am
  
  Abstract: Translation surfaces are surfaces obtained by gluing sides of polygons with translations. The space of these surfaces, their moduli space, lies at the crossroads of dynamical systems and hyperbolic geometry on surfaces. It appears naturally when exploring both billiards trajectories and geodesics in the moduli space of curves. I will give an introduction to these objects and explain these connections. The moduli space of translation surfaces is naturally stratified by prescribing conical singularities, and carries a foliation called the isoperiodic foliation. I will give a characterisation of the leaves that intersect a given stratum.
   
• John Rice (University of Adelaide)
  Title: Bundle gerbes in all dimensions
  Date: 20 October 2023, 10:10am
  
  Abstract: Bundle gerbes were introduced by Michael Murray as a geometric interpretation of integral degree 3 cohomology generalising the interpretation of degree 2 cohomology in terms of vector bundles. It generalises the gauge field interpretation of electromagnetism to some of the fields in string theory. In his thesis Danny Stevenson extended the theory to degree 4 cohomology with the concept of bundle 2-gerbes. In this talk I will show how to define bundle gerbes in all dimensions in a way that recovers the key results of the theory. This work was developed in conjunction with Michael and Danny.
   
• Dionne Ibarra (Monash University)
  Title: Change in framing of links in 3-manifolds via ambient isotopy
  Date: 1 September 2023, 10:10am
  
  Abstract: In this talk we will present work on the change of framing of knots and links via ambient isotopy in 3-manifolds by using D. McCullough's results on a generalized definition of Dehn twist homeomorphisms. In particular, we will discuss work by P. Cahn, V. Chernov, and R. Sadykov for framed knots and then the generalization of their work (joint work with R. P. Bakshi, J, H. Przytycki, G. Montoya-Vega, and D. Weeks) to links by showing that the only way of changing the framing of a link by ambient isotopy in an oriented 3-manifold is when the manifold admits a properly embedded non-separating S^2. We will then use spin structures to show that the ambient isotopy is a composition of even powers of Dehn homeomorphisms along the disjoint union of non-separating 2-spheres.
   
• Marcos Orseli (The University of Adelaide)
  Title: Equivariant index on toric contact manifolds
  Date: 25 August 2023, 10:10am
  
  Abstract: I will discuss the equivariant index of the horizontal Dolbeault operator on compact toric contact manifolds of Reeb type. This operator is transversally elliptic to the Reeb foliation and it features notably in calculations of partition functions of cohomologically twisted gauge theories. I will describe how to evaluate the index in general and give an explicit expression for it in terms of the moment cone. This is joint work with Pedram Hekmati.
   
• Miles Simon (Otto von Guericke University Magdeburg)
  Title: Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds
  Date: 2 June 2023, 10:10am
  
  Abstract: In this talk we examine the Ricci flow of initial metric spaces that are Reifenberg and locally bi-Lipschitz to Euclidean space. We show that any two solutions starting from such an initial metric space, whose Ricci curvatures are uniformly bounded from below and whose curvatures are bounded by $c\cdot t^{-1}$, are exponentially in time close to one another in the appropriate gauge. As an application, we show that smooth three dimensional, complete, uniformly Ricci-pinched Riemannian manifolds with bounded curvature are either compact or flat, thus confirming a conjecture of Hamilton and Lott. This is joint work with Alix Deruelle and Felix Schulze.
   
• Vladimir Matveev (Friedrich Schiller University Jena)
  Title: Separation of variables for spaces of constant curvature
  Date: 26 May 2023, 10:10am
  
  Abstract: I will discuss orthogonal separation of variables for spaces of constant curvature, with the emphasis on pseudo-Riemannian metrics. I will give a local description of all possible separating coordinates, and write the transformation to flat coordinates. The problem was actively studied since at least hundred years: different aspects of separations were considered by Stäckel in the 19th century and by Eisenhart in the beginning of the 20th and in a series of papers and books of E. Kalnins, J. Kress and W. Miller, I will give a historical overview.
  The new results are joint with A. Bolsinov and A. Konyaev and appeared within the Nijenhuis Geometry project and are a by-product of our classification of compatible inhomogeneous geometric Poisson brackets of degree 3+1; separation of variables is closely related to compatible pencils of Poisson brackets of degree 1. They are also closely related to geodesically equivalent metrics; all these will be mentioned if the time allows.
   
• Kyle Broder (University of Queensland)
  Title: Curvature Aspects of Hyperbolicity in Complex Geometry
  Date: 19 May 2023, 10:10am
  
  Abstract: A compact complex manifold X is said to be Kobayashi hyperbolic if every holomorphic map from the complex plane to X is constant. An extension of a conjecture of Kobayashi predicts that all compact Kobayashi hyperbolic manifolds are projective and admit a Kähler--Einstein metric with negative Ricci curvature. We will discuss the curvature aspects of Kobayashi hyperbolic manifolds and their study via Gauduchon connections. We will also present a "positive analog" of the Kobayashi conjecture.
   
