DIAMANT Symposium Spring 2023 & NMC 2023

Non-vanishing theorems for central L-values

Making and breaking post-quantum cryptography from elliptic curves

Most of the public-key cryptography in use today relies on the hardness of either factoring or the discrete logarithm problem in a specially chosen abelian group. Here “hard” does not mean mathematically impossible but that the best known algorithm to solve the problem has complexity (sub-)exponential in the size of the input. However, once scalable quantum computers become a reality, both factoring and the discrete logarithm problem will no longer be hard problems, due to Shor’s polynomial-time quantum algorithm to solve both problems. Post-quantum cryptography is about designing new cryptographic primitives based on different hard problems in mathematics for which there is no known polynomial-time classical or quantum algorithm.

A Unified Framework for Symmetry Handling

Handling symmetries in optimization problems is essential for devising efficient solution methods. In this presentation, we present a general framework that captures many of the already existing symmetry handling methods. While these methods are mostly discussed independently from each other, our framework allows to apply different symmetry handling methods simultaneously and thus outperform their individual effects. Moreover, most existing symmetry handling methods only apply to binary variables. Our framework allows to easily generalize these methods to general variable types. Numerical experiments confirm that our novel framework is superior to the state-of-the-art methods implemented in the solver SCIP.

The moduli space of rational elliptic surfaces of index two

A rational elliptic surface of index two is a rational surface that comes equipped with a (relatively minimal) genus one fibration that has exactly one multiple fiber of multiplicity two. In this talk I will describe how one can construct a moduli space for these surfaces when the choice of a bisection is part of the classification problem. This is based on joint work with Rick Miranda.

Counting reciprocal Littlewood polynomials with square discriminant

A Littlewood polynomial is a single-variable polynomial all of whose coefficients lie in {±1}. It is reciprocal if its list of coefficients forms a palindrome. We establish the leading term asymptotics of the number of reciprocal Littlewood polynomials with square discriminant. This relates to a bounded-height analogue of the Van der Waerden conjecture on Galois groups of random polynomials.

Combinatorial classification of critically fixed rational maps

A rational map on the Riemann sphere is called critically fixed (or conservative) if each of its critical points is also a fixed point. One may naturally associate a canonical combinatorial certificate (given by a planar embedded graph) to each such map. This allows to provide a classification of all (Möbius) conjugacy classes of critically fixed rational maps in combinatorial terms. We will discuss applications of this result in holomorphic dynamics and its analogs in the antiholomorphic setup.

Identifiability of Gaussian Mixtures from their sixth moments

I give a partial answer to an identifiability question from algebraic statistics: When do the truncated moments of a Gaussian mixture distribution uniquely determine the parameters? We will see that generically, a mixture of at most m=O(n^4) Gaussian distributions in n variables is determined by its moments of degree-6 moments. It is possible to answer the question by means of Algebraic Geometry. I show that the resulting Gaussian Moment variety of degree 6 is identifiable up to rank m=O(n^4). The approach relies on classical Terracini analysis coupled with new advances on Fröberg’s conjecture and the theory of secant varieties. The result implies sample-complexity bounds for the parameter estimation problem for Gaussian mixtures.

Approximation for generalized Campana points on projective space

In recent years there has been much interest into Campana points and related notions, such as integral points. For these points we introduce an analog of strong approximation and characterize when this holds on projective space.