16th Panhellenic Geometry Conference

27 Sept 2024, 08:30 → 29 Sept 2024, 13:00 Europe/Athens

Amphitheatre "Konstantin Karatheodory" (Department of Mathematics, NKUA)

Iakovos Androulidakis (National and Kapodistrian University of Athens), Panagiotis Gianniotis (National and Kapodistrian University of Athens)

Friday, September 27 - Sunday, September 29, 2024.

Department of Mathematics

National and Kapodistrian University of Athens

Biennial international research conference on Geometry, organised by Departments of Mathematics of Greek universities. The 16th instalment includes a session dedicated to Dionyssios Lappas and Maria Papatriantafillou.

Plenary speakers

Melanie Rupflin (University of Oxford)

Quantitative estimates for almost harmonic maps

 

Felix Schulze (University of Warwick)

Mean curvature flow with generic initial data

 

Registration deadline: June 30, 2024

Abstracts submission deadline: June 30, 2024

Logo_uoa_blue.png

Friday, 27 September

Registration

Opening

1 A few words in honour of the retirement of Professors Dionysios Lappas and Maria Papatriantafyllou

2 Mean curvature flow with generic initial data

Mean curvature flow is the gradient flow of the area functional where an
embedded hypersurface evolves in direction of its mean curvature vector. This
constitutes a natural geometric heat equation for hypersurfaces, which ideally will
evolve the embedding into a nicer shape. But due to the nonlinear nature of the
equation singularities are guaranteed to form. Nevertheless, a key observation in
geometry and physics is that generic solutions, obtained by small perturbations, can
exhibit simpler singularities. In this direction, a conjecture of Huisken posits
that a generic mean curvature flow encounters only the simplest singularities. We
will discuss work together with Chodosh, Choi and Mantoulidis which together with
recent work of Bamler-Kleiner establishes this conjecture for embedded hypersurfaces
in R^3.

Speaker: Prof. Felix Schulze (University of Warwick)

Talk_Athens-3.pdf

11:00 Coffee break

3 Quasiconformal mappings in the Affine-Translational group

Τhe affine translational group, that is HXR, where H is the hyperbolic plane and R the set of the real numbers, endowed with a natural product, is one of the Thurston 3-geometries. It turns out that, like the Heisenberg group, a theory of quasiconformal mappings may be developed for this group. We briefly discuss the main features of the theory. This is a joint work in progress together with Z. Balogh and E. Bubani.

Speaker: Prof. Ioannis Platis (University of Patras)

athens-talk-aa-sep-2024.pdf

4 Uniformization in higher rank Teichmüller spaces and Fock bundles

Higher Teichmüller theory pertains to the study of special connected components of character varieties sharing analogous properties to the classical Teichmüller space. By fixing a complex structure on the underlying topological surface, one introduces powerful holomorphic techniques via Higgs bundles, the latter corresponding to fundamental group representations through the non-abelian Hodge correspondence. Yet, a rather adverse aspect of this correspondence is that is fails to transfer the action of the mapping class group on character varieties to the moduli space of Higgs bundles. We will introduce a similar class of augmented bundles over a topological surface that we call Fock bundles which does not require fixing any complex structure on the underlying surface. We conjecture that there is an alternative passage to the one given by the non-abelian Hodge correspondence from such pairs to certain higher rank Teichmüller spaces that is independent of the complex structure on the surface. This is joint work with Charles Reid (Leipzig) and Alexander Thomas (Lyon).

Speaker: Prof. Georgios Kydonakis (University of Patras)

12:50 Lunch break

5 Atiyah class and representations up to homotopy

The Atiyah class is an obstruction class for the existence of various geometric structures depending on the context. The original construction of Atiyah measures the obstruction to existence of holomorphic connections on holomorphic vector bundles. Since then the class has seen many applications and generalizations ranging from representation theory to quantum field theory. We construct the Atiyah class for extensions of representations up to homotopy of Lie algebroids using both the classical and the graded geometric picture in a way that recovers the class for ordinary representations. We illustrate the results in 2-term and 3-term representations. This is joint work with Sylvain Lavau.

