## Math Calendar

Supervisor: Wioletta Ruszel

Second reader: Jason Frank

Abstract: Where local boundary value problems are related to classicalderivatives, non-local problems are stated in terms of fractional derivatives.In this talk, we discuss in particular the fractional Laplacian and how it relatesto fractional Gaussian fields. We will define Gaussian random fields on Banachspaces and see that a standard Gaussian field on a Hilbert space cannotexist. A viable alternative is the concept of an abstract Wiener space,which turns out to be equivalent to a centred Gaussian field. This theory isthen applied to the fractional Laplacian to obtain the fractional Gaussianfields. Finally, we look at the examples of the Gaussian free field and thebi-Laplacian field.

In this talk I will summarize what is known about torsion invariants on the Equivariant and Unitary Bordism groups, as well as the mysterious relation between the torsion invariants on surfaces and some birational invariants of finite groups.

Supervisors: Dr. ir. J.M. van denAkker, Dr. S. Dirksen and P. de Bruin, MSc.

**Abstract: **Electric bicycle delivery in cities is often faster, more reliable and more sustainable than using cargo vans. The benefits of these vehicles comes at the price of wind and load dependency of the travel time between customers. The vehicle routing problem models this type of scheduling problem. By taking into account the non deterministic nature of wind and a discretisation of the carried load, schedules can be computed that are expected to be cheaper and on time more often than implementations that ignore these factors or see them as fixed. By sampling from an assumed distribution of wind realisations, a set of arrival time distributions can be obtained that respects the dependency of travel times. By adapting existing code from using deterministic travel times to using the samples to create travel time distributions, an algorithm was built that is shown to produce schedules that behave better in simulations than schedules produced in the deterministic setting. As few as 5 samples was shown to improve both cost and robustness and 20 samples was enough to produce schedules that had an on time percentage of over 99.9 percent in simulations.

Abstract: In this talk, we provide a complete theory of stochastic integration

with respect to arbitrary cylindrical Lévy processes in Hilbert space. Since

cylindrical Lévy processes do not have a semimartingale decomposition,

our approach relies on a limit characterisation of Lévy characteristics and

the theory of decoupled tangent sequences to introduce the notion of the

stochastic integral. Our main result gives both necessary and sufficient

conditions for a predictable Hilbert-Schmidt operator-valued process to

be integrable with respect to an arbitrary cylindrical Lévy process in a

Hilbert space. As it turns out, our integrability conditions can be explicitly

expressed in terms of the cylindrical characteristics of the integrator,

thus establishing a direct relationship between existence of the stochastic

integral and properties of the cylindrical integrator.

Project supervisor: Gunther Cornelissen

Daily supervisor: Mikhail Hlushchanka

Second examinor: Valentijn Karemaker

**Abstract:** Iterated monodromy groups (in short, IMGs)are groups naturally associated to iterations of (anti-)rational maps on theRiemann sphere. In this thesis, we study the properties of the IMGs ofcritically fixed (anti-)rational maps; critically fixed maps being those mapswhose critical points are also fixed points. In particular, we prove that theIMGs of critically fixed (anti-)polynomials are regular branch on the subgroupof group elements with even permutational part. In the case of polynomials, wemake use of the one-to-one correspondence between the conformal conjugacyclasses of critically fixed polynomials and the isomorphism classes ofconnected planar embedded graphs. Similarly, in the case of anti-polynomials,we use that there is a one-to-one correspondence between the Möbius conjugacyclasses of critically fixed anti-rational maps and the equivalence classes ofunobstructed topological Tischler graphs. Furthermore, we discuss why the toolswe use in the case of critically fixed (anti-)polynomials are insufficient forproving the same statement in the more general case of critically fixed(anti-)rational maps.

The second part of the talk is about source Ehresmann connections. In particular, we will see that a Riemannian groupoid gives rise to a special type of source Ehresmann connection, called a weak Cartan connection. Using the cotangent groupoid, we can then construct an entirely new type of connection, whose dual is like a weak Cartan connection, these connections are then dubbed weak * connection.

Higher categorical models of HoTT

Supervisor: Paige Randall North

Second reader: Gijs Heuts

Abstract:

Martin Löf Dependent Type Theory (MLDTT) is a formal system forconstructive mathematical reasoning. The Univalent Foundations (UF) programenhances it with the Univalence Axiom and Higher Inductive Types, and proposesit as an alternative to ZFC for the foundations of mathematics. My thesisfocuses on the categorical semantics of UF.

The semantics of the elimination principle of the identity typecorrespond to lifting properties, a fundamental component of model categories,which serve as an abstract framework for homotopy theory. It has also beenshown that Kan Complexes, or equivalently ∞-groupoids, can support a model ofUF. These findings suggest a profound connection between logic, homotopy theory& higher category theory. Over the past decade, numerous attempts have beenmade to expand on these results.

In my thesis I look into two constructions relating type theories tomodel categories and higher categories. The first asserts that if T is MLDTT,subject to certain rules, then its category of contexts is a fibration categorywhose simplicial localization is an locally cartesian closed ∞-category. Thesecond asserts that any ∞-topos can be presented by a model category thatmodels UF.

Kind regards,

Alkis Ioannidis

The study of connections on Courant algebroids is relevant for the study of metric connections on manifolds with closed skew-symmetric torsion. In this talk we will first recall the basic concepts from generalized Riemannian geometry to then introduce the notions of parallel transport and holonomy on Courant algebroids and to establish their basic properties. To get a feeling for the theory, the second half will focus on a first study of the canonical Levi-Civita connection on exact Courant algebroids, culminating in the classification of the flat ones.

determine the set of rational points on curves. A key input for this method is the values of local height functions. In this talk, we will discuss an algorithm to compute these local heights at odd primes v not equal to p for hyperelliptic curves. Through applications, we will see how this work extends the reach of quadratic Chabauty to curves previously deemed inaccessible. This is joint work with Alexander Betts, Sachi Hashimoto, and Pim Spelier.