Veranstaltungen

We will set the scene by considering Cantor sets in the real line and generated by simple iterated function schemes and estimates on the value of their Hausdorff dimension. This has applications to problems related to the Zaremba conjecture and Lagrange spectra. We will then focus on the problem of estimating the (top) Lyapunov exponent for random matrix products. By way of an application, we will be interested in the value of the drift for Fuchsian groups and implications for the harmonic measure.

We discuss a question of Borman from 2012 on the relation between Entov-Polterovich quasimorphisms on a symplectic manifold and a Donaldson divisor therein.

We define integer valued invariants of an orbifold Calabi-Yau threefold X with transverse ADE orbifold points. These invariants contain equivalent information to the Gromov-Witten invariants of X and are related by a Gopakumar-Vafa like formula which may be regarded as a universal multiple cover / degenerate contribution formula for orbifold Gromov-Witten invariants. We also give sheaf theoretic definitions of our invariants. As examples, we give formulas for our invariants in the case of a (local) orbifold K3 surface. These new formulas generalize the classical Yau-Zaslow and Katz-Klemm-Vafa formulas. This is joint work with S. Pietromonaco.

When do equations have a rational solution? Even if you don't really care about this question (just like me), the theory behind it will fascinate you! I'll give an introduction to p-adic numbers and explain the local-global principle (Hasse principle). For advanced students, I'll conclude by explaining the Brauer-Manin obstruction.

More information: https://zucmap.ethz.ch/call_made

I will explain some new results showing that the Chow and cohomology rings of moduli spaces of stable curves in relatively low genus and low number of marked points are isomorphic and equal to the tautological ring. These computations involve both concrete geometric techniques in order to explicitly study various strata in the moduli spaces and more abstract techniques relating the computations in the Chow ring to those in cohomology. This part is joint work with Hannah Larson. Next, I will explain a surprising extended application of these results to the vanishing of the eleventh cohomology of moduli spaces of pointed stable curves of genus g at least 2. This part is joint work with Hannah Larson and Sam Payne.

This talk reviews 50-60 years of the theory of negative dependence of binary random variables, beginning with origins in mathematical statistics and statistical mechanics. This culminates in the Borcea-Branden-Liggett theory, which connects negative dependence to the geometry of zero sets of polynomials. It provides a reasonably checkable condition, which is satisfied in many examples, and has strong consequences such as negative association. The last part of the talk focuses on more recent (in the last ten years) work of various people. This concerns concentration inequalities, Lorentzian measures, and a CLT based on the geometry of zeros. The talk will end with some open problems.

Solutions of the classical Yang-Baxter equation (CYBE) are important elements in the theory of integrable systems and the theory of quantum groups, notably for their connection with Lie bialgebra structures. In this talk we will present a procedure that assigns a coherent sheaf of Lie algebras on a projective curve to any non-degenerate solution of the CYBE. This approach enables the use of algebro-geometric methods in the study of the CYBE. We will explain how these methods can be used to give a new proof of the Belavin-Drinfeld trichotomy, which states that non-degenerate solutions of the CYBE are either elliptic, trigonometric, or rational.

The classical uniformization theorem states that every simply connected Riemann surface is conformally diffeomorphic to the unit disk, the complex plane or the 2-sphere. The uniformization problem for metric surfaces now asks to generalize this to a non-smooth setting. After motivating this problem with an example arising from geometric group theory, I will introduce the necessary notions and present the main results on uniformization in metric spaces.

By assuming the existence of the growth optimal portfolio (GP) and maximizing the entropy for a hierarchically structured stock market, the paper derives the GP dynamics as those of time-transformed squared Bessel processes of dimension four. The average of each of their squared volatility components is shown to converge toward a common level. The initial values of basis security accounts turn out to be gamma-distributed. The risk-adjusted return of the GP is not depending on a savings account and can be significantly higher than classical assumptions allow to explain.

Statistical predictions should provide a quantification of forecast uncertainty. Ideally, this uncertainty quantification is in the form of a probability distribution for the outcome of interest conditional on the available information. Isotonic distributional regression (IDR) is a nonparametric method that allows to derive probabilistic forecasts from a training data set of point predictions and observations, solely under the assumption of stochastic monotonicity. IDR does not require parameter tuning, and it has interesting properties when analyzed under the paradigm of maximizing sharpness subject to calibration. The method can serve as a natural benchmark for postprocessing forecasts both from statistical models and external sources, which is illustrated through applications in weather forecasting and medicine.

Serge Lang has proposed several influential conjectures relating different notions of hyperbolicity for proper complex algebraic varieties. For example, he asked whether the locus swept out by entire curves coincides with the locus swept out by subvarieties not of general type. I will explain that some of these conjectures (including the one above) are true for varieties admitting a large complex local system.