Speaker: Prof. Panagiotis Batakidis (Aristotle University of Thessaloniki)

PCG 2024 Talk.pdf

6 On the twisted Ruelle zeta function and the Ray-Singer metric

In this talk we will present some results concerning the Fried's conjecture, i.e., the relation of the twisted dynamical zeta function of Ruelle at zero and spectral invariants for a hyperbolic manifold X. In particular, we consider the twisted Ruelle zeta function twisted by an arbitrary representation of the lattice. We study then its relation to the Ray-Singer norm of the refined analytic torsion. The refined analytic torsion is an element of the determinant line of the cohomology of X with coefficients in the flat complex vector bundle associated with the representation.

Speaker: Dr Polyxeni Spilioti (Universität Göttingen)

Spylioti_talk.pdf

16:20 Break

7 Witten instanton complexes on stratified pseudomanifolds

In his groundbreaking paper, "Supersymmetry and Morse theory", Witten categorified the classical Morse polynomials by constructing instanton complexes, and extended them to Dolbeault complexes on spaces with Kahler Hamiltonian actions in a follow up article, generalizing Lefschetz-Riemann-Roch formulas. We recently extended these results to stratified pseudomanifolds with wedge (asymptotically iterated conic) metrics, in particular deriving Morse inequalities for various Dirac operators in L2-cohomology. I will explain these results and some applications in mathematics and physics. This is based on work in arXiv:2404.13481, arXiv:2309.15845.

Speaker: Dr Gayana Jayasinghe (University of Illinois Urbana Champaign)

Athens slides Gayana.pdf

8 The $L^p$-spectrum of the Laplacian on forms over warped products and Kleinian groups.

In this talk we will present some recent results concerning the $L^p$-spectrum of the Laplacian on $k$-forms. We will generalize the set of manifolds over which the $L^p$-spectrum of the Laplacian on $k$-forms depends on $p$. We will consider the case of manifolds that are warped products at infinity and certain quotients of Hyperbolic space. In the case of warped products at infinity we will see that the $L^p$-spectrum of the Laplacian on $k$-forms contains a parabolic region which depends on $k$, $p$ and the limiting curvature $a_0$ at infinity. For quotients $M=\mathbb{H}^{N+1}/\Gamma $ with $\Gamma$ a geometrically finite group such that $M$ has infinite volume and no cusps, we will show that the $L^p$-spectrum of the Laplacian on $k$-forms is a exactly a parabolic region together with a set of isolated eigenvalues on the real line. This talk is based on my PhD thesis at the University of Cyprus.

Speaker: Dr Petros Siasos (University of Cyprus)

Saturday, 28 September

9 Quantitative estimates for almost harmonic maps

In the analysis of variational problems it is often important to understand not only the behaviour of exact minimisers and critical points, but also of maps that almost minimise the energy or that almost solve the associated Euler-Lagrange equation.

It is in particular natural to ask whether the distance of an almost minimiser to the nearest minimising state is controlled in terms of the energy defect and whether such a result not only holds in a qualitative, but in a sharp quantitative way.

In this talk we will discuss this and related questions for the classical Dirichlet energy of maps from surfaces into manifolds, in particular in the simple model problem of maps from the sphere S^2 to itself, for which minimizers (to given degree) are simply given by meromorphic functions in stereographic coordinates.

Speaker: Prof. Melanie Rupflin (University of Oxford)

Rupflin_Athens.pdf

11:00 Coffee break

10 Cartan's homotopy formula and the Linking number in rank-1 symmetric spaces

The linking number of two disjoint oriented null-homologous submanifolds K,L in an oriented manfold M is the primary obstruction to disentangling them. By means of the Thom isomorphism in deRham theory, the linking number can be rewritten as an integral over the product of the two submanfolds, analogous to Gauss' famous linking integral for disjoint loops in euclidean space.

