Weekly Bulletin

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.

In the era of big data, there has been an ever-​growing interest in designing fast algorithms. In classic algorithm design, the input data is revealed upfront, and the goal is to design algorithms that run in near-​linear time. However, in many real-​world applications involving graphs, the input is subject to frequent changes. This motivates the study of dynamic graph algorithms, which are data structures that maintain relevant graph information under vertex/edge updates. In this talk, we’ll discuss a general algorithmic tool for designing dynamic graph algorithms known as vertex sparsification. This is a compression paradigm that reduces large graphs into smaller ones while preserving properties or features of interest. In particular, we show that black-​box efficient constructions of vertex sparsifiers and their data-​structure variants lead to dynamic maintenance of effective resistances on graphs. We will also briefly discuss the implications of this technique in dynamically maintaining Laplacian systems and maximum flows.

More information: https://eth-its.ethz.ch/activities/its-fellows--seminar.htmlcall_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.

Digital signatures are used to confirm the legitimate sender of a message or key of a symmetric cryptosystem. The receiver of a message (or key) is able to verify with a public shared key if the signature, and thus the message has been sent from the person that it claims to be from. To do so one uses asymmetric cryptosystems, and therefore trapdoor one-way functions. In my master thesis two connected signature schemes are introduced: The unbalanced) Oil & Vinegar and Rainbow signature scheme; both of which use multivariate quadratic polynomials to produce signatures. The (unbalanced) Oil & Vinegar scheme uses a trapdoor which allows the signer to solve a linear set of multivariate equations, while a forger trying to attack the system needs to solve a set of multivariate quadratic equations. For well chosen parameters, the forgers task is way harder. The Rainbow signature scheme is based on (unbalanced) Oil & Vinegar and uses multiple layers of this scheme to make it more complex and more efficient. This talk will give a brief overview at different approaches to attack the signature schemes and explains which parameters are the most effective dependant on the attacks. Moreover, we'll see that implementation wise, both signature schemes can be kept short in the programmin language python and compare the computation times of both algorithms while discussing other aspects of security in foresight of Rainbow being a third-round finalist at NIST for the standardizising process in post-quantum cryptography. <BR> <BR> (**This eSeminar will also be live-streamed on Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact simran.tinani@math.uzh.ch **)

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.

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.

I will explain the sense in which Grothendieck toposes can act as unifying 'bridges' for relating different mathematical theories to each other and studying them from a multiplicity of points of view. I shall first present the general techniques underpinning this theory and then discuss a number of selected applications in different mathematical fields.

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.