Geometry, Dynamics, and Computer-Assisted Proofs

Program and Abstracts

Place: Hörsaal, Ground floor - Mathematikon Building, Im Neuenheimer Feld 205.

Coffee breaks in the Foyer, located on level -1 of the Mathematikon Building.

Week Schedule

Click here to download the schedule

Titles and abstracts

Wednesday: Keynote speakers

Modern Mathematics in the times of LLMs

Javier Gómez Serrano

In this talk, I will show what the combination of computer-assisted proofs, mathematical expertise, and modern mathematics via AI and LLMs can do. This paradigm will completely reshape the field of mathematics.

The math setting for Newton's Laws

Dennis Sullivan

There is more flexibility in the formulation of 3D Newtonian fluid motion in triple-periodic Euclidean space. This is related to the symmetry group of 3x3 integer matrices of determinant one. Combining this arithmetic symmetry with random products of these matrices suggests new tools and new results.

Towards the Nielsen-Thurston classification for surfaces of infinite type

Mladen Bestvina

The fundamental theorem of Thurston states that any homeomorphism of a surface of finite type can be isotoped so that some multi-curve is invariant and so that for every complementary component the first return map is either periodic or pseudo-Anosov. Homeomorphisms of infinite type surfaces are much more complicated. In this work, we

focus on the class of tempered homeomorphisms -- these are the ones that do not have any mixing behavior. We show that up to isotopy, there is an invariant geodesic lamination so that the first return maps display one of three qualitatively different behaviors. This work is joint with Federica Fanoni and Jing Tao.

3-Manifold groups, profinite rigidity, and finiteness properties

Martin Bridson

Concentrating on specific 3-manifold groups and trying to keep Rich entertained, I shall discuss aspects of the struggle to determine which groups are uniquely determined by their finite quotients and which are not. The distinction between finitely generated groups and finitely presented ones plays a surprisingly subtle role..

Wednesday: Lightning talks

Formalizing non-discrete trees in Lean

Raphael Appenzeller

Lambda-trees are a generalization of combinatorial trees and R-trees (0-hyperbolic geodesic metric spaces) and they are defined by three axioms. We investigate the (in)dependence of these axioms and how this was formalized in Lean.

Computer-assisted detection of integrability of birational maps

Yuri Suris

I will discuss how to detect and to prove integrability of birational maps in situations where the rational integrals of motion are prohibitively complex.

Black Holes and Glass Blocks

Steve Trettel

A spacetime containing multiple black holes is generically dynamic — gravitational radiation prevents any closed-form metric, and ray-tracing must proceed numerically on a manifold known only through simulation. But there's a beautiful exception: the Majumdar-Papapetrou solutions to Einstein-Maxwell, describing extremally charged black holes in static equilibrium, where gravitational attraction and electrostatic repulsion exactly cancel. In this lightning talk we exploit the extra symmetry of this system to build an exact analog of this curved spacetime inside the classical optics, and render accurate relativistic images by tracing through a medium of varying index of refraction.

Magnetic geodesics on the 2-sphere

Lina Deschamps

We will introduce through visualization what are magnetic deformations of the geodesic flow (or rather, what they look like!), and discuss quaternionic symmetries.

Finding Periodic Orbits via Homotopy Stretching

Tom Stalljohann

Starting from a Morse-Bott functional of loops we consider a sufficiently C^0-close perturbed functional. Arguing via homotopy stretching, which eventually boils down to a typical cobordism argument for moduli spaces, we obtain existence of critical points of the perturbed functional. Interestingly, if the perturbed functional equals the original Morse-Bott functional outside a prescribed set of loops, it must also have a critical point inside the prescribed set. We apply the above existence result to the setting of periodic solutions for electromagnetic Lagrangians on the sphere (and more generally on Zoll manifolds).

A Poincaré-Birkhoff theorem for crossed, possibly non-convex billiard tables

Tobias Witt

In this lightning talk, I will explain the following result: Given a smooth, possibly non-convex billiard table with two intersecting 2-bounce orbits A and B, one can – for every p/q in an interval specified by the endpoint curvatures- find a periodic orbit that links p times with A and q times with B. The proof combines holomorphic curve techniques with a simple quantum mechanical interpretation of the Morse index.

Which generically immersed circles in the plane bound an immersed disk?

Austin Konkel

In his 1967 thesis, Sandy Blank formulated and proved a combinatorial if and only if condition that answers which generically immersed circles in the plane bound immersed disks; Blank's solution also tells one how many suitably inequivalent bounding immersed disks there are. In this lightning talk, I will convey Blank's wonderful solution.

