## Math Calendar

**Title:** Challenges in modeling the transmission dynamics ofchildhood diseases

**Abstract:** Mathematical modelsof childhood diseases are often fitted using deterministic methods under theassumption of homogeneous contact rates within populations. Such models canprovide good agreement with data in the absence of significant changes inpopulation demography or transmission, such as in the case of pre-vaccine erameasles. However, accurate modeling and forecasting after the start of massvaccination have proved to be more challenging. This is true even in the caseof measles which has a well understood natural history and a very effectivevaccine. We demonstrate how the dynamics of homogeneous and age-structuredmodels can be similar in the absence of vaccination, but diverge after vaccineroll-out. We also present some fundamental differences in deterministic andstochastic methods to fit models to data, and propose techniques to fit longterm time series with imperfect covariate information. The methods we developcan be applied to many types of complex systems beyond those in diseaseecology.

**Speaker:** Felicia Magpantay (Queen’s university, Ontario, Canada)

**Title:** Challenges in modeling the transmission dynamics of childhood diseases

**Abstract:** Mathematical models of childhood diseases are often fitted using deterministic methods under the assumption of homogeneous contact rates within populations. Such models can provide good agreement with data in the absence of significant changes in population demography or transmission, such as in the case of pre-vaccine era measles. However, accurate modeling and forecasting after the start of mass vaccination have proved to be more challenging. This is true even in the case of measles which has a well understood natural history and a very effective vaccine. We demonstrate how the dynamics of homogeneous and age-structured models can be similar in the absence of vaccination, but diverge after vaccine roll-out. We also present some fundamental differences in deterministic and stochastic methods to fit models to data, and propose techniques to fit long term time series with imperfect covariate information.The methods we develop can be applied to many types of complex systems beyond those in disease ecology.

This thesis exploresthe application of stochastic control techniques in a parameter study toexamine the implications of climate taxes on a pension fund. The problem athand involves modeling a pension fund comprising various assets, with aspecific focus on the portfolio's emissions and the associated taximplications. This research aims to find the optimal allocation strategies formanaging the pension fund, considering both the portfolio's performance and theimpact of taxes. By utilizing stochastic control, we explore how variousfactors, such as risk preferences, tax regulations, and emissionconsiderations, influence the optimal investment and consumption policies.

To solve this complex problem, the BCOS method isemployed to numerically tackle the fully coupled Forward-Backward StochasticDifferential Equations (FBSDEs) arising from the control problem. The BCOSmethod, known for its effectiveness in solving coupled FBSDEs, is applied totackle the complexity of the problem. This numerical approach enables theexploration of different control parameters, allowing for a comprehensiveparameter study. Numerical experiments are presented to illustrate thedifferent outcomes by varying the parameters, thus revealing the dynamic natureof the solutions.

In this talk we will introduce the notion of arboreal skeletal in smooth manifold, discuss how the gradient can be perturbed to obtain an arboreal skeleton and touch on how the manifold can be recovered from the skeleton.

Second talk: Lie algebroid connections and holonomy were introduced by Fernandes in 2000. These present some remarkable features. First of all, the classical Ambrose-Singer theorem does not hold anymore: the holonomy algebra is not spanned by parallel transported curvature endomorphisms only, there are some additional terms concerning the isotropy algebras of the Lie algebroid that can make a flat connection have non-discrete holonomy. Secondly, since the definition of the Lie algebroid holonomy requires the use of paths lifted to the Lie algebroid, it is essentially a leafwise property, and it can jump from leaf to leaf. We will give examples of these two phenomena and we will sketch a new proof for the Ambrose-Singer-Fernandes theorem.

**Title****:** Predictive maintenance – from sensor measurements to prognostics, to maintenance planning

**Speaker****:** dr. Mihaela Mitici, Algorithmic Data Analysis group, Information and Computing Sciences department

**Bio****:** Mihaela Mitici has a PhD in Stochastic Operations Research, Department of Applied Mathematics, University of Twente. From 2016-2022 she was an Assistant Professor at Aerospace Engineering Faculty, TU Delft. She specializes in Operations Research, with a focus on stochastic processes, decision-making under uncertainty, applied probability theory, machine learning. Her main application domains are predictive asset maintenance and mobility. Her work has been awarded Best Paper Award 2nd prize at the 2022 Prognostics and Health Management Europe (PHMe), Thomas L. Fagan Award at the 2021 Reliability and Maintainability Symposium (RAMS), and Best Innovation Award at the 2021 AGIFORS Aircraft Maintenance Operations Special Session.

**Second talk**

**Title**: Conformal Probability: A personal Perspective. II. Percolation.

**Abstract**: The last two decades have seen the emergence of a new area of probability theory concerned with certain random fractal structures characterized by their invariance under conformal transformations. The study of such structures has had deep repercussions on both mathematics and physics, generating tremendous progress in probability theory, statistical mechanics and conformal field theory. In this series of talks, I will give a personal perspective on some aspects of this new area, focusing for concreteness on three specific examples: the Ising model, percolation, Brownian loops. The three talks will be independent and self-contained.

