Place: Konferenzraum, 5th floor - Mathematikon Im Neuenheimer Feld 205.
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.
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.
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.
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.
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.
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.