----Slides for this day----

Thursday: Keynote speakers

Rigidity of commensurators and groups acting on products of trees

David Fisher

I will discuss a problem that has frustrated me and many other people in recent years. It is something I first ran across in 2015, but it has evolved over time and admits multiple formulations, applications, approaches. I mention two here.

Can a surface group act properly on a finite product of bounded valence trees?
Are all discrete subgroups of lattices with dense commensurator lattices?

I hope to give a tour of these problems, including the connection between them and some account of the current results on the problems.

Complex hyperbolic triangle groups

John Parker

A complex hyperbolic triangle group is the group generated by three complex reflections, each fixing a complex line in complex hyperbolic space. Unlike the case of constant curvature, the angles between the lines do not determine the triangle: there is one extra real parameter. In his 2001 ICM talk, Rich gave a series of conjectures about when such groups are discrete. I will discuss these conjectures and other related work of Rich, and then I will survey later work that his results have inspired. Finally, I will report on two ongoing projects that refine and extend Rich’s conjectures.

The algebra and geometry of bordered Floer homology

Peter Ozsváth

Bordered Floer homology is a version of Heegaard Floer homology for three-manifolds with boundary. I will describe some of the algebraic and combinatorial objects which go into this construction.

Heegaard Floer homology is joint work with Zoltan Szabo; bordered Floer homology is joint with Robert Lipshitz and Dylan Thurston.

Reduction theory for Fuchsian groups with cusps

Svetlana Katok

We study a family of Bowen–Series-like maps associated to any finitely generated Fuchsian group of the first kind with at least one cusp. These maps act on the boundary of the hyperbolic plane in a piecewise manner by the generators of the group. We show that the two-dimensional natural extension (reduction map) of the boundary map has a domain of bijectivity and a global attractor with a finite rectangular structure. This provides a positive answer to a conjecture by Don Zagier, motivated by the problem of determining modular forms from their period polynomials.

Our work is based on the construction of a special fundamental polygon, related to the free product structure of the group, whose combinatorial structure and marking are preserved by “Teichmüller deformation.”

This is joint work with I. Ugarcovici and A.Abrams.

Billiard, Teichmuller, and Homogeneous Dynamics

John Smillie

The investigation of closed billiard trajectories on triangles has led to some interesting directions. We will discuss connections with work by Veech, Masur, and Eskin-Mirzakhani-Mohammadi. This last work was inspired by the connection with homogeneous dynamics and we will investigate the extent to which this analogy can be extended.

The illumination problem and the “Magic Wand” theorem

Barak Weiss

The “Magic Wand” theorem of Eskin-Mirzakhani and Eskin-Mirzakhani-Mohammadi is a far-reaching result regarding the dynamics of an action of the group SL(2, R) on the moduli space of translation surfaces. Its proof (in 2013) was the culmination of many years of work by many authors and employed tools in ergodic theory, probability, group theory, Teichmuller theory, and more. Surprisingly, this result has significant implications for the illumination problem, which is an elementary problem in plane geometry. I will present what is known about the illumination problem, give a (somewhat impressionistic) overview of the Magic Wand theorem, and explain the connection between the two. I will formulate a conjecture which might be within reach, given the present state of the art in the field.

Thursday: Lightning talks

Periodic geodesics on manifolds of mappings

Levin Maier

In this we will prove the Lysternik-Fet theorem on several infinite dimensional manifolds to be precise on manifolds of mappings like certain diffeomorphims groups or the space of immersed surfaces into R^d.

Billiard Table Maps

Lael Costa

When working with polygonal billiards, one typically asks about the orbit structure of a given table. In this talk, I explore the inverse problem: given an orbit, what table could have produced it? This produces a map, which one can iterate, on the space of polygons. I will discuss what is known (and what is not!) about these so-called billiard table maps for several billiard systems. These maps bear a striking resemblance to the beautiful pentagram map.

Thurston's asymmetric metric on half-translation metrics

Jiajun Shi

We generalise Thurston's asymmetric metric to the space of flat metrics coming from holomorphic quadratic differentials, by considering length ratio of simple closed curves.

All countable groups are QI groups

Paula Heim

Determining the quasiisometry group (QI group) of a metric space is, in general, hard and few QI groups are known explicitly. We present a construction which, given a countable group G, produces uncountably many pairwise non-quasiisometric metric spaces X with QI(X)=G. This talk is based on joint work with Joe MacManus and Lawk Mineh.