The integrand of such linking integrals is closely related to the fundamental solution of the Cartan differential, i.e. to the integral kernel of a right inverse. In R^n such a fundamental solution is provided by the Riesz potential, which coincidentally also solves the adjoint equation. In magnetostatics, this is known as the Biot-Savart law which computes the magnetic field caused by a stationary current. In a series of papers, DeTurck, Gluck and Cantarella have constructed such a "Biot-Savart Operator" for space forms. We extend part of this to negatively curved symmetric spaces exploiting their harmonicity.

For the linking number however, one may ignore the adjoint equation, only a right inverse of the Cartan differential is needed. On the rank-1 symmetric spaces we apply Cartan's homotopy ("magic") formula to the gradient flows of the Busemann functions to construct this.

Speaker: Prof. Stefan Bechtluft-Sachs (University of Maynooth)

SBS-CartanFormula.pdf

11 Harmonic diffeomorphisms between pseudo-riemannian surfaces

We study locally harmonic maps between pseudo-Riemannian surfaces.
These maps are classified by the classification of the solutions of a
generalized sine-Gordon equation. We then study the one-soliton solutions
of this equation and we find the corresponding harmonic maps in a unified
way. Finally, we discuss a Backlund transformation of the harmonic map
equations that provides a connection between the solutions of two sineGordon type equations.

Speaker: Prof. Anestis Fotiadis (Aristotle University of Thessaloniki)

pseudosurfaces-2.pdf

12:50 Lunch break

12 New examples of harmonic maps to the hyperbolic plane via Bäcklund transformation

We study harmonic maps from a subset of the complex plane to a subset of the hyperbolic plane. In Fotiadis and Daskaloyannis (Nonlinear Anal 214, 112546, 2022), harmonic maps are related to the sinh-Gordon equation and a Bäcklund transformation is introduced, which connects solutions of the sinh-Gordon and sine-Gordon equation. We develop this machinery in order to construct new harmonic maps to the hyperbolic plane.

Speaker: Dr Giannis Polychrou (Aristotle University of Thessaloniki)

presentation.pdf

13 Smooth approximation of CMC hypersurfaces with isolated singularities

It is known that constant mean curvature (CMC) hypersurfaces, with or without boundary, in a smooth Riemannian manifold $(M, g)$ can develop singularities if $n\geq 7$. For a CMC hypersurface $S$ that minimises the prescribed mean curvature functional in a ball $B_r(0)$ with boundary $\partial S$ in $\partial B_r(0)$ and an isolated singularity at the origin we construct in enter link description here a sequence $S_j$ of smooth CMC hypersurfaces such that $S_j \to S$ in the Hausdorff distance sense. We employ techniques from both Geometric measure theory and Elliptic PDE theory to set up a perturbation problem and first construct a sequence of possibly singular CMC hypersurfaces that converge to the initial hypersurface $S$. In order to prove that these hypersurfaces are actually smooth we develop a singular maximum principle. This then allow us to blow-up along the possible singularities of the constructed sequence and come to a contradiction using the well-known Hardt-Simon foliation for area minimising tangent cones with isolated singularities.

Speaker: Dr Kostas Leskas (National and Kapodistrian University of Athens)

16:20 Break

14 Double Contact Conics In Involution

In a previous study of ours we considered two concentric conics $C_1,C_2$ intersecting at four points and we searched all conics having double contact with these two. As a solution we found an one-parameter family of conics, the so-called double contact conics of $C_1,C_2$. We noticed that this family creates a hyperbolic involution $f_{AB}$ on the pencil of lines through their common centre $O$, with double lines the lines of the common diameters $AC,BD$ of $C_1,C_2$. The lines of the contact diameters of every double contact conic $C_3$ with $C_1,C_2$ also correspond through $f_{AB}$.

