Veranstaltungen

We know from a result by Avila and Forni that almost every interval exchange transformation is weakly-mixing. Until very recently, no (non trivial) example with discrete spectrum was known. In this talk, I will describe two families of interval exchange transformations which are conjugate to rotations of the two dimensional torus.

In the first part of the talk we will discuss some aspects of a celebrated theorem due to Bangert and Hingston which says the following: on any closed manifold Q, which is not a circle and has fundamental group Z, there exist prime-many geometrically distinct closed geodesics. In the second part we will explain how Bangert and Hingston's theorem can be restated in terms of Hamiltonian dynamics on S*Q and discuss the natural generalization from geodesics to Reeb orbits. Under an additional circle action assumption and the use of Floer theory, we proceed to give a proof of a Bangert and Hingston type result for closed Reeb orbits on non-degenerate starshaped hypersurfaces inside T*Q.

We consider the linear BCS equation, determining the BCS critical temperature, in the presence of a boundary, where Dirichlet boundary conditions are imposed. In the one-dimensional case with point interactions, we prove that the critical temperature is strictly larger than the bulk value, at least at weak coupling. We also show that the relative shift in critical temperature vanishes if the coupling constant either goes to zero or to infinity.

The P=W conjecture, first proposed by de Cataldo-Hausel-Migliorini in 2010, gives a link between the topology of the moduli space of Higgs bundles on a curve and the Hodge theory of the corresponding character variety, using non-abelian Hodge theory. In this talk, I will explain this circle of ideas and discuss a recent proof of the conjecture for GL(n) (joint with Junliang Shen).

In this series of talks, I will summarize some recent progress on degenerations and mirror symmetry. The first talk is to relate relative Gromov--Witten theory with absolute orbifold Gromov--Witten theory. The second talk is about structures in relative Gromov--Witten theory. The third talk is about relative mirror symmetry and applications.

Many models in machine learning and PDE-based inverse problems exhibit intrinsic spectral properties, which have been used to explain the generalization capacity and the ill-posedness of such problems. In this talk, we discuss weighted training for computational learning and inversion with noisy data. The highlight of the proposed framework is that we allow weighting in both the parameter space and the data space. The weighting scheme encodes both a priori knowledge of the object to be learned and a strategy to weight the contribution of training data in the loss function. We demonstrate that appropriate weighting from prior knowledge can improve the generalization capability of the learned model in both machine learning and PDE-based inverse problems.

In 1992, V. Turaev and O. Viro defined an invariant of smooth oriented closed \(3\)-manifolds consisting of labelling the edges of a triangulation of the manifold with representations of \(\mathcal{U}_{q}\!\left(\mathfrak{sl}_{2}\!\left(\mathbb{C}\right)\right)\) (\(q\) being a root of unity), associating a (quantum) \(6j\)-symbol to each tetrahedron of the triangulation, taking the product of the \(6j\)-symbols over all the tetrahedra of the manifold, then summing over all the admissible labelling representations. It is commonly admitted that this construction is a regularization of a path integral occurring in quantum gravity, the so-called "Ponzano-Regge model", which is a kind of \(\mathrm{SU}\!\left(2\right)\) BF gauge theory. A naive question is: Is it possible to define an abelian version of this invariant? If yes, is there a relation with an abelian BF gauge theory? These questions were answered positively in 2016, and the corresponding Turaev-Viro invariant is built from \(\mathbb{Z}/k\mathbb{Z}\) labelling representations (the equivalent of \(6j\)-symbols being "modulo \(k\)" Kronecker symbols) while the associated gauge theory is a particular \(\mathrm{U}\!\left(1\right)\) BF theory (with coupling constant \(k\)). This \(\mathrm{U}\!\left(1\right)\) BF theory can be straightforwardly extended to any finite dimensional closed oriented manifold, and so can be the Turaev-Viro construction built from \(\mathbb{Z}/k\mathbb{Z}\) labelling representations. A natural question is thus: Are these extensions still related? I will answer this question during the talk.

A classical object of study in low dimensional topology, and specifically knot theory, is the knot concordance group. The study and understanding of this group has a strong relation with the 4-dimensional topology of the knot, like for example its 4-genus. After an introduction on classical knot concordance, we will see an analogue construction of this group for strongly invertible knots (i.e. knots with a special symmetry), called the equivariant concordance group. We will also see some differences in the study of the 4-genus in the classical and equivariant context.

Der Vortrag beschäftigt sich mit Zauberkunststücken, die deutlich mehr Mathematik als simple, algebraische Umformungen benötigen. Dabei werden nicht nur die mathematischen Hintergründe erläutert, es wird auch tatsächlich gezaubert. Und schließlich gibt es noch Anregungen für die Integration von Magie in den Mathematikunterricht in der Schule. <br /> <a href="https://www.math.ethz.ch/content/dam/ethz/special-interest/math/math-ausbildung-dam/documents/kolloquium/Vortrag_ETHZ_Weng.pdf">Präsentation als PDF</a><br />

I will describe recent work giving an asymptotic formula for a count of primitive integral zeros of an isotropic ternary quadratic form in an orbit under integral automorphs of the form. The constant in the asymptotic is given explicitly in terms of local data determined by the orbit. Comparison with the well-known asymptotic for the corresponding count of all primitive zeros yields information on the distribution of the zeros in different orbits, the number of orbits r and the class number h of the genus of the form. For a certain special class of forms the distribution is shown to be uniform and a simple explicit formula is given for hr.

The minimal stratum H(2g-2) of the Hodge bundle over the moduli space of complex curves admits a natural volume form called the Masur--Veech volume. The total mass with respect to this volume, called the Masur--Veech volume, is of great importance in combinatorics, geometry and dynamics on Riemann surfaces. Zorich showed how to express the Masur--Veech volume as the asymptotic count of so-called square-tiled surfaces. Sauvaget found an explicit formula for the generating series of the Masur--Veech volumes of H(2g-2) via an orthogonal approach using intersection theory. We prove a refinement of Sauvaget formula via Zorich combinatorial approach. This refinement involves counting square-tiled surfaces with a fixed number of cylinders. It relies ultimately on the study of counting functions for certain families of metric ribbon graphs (equivalently -- double Hurwitz numbers with a single non-completed cycle). The top-degree terms of these counting functions turn out to be polynomials whose coefficients have a simple combinatorial description. Our result raises the question of intersection-theoretic interpretation of the volume contributions and of the coefficients of these polynomials.