## KWG-PhD Prize

#### Factoring in number fields

The fundamental theorem of arithmetic states that every positive integer can uniquely be factored as a product of primes. Nowadays, factoring plays an important role in many fields of mathematics ranging from number theory to cryptology. In this talk we study factorization properties of other rings than the integers. Factorizations may no longer be unique in this new setting. We then introduce the class group, which measures the failure of unique factorization, and discuss its statistical properties.

#### Breaking of ensemble equivalence in complex networks

In random graph theory researchers often rely on machinery and concepts from statistical physics and interacting particle systems. A widely applied method uses mathematical models that have maximal entropy given the observed information. A (crucial) subtlety in this respect is that one should distinguish between the case in which the network information is exact’ (i.e. covers detailed information on individual network components) or on average’

(i.e. describes average properties of the underlying network). We obtain thus two mathematical models, viz. the microcanonical ensemble corresponding to exact observations’ and the canonical ensemble corresponding to average observations’. Traditionally, it was generally believed among physicists that both ensembles are equivalent (which we call ensemble equivalence).

Importantly, it was recently shown that one can construct examples in which there is no equivalence. In my work I made substantial progress in understanding under what conditions ensemble equivalence breaks. For the situation that the measurements correspond to the total number of edges in combination with the total number of triangles, I have developed an in-depth assessment of ensemble equivalence. Informally, I have proven the result that if the number of edges is somehow `misaligned’ with the number of triangles, there is breaking of ensemble equivalence. In addition, for a small perturbation around the typical values I have succeeded in computing the entropy of both ensembles, and have shown that they correspond to drastically different network structures. Specifically, my results show that decreasing the number of triangles in a network, while keeping the total number of edges fixed, is much more costly than increasing it.

#### Invariants of global fields: From Artin L-functions towards anabelian geometry

Let $$K$$ be a global field i.e. either a number field or global function field. These fields play the crucial role in every branch of modern mathematics from abstract algebra and number theory to applied cryptography and information theory. In order to study different properties of $$K$$ one attaches to it a list of different invariants. Some invariants such as degree, discriminant, regulator or Dedekind $$\zeta$$-function $$\zeta_K(s)$$ can be represented as numerical data. Meanwhile other invariants like normal closure, group of units, ideal class-group or the absolute Galois group $$\mathcal{G}_K= \text{Gal}(K^{sep}/K)$$ represent more sophisticated data attached to $$K$$.

Of course, one of the basic questions is to understand how much information about $$K$$ one can recover from those invariants mentioned above. For instance, an interesting problem could be stated as follows, “when does the Dedekind zeta-function $$\zeta_K(s)$$ recover the isomorphism class of $$K$$?”. This is known to be true, for example for normal extensions of $$\mathbb{Q}$$, but in general this is not the case and the answer is given by a beautiful group theoretical construction known as Gassmann triples. This problem is often called the isospectral problem for number fields, since it has strict parallel with the differential geometry phenomena known as isospectral manifolds. It turns out that $$\zeta_K(s)$$ determines a lot of other invariants of $$K$$ for example degree, discriminant, group of units, normal closure and in some sense almost determines regulator, ideal class-group. Meanwhile the famous Neukirch–Uchida Theorem states that the isomorphism class of $$K$$ is determined by the isomorphism class of $$\mathcal{G}_K$$, so $$\zeta_K(s)$$ allows us to know a lot about $$K$$, but not everything. Note that originally Neukirch proved this result for the case of normal extensions by recovering $$\zeta_K(s)$$ from $$\mathcal{G}_K$$. After that Uchida extended it to the case of all global fields, but he used a different and more sophisticated technique.

We will explain what kind of numerical information one can add to $$\zeta_K(s)$$ such that this data will determine the isomorphism class of $$K$$. This numerical information is encoded in the so-called Artin L-functions of Galois representations of $$K$$.

#### Large deviations for random walks in Riemannian manifolds

Large deviations is the area where one quantifies the exponentially small probabilities of deviations on the scale of the Law of Large Numbers. It finds its applications in (among others) statistical physics and stochastic control.

For random walks and diffusions, large deviation properties have mostly been studied in a vector space context. We initiate the study of large deviations in a geometric setting such as Riemannian manifolds. This is motivated for example by nanobiology, where one wants to understand the (random) movement of proteins through a cell-membrane. Because of the small scale, it is necessary to take into account the curvature of the cell-membrane. Additionally, it is also motivated by more applied fields such as geometric data analysis and computer vision.

The aim of this talk is to introduce a first result in this direction, namely a generalization of Cramér’s theorem on large deviations for a sequence of rescaled random walks. Following Erik Jørgensen’s work, “The central limit problem for geodesic random walks,” published in 1975, we introduce the generalization of the concept of a random walk to Riemannian manifolds, so called geodesic random walks. Using this, we explain the statement of the generalization of Cramér’s theorem to Riemannian manifolds and compare this result to the Euclidean setting.