Title Promoter Affiliations Abstract "Dlt zeta functions" "Wim Veys" Algebra "Xu (2016) defines the dlt motivic zeta function associated to a regular function f on a smooth variety X over a field of characteristic zero.This is an adaptation of the classical motivic zeta function that was introduced by Denef and Loeser (1998). The dlt motivic zeta function is defined on a dlt modification via a Denef-Loeser-type formula, replacing classes of strata in the Grothendieck ring of varieties by stringy motives. This dlt motivic zeta function is, together with the dlt topological zeta function, a specialisation of the aforementioned zeta function, the protagonist of this thesis. We show that Xu's dlt motivic zeta function is, unfortunately, not well-defined: it depends on the choice of dlt modification. However, it is possible that the dlt topological zeta function is well-defined. To this end, we provide an explicit construction that yields a dlt modification for non-degenerate polynomials, based on ideas of Ishii (1997,1999). Moreover, we give an explicit formula that computes the dlt motivic zeta function and the dlt topological zeta function on such models. So far, we are not able to convert these ideas into a proof that the dlt topological zeta function is well-defined for non-degenerate polynomials. Using other techniques, we provide some evidence that suggests that the dlt topological zeta function is well-defined in dimension three. In particular, we show that for non-degenerate polynomials in three variables, the local dlt topological zeta function does not depend on the choice of toric dlt modification." "Dlt zeta functions and valuation spaces" "Johannes Nicaise" Algebra "This proposal lies at the intersection of three major fields in algebraic geometry: Igusa zeta functions for hypersurface singularities, the Minimal Model Programme, and non-archimedean geometry (in the form developed by Berkovich). The motivating problem is the monodromy conjecture for Igusa zeta functions, which remains unsolved after more than 40 years, in spite of important advances over the years and some recent breakthroughs. The conjecture predicts a connection between the poles of the zeta function (geometric information about the singularity) and local monodromy eigenvalues (cohomological information). We will build upon new insights from the Minimal Model Programme and its interactions with non-archimedean geometry to open a new line of attack on the conjecture." "Igusa type zeta functions and the monodromy conjecture" "Wim Veys" Algebra "For a polynomial f over the integers, there are important results and open questions relating its arithmetic and geometric properties. The famous monodromy conjecture is such an open problem. Here the arithmetic part consists of numbers of solutions of congruences, that is, solutions of the equation f=0 over the integers, modulo some given number. Technically, one gathers this information in a generating series, called the Igusa zeta function. The geometric part consists of the monodromy eigenvalues of f, which are invariants of the solutions of f=0 over the complex numbers, more precisely related to the non-smooth or singular points of this solution set. The conjecture predicts a precise link between these two a priori unrelated packages of information. Our main project consists in studying this conjecture, especially for polynomials in three variables. As a side project, we apply techniques from the study of Igusa zeta functions to questions in group theory." "Motivic zeta functions and Bernstein polynomials." "Wim Veys" Algebra "The monodromy conjecture is one of the most important open problems in singularity theory. Igusa formulated the conjecture in the seventies, motivated by his study of p-adic zeta functions. It predicts a precise relation between arithmetic properties of an integer multivariate polynomial f and geometric properties of the solutions of the equation f=0 over the complex numbers. The goal of the project is to incorporate the expertise of the research group 'Algebraic geometry and number theory' in high level international collaboration in order to achieve a more profound understanding, and in the long run a solution, of the conjecture. The foreign research partners are established specialists in singularity theory, motivic integration and non-archimedean geometry, domains playing a key role in the study of the monodromy conjecture." "Motivic integration and motivic Haar measure, zeta functions, and p-adic groups with the Howe - Moore property." "Johannes Nicaise" Algebra "In this research project, we study applications of p-adic and motivic integration to number theory, group theory and algebraic geometry. The main goals are the following:developing a theory of motivic integration with characters, with applications to p-adic representations and the Langlands program,applying motivic integration of Haar measures to the study of arithmetic properties of semi-abelian varieties,studying p-adic and motivic zeta functions associated to equivalence relations on group-theoretic objects,classification of p-adic algebraic groups with the Howe-Moore property. The project will be carried out in collaboration with Prof. F. Loeser (Ecole Normale Supérieure de Paris), Prof. J. Gordon (University of British Columbia), Prof M. du Sautoy (University of Oxford) and Prof. A. Valette (Université de Neuchâtel)." "Invariants of singularities: Poincaré series, Zeta functions and related problems." "Jan Denef" Algebra "The following topics are studied: the topological zeta function and its poles, the monodromy conjecture, Poincaré series, toric geometry, Newton polyhedra, combinatorics, multiplier ideals and jumping numbers, clusters of infinitely near points." "Absolute geometry and zeta functions" "Koen Thas" "Department of Mathematics: Algebra and Geometry" "The Deninger-Manin program aims at an Algebraic Geometry over the “ield with one element F1”which allows one to mimic Weil’ solution of the Riemann Hypothesis for function fields of curves in positive characteristic, to the classical Riemann Hypothesis. This mysterious “ield”already was mentioned in a 1957 paper of Tits, in which symmetric groups are seen as linear groups over F1. In 1992, Deninger described of a category of motives that would admit a translation