• Kyle Broder (University of Queensland)
  Title: Complex Manifolds of Hyperbolic and Non-Hyperbolic-Type
  Date: 18 May 2023, 11:10am
  
  Abstract: A general overview of the role played by curvature in the study of complex manifolds of hyperbolic-type and non-hyperbolic-type.

• Damodar Rajbhandari (Jagiellonian University Krakow)
  Title: Generalised spectral dimensions in non-perturbative quantum gravity
  Date: 12 May 2023, 10:10am
  
  Abstract: The crucial tasks of quantum gravity research is to find observables that show possible deviations from classical gravity. One such example is the scale dependent effective dimension of spacetime. It was first discovered in Causal Dynamical Triangulations (CDT) which is a model of non-perturbative quantum gravity. Subsequently, it has been discovered to be present in many different theories of quantum gravity. This seemingly universal phenomenon of scale-dependent effective dimensions in non-perturbative theories of quantum gravity has been shown to be a potential source of quantum gravity phenomenology. The scale-dependent effective dimension from quantum gravity has only been considered for scalar (and dual-scalar) fields. It is however possible that the non-manifold like structures, that are expected to appear near the Planck scale, have an effective dimension that depends on the type of field under consideration. To investigate this question, we have studied the spectral dimension associated to the Laplace-Beltrami operator generalised to k-form fields on spatial slices of the CDT spacetime. We have found that the two-form, tensor and dual scalar spectral dimensions exhibit a flow between two scales at which an effective dimension appears. However, the one-form and vector spectral dimensions show only a single effective dimension. The fact that the one-form and vector spectral dimension do not show a flow of the effective dimension can potentially be related to the absence of a dispersion relation for the electromagnetic field, but dynamically generated instead of as an assumption.

• Marcy Robertson (University of Melbourne)
  Title: Loop spaces, Lie bialgebras and variations on Teichmüller space
  Date: 5 May 2023, 10:10am
  
  Abstract: A fundamental idea in “Grothendieck-Teichmüller theory” is that one can study the absolute Galois group of the rationals by studying the actions of the Galois group on the “Teichmüller tower”, i.e. the collection of (geometric) fundamental groups of all the moduli spaces $\mathcal{M}_{g,n}$ and the natural maps between them. This transforms a difficult arithmetic problem into a geometric problem.
  The goal of this talk is to explain an extension of this idea where we replace one mysterious group (the absolute Galois group) with a family of mysterious groups (the higher genus Kashiwara-Vergne groups). The associated “KV tower” is a tower of fundamental groups of the free loop spaces on various punctured Reimann surfaces. We aim to describe how this variation on Teichmüller space turns a problem in Lie theory (the Kashiwara-Vernge problem) into a problem in topology.

• Timothy Buttsworth (University of Queensland)
  Title: Cohomogeneity One Ancient Ricci Flows
  Date: 31 March 2023, 10:10am
  
  Abstract: The construction and classification of ancient solutions to the Ricci flow is a popular subject in geometric analysis which has arisen out of the need to provide useful models of finite-time singularities of Ricci flow. In this talk, I will discuss classical examples of ancient Ricci flows which are rotationally-invariant, as well as some more recent constructions of ancient flows on spheres which are invariant under a product of two orthogonal groups.

• Valentina-Mira Wheeler (University of Wollongong)
  Title: Arbitrarily high order concentration-compactness for curvature flow
  Date: 17 March 2023, 10:10am
  
  Abstract: We extend Struwe and Kuwert-Sch\"atzle's concentration-compactness method for the analysis of geometric evolution equations to flows of arbitrarily high order, with the geometric polyharmonic heat flow (GPHF) of surfaces, a generalisation of surface diffusion flow, as exemplar. For the (GPHF) we apply the technique to deduce localised energy and interior estimates, a concentration-compactness alternative, pointwise curvature estimates, a gap theorem, and study the blowup at a singular time. This gives general information on the behaviour of the flow for any initial data. Applying this for initial data satisfying $||A^o||_2^2 < \varepsilon$ where $\varepsilon$ is a universal constant, we perform global analysis to obtain exponentially fast full convergence of the flow in the smooth topology to a standard round sphere.The talk will focus mostly on the concentration compactness duality and how it translates from its initial appearance in elliptic setting in the literature to the parabolic setting of flows and finally to our geometric polyharmonic heat flow.
  This is joint work with James McCoy, Scott Parkins, and Glen Wheeler.

• Volker Schlue (University of Melbourne)
  Title: Scattering for wave equations with sources close to the light cone
  Date: 3 March 2023, 10:10am
  
  Abstract: For the classical wave equation there is an isometry from the space of initial data to the space of radiation fields. I will review an improved result which allows us to construct global solutions to the wave equation with detailed asymptotics from knowledge of the radiation field, when the latter is known to decay. For wave equations with sources near the light cone, the radiation field has a logarithmic singularity, and solutions have non-trivial decay properties in the interior of the light cone. I will discuss the scattering problem in this setting, and focus on the role of homogeneous solutions in the interior, and exterior of the light cone, and their role in the construction of matching approximate solutions. The sources we consider have applications to semi-linear wave equations satisfying the weak null condition as they appear in the study of Einstein's equations in harmonic coordinates. This is joint work with Hans Lindblad.