For more information please see the relevant PhD programme pages. Keeping this origin of the field in mind, my research merges the rapid development in structure theory of operator algebras that was achieved over the past decade with the needs of an emerging general theory of locally compact groups beyond Lie theory. Then $R$ is graded, and the algebra R can be decomposed as $R= R_0\oplus R_1 \oplus R_2 \oplus R_3$, and one can check by hand that the map $R_i \to R_{i+1}$ induced by multiplication by $x+y+z$ has maximal rank in every degree. One of these deloopings, due to Dwyer-Hess, is homotopy-theoretic in nature and is given in terms of mapping spaces between operads. When complex analysis and functional analysis meet, many interesting things can happen. A few suggestions for PhD topics are presented below. This problem is connected both to the Lefschetz properties, and to a conjecture by Iarrobino from 1997. Tools such as Koszul duality theory, A-infinity algebras and Hochschild cohomology can be used to construct tractable algebraic models for free loop spaces. This kind of behaviour is common in many other examples of sequences of polynomials, that, as here, are solutions to parameter dependent differential equations. One such topic are questions involving so-called Herglotz-Nevanlinna functions, these are functions mapping the complex upper half plane analytically onto itself. These spaces have become central objects of study in modern algebraic geometry. To clarify the relationship of the first derivative to topological cyclic homology and to Waldhausen's algebraic K-theory. In a recent series of papers, my coauthors and I have started making headway on the problem of identifying cyclic vectors in weighted Dirichlet spaces in the bidisk, and we have found techniques for checking membership in such spaces of functions having singularities on the boundary of the bidisk. Two possible PhD projects derived from this line of research are: Fine structure of operator algebras associated with groups acting on trees. 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. Develop a machinery to analyze integrability and regularity of rational functions by examining the geometry of their singularities. My recent research interests have been dealing with questions regarding both linear and multilinear operators of those described above, and in particular with those of rough type. In recent years, I have been particularly interested in weighted Dirichlet spaces, which can be defined in terms of area-integrability of partial derivatives of an analytic function. The OfS aims to help students succeed in Higher Education by ensuring they receive excellent information and guidance, get high quality education that prepares them for the future and by protecting their interests. Topics in Quantum Foundations: Ontological Models, and Distinguishability as a Resource: Michael Dumphy: The Influence of Mesoscale Eddies on the Internal Tide: Robert Huneault: Time-Optimal Control of Closed Quantum Systems: Colin Turner: Mathematical Modelling of Cancer Stem Cells The research facilities include one of the finest libraries in the country, the John Rylands University Library. 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. Since the spectra of most models cannot be calculated explicitly, there is a strong need for qualitative and quantitative estimates. The aim of the programme is to turn you into a competent independent researcher and the thesis you will write at the end of your programme should evidence this fact. As a MPhil student you will be assigned to a supervisory team. Roughly speaking, these operators act on functions (or signals) by filtering (attenuating or amplifying) specific frequencies of those. Study multilinear end-point results and results of minimal regularity assumptions, for paraproducts and their application to the study of boundedness properties of multilinear pseudodifferential and Fourier integral operators. Let $\text{Emb}_c(\mathbb{R}^m, \mathbb{R}^{m+i})$ be the space of all such knots. When does $k[x_1,\ldots,x_n]/(x_1^{d}, \ldots, x_n^{d}, m)$ have the WLP? project and research students can attend any of the postgraduate courses offered by the MAGIC consortium, Opportunities for Postgraduate research are available in a wide range of topics in Mathematics. I have often used concrete counts over small finite fields using the computer to find such information. dynamics is described by differential equations coupled at the vertices. Algebra, Geometry and Combinatorics: Gregory Arone, Jörgen Backelin, Alexander Berglund, Jonas Bergström, Rikard Bøgvad, Wushi Goldring, Samuel Lundqvist, Dan Petersen, Sven Raum, Boris Shapiro. Possible directions for a PhD project might be to: Last updated: Many research seminars Study spectral properties of continuous quantum graphs in relation to their connectivity and complexity (this question is well-understood for discrete graphs); Investigate relations between quantum graphs and quasicrystals; Develop new models combining features of discrete and continuous graphs and  study their properties; Transport properties of networks and their complexity. (B) Grothendieck's motives program in algebraic geometry, including the structures on the known realizations (Hodge theory, Galois representations, crystals etc. The reason for this is that one delooping has the Pontryagin classes in its rational homotopy, while the other one does not. The goal of the project is to compare dynamics given by discrete equations associated with (discrete) graphs with the evolution governed by quantum graphs. High-dimensional long knots constitute an important family of spaces that I am currently interested in. Indeed, associated with a locally compact group is its group C*-algebra and its group von Neumann algebras, which hence allows to study representation theory by algebro-analytic methods. Due to their rearrangement-invariant nature, these spaces are blind to the description of where solutions are concentrated, and thus the consideration of Lebesgue spaces with weights appears naturally. Here are nevertheless some suggestions for possible research topics, which the department is particularly qualified to supervise. For details of the next University Postgraduate open day, visit, Undergraduate open days, visits and fairs, Postgraduate research open days and study fairs, Find out more about Science and Engineering at Manchester, 1st or upper 2nd class 4 year undergraduate degree (e.g.

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