In the present study we consider three concentric ellipses, mutually conjugate (i.e. starting with three coplanar line segments $OA,OB,OC$, we consider three concentric ellipses $C_1,C_2,C_3$ defined, so that every two of the line segments are conjugate semi-diameters of one ellipse) and we search all conics having double contact with these three ellipses. The problem of finding a fourth concentric ellipse circumscribed to all three is solved through the three-dimensional space by G.A.Peschka (1879) in his proof of K.Pohlke's Fundamental Theorem of Axonometry. Previous studies of ours, dealing with the problem as a two-dimensional one, proved with Analytic Plane Geometry that at most two solutions to the problem exist. The primary solution $T_1$ always exists and it is an ellipse. That's why the problem is referred as the Four Ellipses Problem. The secondary solution $T_2$, i.e. the second concentric conic circumscribed to all three, (if it exists) is an ellipse or a hyperbola. Furthermore, only $T_1$ was constructed using Synthetic Projective Plane Geometry in these studies.

The present study focuses on the investigation of the existence and on the construction of the secondary solution $T_2$ of the Four Ellipses Problem using methods of Synthetic Projective Plane Geometry, in particular the Theory of Involution.

First we proved with Analytic Plane Geometry that the common diameters of every couple of $C_1,C_2,C_3$ correspond through an involution $f$. In the sequel, criteria of Synthetic Projective Plane Geometry determine whether $f$ is hyperbolic or elliptic.

If $f$ is hyperbolic, then there exist exactly two conics $T_1,T_2$ concentric to $C_1,C_2,C_3$, that circumscribe $C_1,C_2,C_3$. The primary solution $T_1$, is always an ellipse, while the secondary solution $T_2$ is an ellipse, a hyperbola or a degenerate parabola, i.e. a pair of parallel lines. In any case, the common diameters of $T_1,T_2$ define the double lines of $f$.

If $f$ is elliptic, then there still exist two conics $T_1,T_2$ concentric to $C_1,C_2$, $C_3$, that have double contact with $C_1,C_2,C_3$. But this time only the primary solution $T_1$ is an ellipse circumscribed to $C_1,C_2,C_3$, while $T_2$ is an ellipse inscribed to $C_1,C_2,C_3$.

Regardless of whether $f$ is hyperbolic or elliptic, $T_2$ can now be constructed with Synthetic Projective Plane Geometry, using the already constructed $T_1$ and the involution $f$, since the contact diameters of $T_1,C_i$ and $T_2,C_i$, $i=1,2,3$ also correspond through $f$.

Speaker: Dr Anastasia Taouktsoglou (Department of Production and Management Engineering, Democritus University of Thrace)

Double Contact Conics in Involution (G. Lefkaditis-A. Taouktsoglou).pdf

Sunday, 29 September

15 Some results on the $L_p$-Brunn-Minkowski inequality for intrinsic volumes and the $L_p$-Christosffel-Minkowski problem

Speaker: Dr Konstantinos Patsalos (University of Ioannina)

talk_16th_panhellenic_geometry_conference_NKUA_2024_PATSALOS.pdf

10:40 Coffee break

16 Stability of the Moebius group and simple bubbles of the (2-dim) H-system

Speaker: Dr Konstantinos Zemas (Hausdorff Center for Mathematics Bonn)

Zemas_Abstract.pdf

17 Gradient Bounds and Liouville theorems for Quasi-linear equations on compact manifolds with nonnegative Ricci curvature

In this talk we establish a gradient bound and Liouville-type
theorems for solutions to Quasi-linear equations on compact Riemannian
manifolds with nonnegative Ricci curvature. Moreover, we obtain a
Harnack-type inequality and an Alexandrov-Bekelman-Pucci type estimate for
the gradient of solutions in domains contained in the manifold.
(This is joint work with George Zacharopoulos, EKPA)

Speaker: Dr Dimitrios Gazoulis (National and Kapodistrian University of Athens)