Schedule and abstracts

Classically, algebraic K-theory is an instrument that is used to study categories of objects, often of an algebraic flavour such as modules, by keeping track of how the objects decompose and asseble. More recently advancements have been made towards a version of this instrument that also applies to categories such as manifolds and algebraic varieties, which decompose and assemble through cutting and pasting. I will explain how this works, with the category of finite sets as a leading example. If time permits I will also talk about a result of my own: with just compact varieties, one can build the K-theory of the category of all varieties.

PDEs which model physical phenomena in nature often include a divergence condition coming from a physical conservation law. In a Finite Element Method (FEM) this condition should ideally hold to machine precision if the data is in the finite element space. However, in general the divergence condition is often violated in the discrete setting. I will talk about how we can save the divergence condition of a variant of the FEM called the Cut Finite Element Method.

We study the problem of the existence of community structure in graphs - such as for example friend groups in a Facebook friend-graph, functional units in the neuron graph of the brain, or scientific subfields in the coauthorship graph of papers. For sparse graphs, we show a lower bound on how much community structure any graph must have, as measured by a popular metric of community structure called modularity. This can be interpreted as a statement about any null model without actual community structure.

A general multi-parametric gradient boosting machine (GBM) approach is introduced. The starting point is a standard univariate GBM, which is generalised to higher dimensions by using cyclic coordinate descent. This allows for different covariate dependencies in different dimensions.Further, when having d-parametric distribution functions, it is important to design appropriate early stopping schemes. A simple alternative is introduced and more advanced schemes are discussed. The flexibility of the method is illustrated on simulated data examples using different multi-parametric distributions. Based on joint work with Mathias Lindholm, SU, and Łukasz Delong, Warsaw School of Economics.

Modular forms are functions on the complex upper half-plane satisfying a certain transformation law under elements from SL_2(Z). One may also see them as sections of a certain line bundle on the moduli space of elliptic curves. To obtain an analogous situation over function fields, we want to replace Z by A = F_q[T] and look at functions on the Drinfeld upper half-plane satisfying a transformation law under GL_2(A). These so-called Drinfeld modular forms can also be viewed as sections of a line bundle on the moduli space of Drinfeld modules. The talk will be an introduction to this circle of ideas.

The study of rare events is important in many fields, such as studying market crashes in finance, default probabilities in insurance and simulating extreme weather occurrences in climate models. The theory of large deviation estimates the tail behaviour  of stochastic processes by characterising the exponential decay rate of the probability of such rare events. In this talk I will give you an introduction to the field of large deviation and show some applications of the theory.

Do all artinian complete intersections have the weak and strong Lefschetz property? In the case of monomials, the answer is yes, but other than that not much is known. In this talk I will explain a little bit about this question and give a possible strategy for the case of a specific family of binomial complete intersections. Mainly, we will consider a directed labeled graph associated to such binomial complete intersections from which one can determine its vector space basis, its resultant and a Macaulay dual generator for it. Based on joint work with Samuel Lundqvist and Lisa Nicklasson.

In this talk I summarise the key results regarding sequential testing of Brownian Motion inspired by Shiryaev (1978), Lipster and Shiryaev (2001), and Peskir and Shiryaev (2006). While excellent materials, they do not present self-contained or exhaustive formulations of the optimal stopping problem derived from the Bayesian testing set-up, or the free-boundary problem that follows. The talk starts with motivating key concepts such as likelihood ratio process, innovations process, and filtering theory from the Bayesian decision rules, and continues with deriving the related optimal stopping problem and the respective free boundary problem. In addition, I present some standard applications for these problems, such as sequential testing between two drifts. If time permits, I will discuss a related quickest detection problem, and/or applications related to economical applications such as in Ekström and Tolonen (2023*).