of Weil’ proof to the hypothetical curve of integers. He showed that a certain Lefschetz-type formula would hold in which a terms ""h2"" appears. Manin proposed that this mysterious term be interpreted as the affine line over F1. The Riemann Hypothesis became a main motivation to search for geometry over F1. Since 2005, this new and emerging field started to grow rapidly, with several scheme theories over F1 being independently initiated by Connes—onsani, Deitmar, Lorscheid and others. Connes and Consani wrote a large number of papers on this subject, and many deep problems arise in their program. In this proposal, I want to attack foundational questions that are central in F1-theory, including a conjecture about zeta functions which is addressed in the Deninger-Manin program, the further development of F1-scheme theory to enable this approach, and questions of Connes and Consani relating the hyperstructure of the adèle class space of a global field to Singer actions of projective spaces." "Motivic zeta functions and the monodromy conjecture." "Johannes Nicaise" Algebra "The monodromy conjecture, formulated in the seventies by the Japanese mathematician Igusa, is one of the most important open problems in the theory of singularities. It predicts a remarkable connection between certain geometric and arithmetic invariants of a polynomial f with integer coefficients. The conjecture describes in a precise way how the singularities of the complex hypersurface defined by the equation f = 0 influence the asymptotic behaviour of the number of solutions of the congruence f = 0 modulo powers of a prime p. Some special cases have been proven, but the general case remains wide open. A proof of the conjecture would unveil profound relations between several branches of mathematics, in particular singularity theory and number theory. In the past years, we have developed a new interpretation of the monodromy conjecture, based on non-archimedean geometry, and we have generalized it to a larger framework. A significant success of this approach was our proof of the monodromy conjecture for one-parameter degenerations of abelian varieties. The aim of our proposal is to generalize this proof to degenerations of Calabi-Yau varieties, and to adapt the arguments to the local case of the conjecture (hypersurface singularities). Degenerations of Calabi-Yau varieties play a central role in the theory of Mirror Symmetry, and we will explore in detail the connections between the monodromy conjecture and recent developments in Mirror Symmetry (tropical constructions of degenerating Calabi-Yau varieties). We hope to achieve these goals by combining advanced tools from several research domains, in particular: motivic integration, non-archimedean geometry, Hodge theory, logarithmic geometry and tropical geometry." "Relation between residual beta cell function and glycemic variability in (pre)type 1 diabetes." "Pieter Gillard" "Clinical and Experimental Endocrinology" "Type 1 diabetes develops when 60 to 90% of insulin-producing beta cells ahve been destroyed. This cell loss leads to greater variability of blood glucose levels both before and after diagnosis. This variability is predictive of progression to clinical onset of diabetes in risk groups and of frequenc of hypoglycemic events in patients. Novel beta cell therapy trials aim to prevent or cure diabetes by trying to preserve or restore functional beta cell mass. In preparation of future trials the collaborating teams of the present application have validated dynamic tests to measure functional beta cell mass in vivo through prolonged stimulation of beta cells by elevated glucose levels (hyperglycemic clamp tests). The present application proposes to measure glycemic variability by continuous glucose monitoring (CGM) and frequent self-monitoring of blood glucose (SBMG) in 40 recent-onset type 1 diabetic patients and in 40 high-risk relatives (>50% 5-year risk of diabetes) (age 12-39 years) as a funtion of their residual functional beta cell mass as determined by hyperglicemic clamp. The participant will undergo 5 clamp tests and 5 periods of glycemic monitoring during a 2-year follow-up. Various parameters of glycemic variability will be derived from CGM and SMBG measurements and correlated with corresponding values of residual beta cell function (anticipated to vary between 10 and 100% of healthy controls in the proposed study groups) and parameters of metabolic control. This should allow to identify treatment goals for functional beta cell mass to be reached in therapy trials in order to avoid frequent hypoglycemia in patients and dysglycemia in risk groups." "Uncovering the role of iron metabolism in beta cell differentiation, maturation, and function." "Willem Staels" "Pathology/molecular and cellular medicine, Growth and Development" "Diabetes is a pandemic metabolic disorder. Current therapy strives for glycemic control but fails to tackle the underlying beta cell defect. Transplantation of pancreatic islets has shown that cell therapy can result in insulin independence, but donor islets are scarce. Stem cellderived beta cells (SC-beta) may offer a reliable cell source for transplantation. However, improved understanding of beta cell differentiation, maturation and function is needed to generate functional SC-beta cells. Beta cells specifically express high levels of the transferrin receptor (TFRC), which mediates cellular iron uptake from transferrin, compared to alpha and delta cells – the main endocrine cell types in pancreatic islets. It is currently unclear why this is the case and whether it is relevant for beta cell biology. In my proposal, I aim to uncover the role of iron metabolism in beta cell differentiation, maturation, and function. We will characterize changes in iron metabolism over time in the main endocrine islet cell types and disclose if in beta cells these changes relate to metabolic and signaling shifts key for maturation. We will study how iron deficiency affects beta cell development using novel mouse lines and cutting-edge technologies with the aim of improving protocols to generate SC-betas. This work will improve our understanding of diabetes pathophysiology by disclosing new intricacies of beta cell development and hopefully contribute to the development of a cellbased cure."