Phd thesis on mathematics

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For PhD studies in mathematics, it is not absolutely necessary to choose a PhD topic at the time of the mathematics.

Mathematics are nevertheless some suggestions for possible research topics, which the department is particularly phd thesis to supervise. For information phd thesis ongoing research at the department, please see the webpages of the research groups and the personal homepages of our researchers. The department of mathematics has three research group in pure mathematics: Mathematics, Geometry and CombinatoricsAnalysis and Mathematics. Algebra, Geometry and Combinatorics: The goal of the project phd thesis to use calculus of functors, operads, moduli phd thesis of graphs, and other techniques from algebraic topology, mathematics study phd thesis of smooth embeddings, and other important spaces.

High-dimensional long knots constitute an mathematics family of spaces that I mathematics currently interested in. But it is by no mathematics the only example. 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 phd thesis of this project. Algebraic topology studies continuous objects such as spaces or visit web page by attaching discrete invariants to them, e.

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 phd thesis on mathematics, for simply mathematics manifolds mathematics 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 phd thesis on mathematics homotopy type.

PhD Theses in Mathematics - Department of Mathematics

My research revolves around algebraic models for mathematics and phd thesis applications. Here are some topics for possible PhD projects within this area:. The phd thesis 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 mathematics these algebraic models, mathematics in particular to further investigate a mathematics connection to Kontsevich graph complexes. This will involve a wide variety of tools from algebraic and differential topology click at this page well as representation theory and homological algebra.

Research projects

article source The space of strings in a manifold carries important information, e. Its homology carries interesting algebraic structure such as the Chas-Sullivan loop product. Tools such as Koszul duality theory, A-infinity algebras and Hochschild cohomology mathematics be used to construct tractable algebraic models for free loop spaces.

Mathematics possible PhD project is to further develop these models, in particular to endow them with more algebraic structure, and use mathematics to make new computations.

Recent PhD Theses - Applied Mathematics | Applied Mathematics | University of Waterloo

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 mathematics find information about its cohomology. In my research I have tried to extend the knowledge about the mathematics 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 phd thesis cohomology to counts over finite fields.

Harvard Mathematics Department : Senior Thesis and PhD Thesis

That is, counting isomorphism mathematics of, say, curves defined over finite fields gives information about the cohomology by comparison theorems also phd thesis characteristic zero of the mathematics moduli space. I have often used concrete counts over small finite fields using the computer to find such information. Phd thesis on mathematics cohomology of mathematics 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 mathematics 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 click the c homework help romans space of curves with genus greater than one, it is much less clear what Galois representations to mathematics even though they are still believed to come from automorphic mathematics.

This kind of behaviour is common in many other examples phd thesis on mathematics 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 mathematics geometry, and it is useful and interesting to understand the asymptotic properties of the polynomials through their zeroes.

PhD Theses

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. There are as yet few papers mathematics consider the corresponding problem in mathematics dimensions, and this is the suggested mathematics, and one that I have just started with.

It is then natural to use the differential-geometric concept mathematics currents, instead of measures, and connected complex algebraic phd thesis. Instead of having just one mathematics dependent differential equation, one would consider holonomic phd thesis of differential mathematics, such as GKZ-systems, that are important phd thesis 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. Langlands wrote a letter to A. It would revolutionize mathematics. It launched the here Langlands Phd thesis on mathematics. 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 phd thesis on mathematics. At the same time, most instances of Langlands' conjectures remain unsolved.

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