In this talk, I will consider the scaling limit of critical site percolation on the triangular lattice. This was the first model where conformal invariance was proved rigorously, thanks to Smirnov’s celebrated proof of Cardy’s formula for the scaling limit of crossing probabilities (between boundary arcs of a bounded domain). Much progress followed swiftly, but the conformal covariance of connection probabilities (between points in the interior of a domain), expected by physicists since the 1980s and explicitly conjectured by Aizenman in the 1990s, remained open. I will discuss a recent proof of this conjecture based on the conformal invariance of the percolation full scaling limit constructed by Newman and myself in the early 2000s.

**Third talk**

**Title**: Conformal Probability: A personal Perspective. III. Conformal Fields from Brownian Loops.

**Abstract**: The last two decades have seen the emergence of a new area of probability theory concerned with certain random fractal structures characterized by their invariance under conformal transformations. The study of such structures has had deep repercussions on both mathematics and physics, generating tremendous progress in probability theory, statistical mechanics and conformal field theory. In this series of talks, I will give a personal perspective on some aspects of this new area, focusing for concreteness on three specific examples: the Ising model, percolation, Brownian loops. The three talks will be independent and self-contained.

In this talk, I will first introduce the Brownian loop soup (BLS), a conformally invariant Poissonian ensemble of loops in two dimensions whose intensity measure is proportional to the unique (up to a multiplicative constant) conformally invariant measure on simple planar loops. The BLS is closely related to the Schramm-Loewner evolution (SLE), conformal loop ensembles (CLE), and the scaling limit of various models of statistical mechanics, such as the Ising model, percolation, the loop O(n) model. I will then discuss several observable quantities of the BLS (e.g., the winding number of loops around a point) and show that, when properly rescaled, they behave like conformal primary fields in conformal field theory (CFT). These fields are the building blocks of any CFT, and our results suggest that we have identified a new family of CFTs with novel properties, but closely related to SLE and CLE and to some of the standard models of statistical mechanics. (Based on joint work with A. Gandolfi, V. Foit and M. Kleban.)

Normalization yields a normal form approximation and the resulting (non-integrable) system can be reduced to 2 degrees of freedom. The non-trivial isotropies of the two coupled 1 : ±2 resonances prevent the reduced phase space from being a smooth manifold but the dynamics on the singular part is in fact easier to understand. On the regular part of the reduced phase space the distribution of equilibria turns out to be determined by a single polynomial of degree 4. These are the relative equilibria that determine the behaviour of the 3 normal modes when passing through the resonance.

Seminar webpage https://utrechtgeometrycentre.nl/ugc-seminar/

**Abstract**: The word entropy is used differently in different fields of science. In dynamical systems entropy measures the extent to which a dynamical system acts chaotically. Roughly speaking, the faster different orbits of the dynamical system mix together, the higher the entropy.

Complex dynamics provides a large variety of non-trivial dynamical systems for which the entropy can be determined rigorously. Examples include polynomials and rational functions in one complex variable, polynomial automorphisms in two complex variables and holomorphic endomorphisms of complex projective space in arbitrary dimensions. The entropy of these systems was described in classical works of Misiurewicz and Przytycki, Gromov, Lyubich, Freire, Lopes and Mane, and Bedford, Lyubich and Smillie. For example, the topological entropy of a polynomial of degree $d$ equals $\log(d)$. Moreover, there exists a unique ergodic probability measure for which the measure theoretic entropy is also $\log(d)$: the measure of maximal entropy. The support of this measure coincides with the Julia set: the set where the dynamical system acts chaotically.

In an ongoing project with Leandro Arosio, Anna Miriam Benini and John Erik Fornaess, we consider the entropy of entire functions: power series that converge on the entire complex plane. Naively, one can consider entire functions as infinite dimensional polynomials. This line of thought leads to the conjecture that these maps should always have infinite entropy. I will discuss the current state of this conjecture, as well as ideas for future work. If time permits I will also discuss entire maps in two complex variables.

Abstract: I will introduce the notion of semi-tautological systems, which are systems of subalgebras with minimal set of functoriality properties of the cohomology rings of the moduli spaces of stable curves. They are designed to study the structure of the cohomology of the moduli spaces of stable curves beyond the tautological ring. I will give a criterion for a given semi-tautological system to span all of cohomology in a given degree. Using this criterion and other results about the moduli space of curves, both topological and algebro-geometric, I will give several applications. These applications include a complete description of the thirteenth cohomology of the moduli space of stable n pointed curves of genus g for all g,n and that all cohomology classes of sufficiently high degree are tautological. This is joint work in progress with Hannah Larson and Sam Payne.

AG seminar webpage: https://webspace.science.uu.nl/~marse004/ag_seminar.html