 The task of preparing a La'JEX-version of the text was extremely daunting, owing to the complexity and diversity of the symbolisms inherent in the various parts of the book. Our mathematics students, having been drilled for years in high-school and college, and having studied the immense edifice of analysis, regrettably come away convinced that they understand the concepts of real numbers, Euclidean space, and algorithm. Show sample text content. First from the point of view of physics: for physical results is it necessary to postulate an independent preexisting geometric substratum, i.

Geometry should one not integrate measurement in some way into physics, possibly in such a way that it cannot be later detached as a separate subconcept of purely "geometric". In the diagrams we use marks on lines in order to indicate sets of parallel lines. Addition Fig. A possible PhD project is to further develop these models, in particular to endow them with more algebraic structure, and use them to make new computations.

Moduli spaces are spaces that parametrize some set of geometric objects. These spaces have become central objects of study in modern algebraic geometry. One way of getting a better understanding of a space is to find information about its cohomology. In my research I have tried to extend the knowledge about the cohomology of moduli spaces when the objects parametrized are curves or abelian varieties.

The main tool has been the so called Lefschetz fixed point theorem which connects the cohomology to counts over finite fields. That is, counting isomorphism classes of, say, curves defined over finite fields gives information about the cohomology by comparison theorems also in characteristic zero of the corresponding moduli space.

## Inconsistent Mathematics

I have often used concrete counts over small finite fields using the computer to find such information. The cohomology of an algebraic variety that is defined over the integers comes with an action of the absolute Galois group of the rational numbers. Such Galois representations are in themselves very interesting objects.

A count over finite fields also gives aritmethic information about the Galois representations that appear. In the case of Shimura varieties at least according to a general conjecture which is part of the so called Langlands program one has a good idea of which Galois representations that should appear, namely ones coming from the corresponding modular and more generally, automorphic forms.

If one is not considering a Shimura variety, as for example the moduli space of curves with genus greater than one, it is much less clear what Galois representations to expect even though they are still believed to come from automorphic forms. Key words: algebraic geometry, moduli spaces, curves, abelian varieties, finite fields, modular forms.

This kind of behaviour is common in many other examples of sequences of polynomials, that, as here, are solutions to parameter dependent differential equations. The sequences occur in different areas, such as combinatorics, or special functions in Lie theory and algebraic geometry, and it is useful and interesting to understand the asymptotic properties of the polynomials through their zeroes.

A large amount of work has been done on this, in particular to determine what kind of curves in the complex plane that arise as asymptotic zero-sets.

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There are as yet few papers that consider the corresponding problem in higher dimensions, and this is the suggested topic, and one that I have just started with. It is then natural to use the differential-geometric concept of currents, instead of measures, and connected complex algebraic geometry.

## Set Theory and Foundations of Mathematics

Instead of having just one parameter dependent differential equation, one would consider holonomic systems of differential equations, such as GKZ-systems, that are important in some parts of algebraic geometry and algebraic topology. Holonomic systems come from the algebraic study of systems of differential equations, so-called D-module theory, and is a nice mixture of commutative algebra and analysis.

In particular I am interested in understanding the relation to the characteristic variety better, since I expect this to also give a better understanding of the one-variable case. Key words: complex algebraic geometry, D-module theory, varieties, hyper-geometric functions, harmonic analysis. Main supervisor: Wushi Goldring. In , R. Langlands wrote a letter to A.

It would revolutionize mathematics. It launched the now-famous Langlands Program. For almost half-a-century, this program has been a driving force in several areas of mathematics, particularly harmonic analysis, representation theory, algebraic geometry, number theory and mathematical physics.

At the same time, most instances of Langlands' conjectures remain unsolved. Fifty years later, all agree that the Langlands Program is indispensable for the unification of abstract mathematics. So what is really at the heart of the Langlands Program?

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The prevailing common view has been that large swaths of the Langlands Program are inherently analysis-bound, that an algebraic understanding of them is impossible. Under this view, the Langlands Program is seen as injecting analytic methods to solve classical problems in number theory and algebraic geometry. My research is focused on inverting the common view: My working algebraicity thesis is that, on the contrary, the Langlands Program is deeply algebraic and unveiling its algebraic nature leads to new results, both within it and in the myriad of areas it impinges upon.

In pursuit of my algebraicity theme, building on joint work with Jean-Stefan Koskivirta and other collaborators, I have begun a program to make simultaneous progress in the following four seemingly unrelated areas, by developing the connections between them:. Main supervisor: Pavel Kurasov. Quantum graphs - differential operators on metric graphs - is a rapidly growing branch of mathematical physics lying on the border between differential equations, spectral geometry and operator theory.

The goal of the project is to compare dynamics given by discrete equations associated with discrete graphs with the evolution governed by quantum graphs. Discrete models can be successfully used to describe complex systems where the geometry of the connections between the nodes can be neglected.

### Mathematical Foundations of Computing

It is more realistic to use instead metric graphs with edges having lengths. The corresponding continuous dynamics is described by differential equations coupled at the vertices. Such models are used for example in modern physics of nano-structures and microwave cavities. Understanding the relation between discrete and continuous quantum graphs is a challenging task leaving a lot of freedom, since this area has not been studied systematically yet. In special cases such relations are straightforward, sometimes methods originally developed for discrete graphs can be generalized, but often studies lead to new unexpected results.

To find explicit connections between the geometry and topology of such graphs on one side and spectral properties of corresponding differential equations on the other is one of the most exciting directions in this research area. As an example one may mention an explicit formula connecting the asymptotics of eigenvalues to the number of cycles in the graph, or the estimate for the spectral gap the difference between the two lowest eigenvalues proved using a classical Euler theorem dated to !