Compact character varieties of Right Angled Artin Groups (RAAGs)

Anunoy Chakraborty

In this talk, we study the character varieties of right-angled Artin groups (RAAGs) into compact Lie groups and analyze how the algebraic structure of the target group influences the topology of these moduli spaces. For a broad class of compact, connected Lie groups, including the Special Unitary Group SU(n), Symplectic Group Sp(n), and Unitary Group U(n), we prove that the corresponding representation varieties of RAAGs are connected. In contrast, we show that for other large families, such as the Special Orthogonal Group SO(n) and the Spin Group Spin(n) for n ≥ 3, these varieties are typically disconnected, revealing a sharp distinction driven by the topology of the target group.

We further examine star-shaped RAAGs with n leaves and describe the geometry of their SU(2) character varieties. We show that the smooth locus forms a trivial bundle over an interval, with fibers homeomorphic to the SU(2) character variety of a free abelian group of rank n. In the special case n = 2, we prove that the SU(2) character variety deformation retracts to the 3-sphere S³. These results illuminate how the combinatorics of a RAAG and the topology of the Lie group together shape the global structure of character varieties.

New thin subgroups of SL_3 (Z)?

Joaquin Lema

I'll discuss about a candidate for a thin subgroup of SL_3 (Z) that acts minimally on the space of flags. This candidate comes from a very geometric representation of the ideal triangle group that seems to be faithful up to words of length approx 100.

Willmore stability of minimal surfaces in the round sphere

Rob Kusner

We show the Lawson minimal surfaces in S^3 with area < 8π are minima for the Willmore bending energy W=ÍÍ 1+H^2 da among all nearby surface in S^n for any n≥3. The proof exploits the large symmetry groups of these surfaces to establish a gap in the area-Jacobi-operator spectrum between -2 and 0 using a subtle version of the Courant nodal domain method.

----Slides for this day----

Friday: Keynote speakers

Two or infinity — and beyond

Dan Cristofaro-Gardiner

The “two or infinity” conjecture of Hofer-Wysocki-Zehnder asserts that a Hamiltonian flow on a regular compact connected energy level has either two or infinitely many periodic orbits, when the energy level bounds a four-dimensional star-shaped domain. I will explain the context for this conjecture and some key ideas behind recent joint work resolving it. Then, I will briefly discuss some aspects of a different work, going beyond periodic orbits to establish the dense existence of non-trivial invariant sets. I'll finish with some open questions about where I think the next exciting frontiers lie.

Quadrilateral inequalities and harmonic maps

Peter Smillie

In 1991, Gromov asked: given a map f from a Riemannian manifold X to a nonpositively curved manifold Y, when is there a harmonic map h at bounded distance from f? In joint work with Max Riestenberg, we prove a sufficient condition which we apply to a problem in Higher Teichmuller Theory. In this talk, I will explain the metric geometry behind our proof.

The basic idea is that since the problem is coarse, the differential geometry of Y is not so important as its coarse metric geometry. The famous work of Korevaar and Schoen in the 90s on harmonic maps to nonpositively curved metric spaces showed that key differential geometric arguments can be replaced with quadrilateral inequalities, relationships between the six pairwise distances between four points. We follow the same principle but, in a new twist, with the Ptolemy inequality playing the main role.

Moduli spaces of graphs, surfaces, and handlebodies

Karen Vogtmann

Teichmuller space parameterizes marked Riemann surfaces, and Outer space parameterizes marked metric graphs. If you thicken a graph, you get a handlebody, whose boundary is a closed orientable surface. This picture leads to two different notions of a moduli space of marked complex handlebodies, which interpolate between Teichmuller space and Outer space. I will show how augmented versions of these spaces fit together in a nice way. This is joint work with R. Ramadas, R. Silversmith, and R Winarski.

Periodic orbits for all polygonal billiards

Giovanni Forni

We propose a proof by contradiction that all polygonal billiards, rational or non-rational, have a periodic orbit. The proof is based on a result of Galperin, Krieger, and Troubetzkoy and on the analysis of a one-parameter deformation of a natural (solvable) Riemannian metric on the 3-dimensional phase space of the geodesic flow on a flat surface with conical singularities. The contradiction comes from a lower bound on the first Betti number of the cut-locus (or skeleton) of the phase space with respect to a one-parameter deformation of the Riemannian metric for large parameter values.

Page updated

Google Sites

Report abuse