Department of MathematicsPossible Topics for PhD Theses in Mathematics A PhD student can freely choose the subject for her/his thesis. For PhD studies in mathematics, it is not absolutely necessary to choose a PhD topic at the time of the application.

Here are nevertheless some suggestions for possible research topics, which the department is particularly qualified to supervise.

For information about ongoing research at the department, please see the webpages of the research groups and the personal homepages of our researchers RESEARCH OBJECTIVES AND MERIT OF THE PROPOSAL population. b) Numerical analysis and computer simulations will be undertaken to put theory and .

The department of mathematics has three research group in pure mathematics: Algebra, Geometry and Combinatorics, Analysis and Logic. Current and/or potential PhD advisors areThe goal of the project is to use calculus of functors, operads, moduli spaces of graphs, and other techniques from algebraic topology, to study spaces of smooth embeddings, and other important spaces.

High-dimensional long knots constitute an important family of spaces that I am currently interested in. m+i $ that agrees with the inclusion outside a compact set. The overarching goal of this project is to understand the dependence of the space $\text Emb c(\mathbb R The framework for doing this is provided by orthogonal calculus of functors, that was developed by Michael Weiss.

The following are some of the specific objectives of this project. To analyse the structure to polynomial functors in orthogonal calculus.

To show that it is possible to use orthogonal calculus to study the space of long knots $\text Emb c(\mathbb R To describe explicitly the derivatives (in the sense of orthogonal calculus) of functors such as $\text Emb c(\mathbb R

m+i )$ in terms of moduli spaces of graphs similar to ones introduced by Culler and Vogtmann.

The $n$-th derivative functor should be closely related to the moduli space of graphs that are homotopy equivalent to a wedge of $n$ circles 15 Mar 2010 - Please note:ааThis is a sample PhD thesis proposal for the School of Geography Here we use numerical modelling as a tool to quantify Here, I will discuss the methods and models used in my research before explaining .

### Possible topics for phd theses in mathematics - department of

To compare two known $m+1$-fold deloopings of the space $\text Emb c(\mathbb R One of these deloopings, due to Dwyer-Hess, is homotopy-theoretic in nature and is given in terms of mapping spaces between operads.

The other one is geometric in nature, and is given in terms of "topological Stiefel manifolds" $TOP(m+i)/TOP(i,m)$. I would like to show that the two deloopings are equivalent when $i\ge 3$, and also, by contrast, that they are not even rationally equivalent when $i=0$.

The reason for this is that one delooping has the Pontryagin classes in its rational homotopy, while the other one does not. To clarify the connection between the delooping of $\text Emb c(\mathbb R

m+i )$, and $G i$ -- the group of self-homotopy equivalences of the sphere $S

More precisely I want to show that $G i$ is the limit of the delooping, as $m$ goes to infinity Uses original analysis and literature to support the feasibility of the approach. experimental/numerical/theoretical techniques, and metrics for evaluation. Your thesis proposal should be limited to 6 pages including figures and references. during graduate school, only a select few people will read your thesis proposal..

To clarify the relationship of the first derivative to topological cyclic homology and to Waldhausen's algebraic K-theory.

### Research proposal - what statistical analysis do i use? - researchgate

, the Euler characteristic, the fundamental group, cohomology groups, etc.

As the invariants are refined by adding more algebraic structure, complete classification becomes possible in favorable situations. For example, for closed surfaces the fundamental group is a complete algebraic invariant, for simply connected manifolds the de Rham complex with its wedge product is a complete invariant of the real homotopy type, and for simply connected topological spaces the singular cochain complex with its E-infinity algebra structure is a complete invariant of the integral homotopy type.

My research revolves around algebraic models for spaces and their applications. Here are some topics for possible PhD projects within this area:Automorphisms of manifoldsThe cohomology ring of the automorphism group of a manifold M is the ring of characteristic classes for fiber bundles with fiber M, which is an important tool for classification.

Tractable differential graded Lie algebra models can be constructed for certain of these automorphism groups. A possible PhD project here is to further develop these algebraic models, and in particular to further investigate a newfound connection to Kontsevich graph complexes.

This will involve a wide variety of tools from algebraic and differential topology as well as representation theory and homological algebra. String topology and free loop spacesThe space of strings in a manifold carries important information, e.

Its homology carries interesting algebraic structure such as the Chas-Sullivan loop product I'm wondering if anyone can help. I am currently trying to put together my research proposal and I am having a few difficulties. I am looking to do an onlin..

### Thesis proposal : modeling and numerical simulations of blood flows

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, varieties over finite fields and Galois representationsModuli 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. 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 This proposal concerns the mathematical analysis of global existence, This project will carry out a systematic research on the challenging topics to study the..

### Research proposal doing a literature review what is a literature review?

As the parameter $n$ increases, the n complex zeroes cluster in a very regular manner on a curve, the unit circle.

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. The main idea in these papers is often to consider the zero set as a measure and then use harmonic analysis, related to an algebraic curve, the so-called characteristic curve of the equation.

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.

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 analysisThe quest for algebraicity: The Langlands Program and beyondIn 1967, R.

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.

It has seen spectacular progress and varied applications, such as Wiles' proof of Fermat's Last Theorem and Ng 's proof of the Fundamental Lemma. 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. But many -- including perhaps Langlands himself -- grapple with the ultimate raison d tre of the program.

So what is really at the heart of the Langlands Program?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: (A) Algebraicity of automorphic representations (B) The Deligne-Serre ``interchange of characteristic'' approach to algebraicity, concerning both (A) but also a variety of other questions stratification of Shimura varieties and their Hasse invariants.

(D) Algebraicity of Griffiths-Schmid manifolds 24 Jan 2010 - assume that your research or analysis is a good idea; they will want to be As well as using tables to display numerical data, tables can be .

### Research in numerical analysis | the university of manchester

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.

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 1736!Possible directions of the research project are