Keep in mind that there is considerable overlap between different categories listed below and many academic staff work across areas. We have given cross references in many cases, but you should browse the list carefully and talk with more than one member of staff.
Algebra and Group Theory
Adam Thomas can supervise projects in group theory, Lie theory and representation theory. Topics could include algebraic groups, Lie algebras, permutation groups or a related area mutually agreed upon.
Inna Capdeboscq can supervise projects in group theory.
Derek Holt is willing to supervise one (or at most two) research projects on the topic of Computational Group Theory. The project would involve some programming. Prospective students would have to see him to discuss further details.
Dmitriy Rumynin is interested in Algebra and Representation Theory. Prospective students would have to see him to discuss further details.
Marco Schlichting can supervise projects in Homological/Homotopical Algebra (algebraic K-theory, homology of classical groups, derived categories, Quillen model categories), Algebra (quadratic forms and central simple algebras, projective modules, Milnor K-theory), Algebraic Geometry (algebraic cycles, (oriented) Chow groups, motivic cohomology, A1-homotopy theory) and Algebraic Topology (homotopy theory, topological K-theory)
Christian Boehning is interested in algebraic geometry, representation and invariant theory, derived category methods in birational geometry, birational automorphism groups.
Gavin Brown can supervise projects in algebraic geometry, including the projective geometry of curves and surfaces and applications of Riemann-Roch.
Chunyi Li is interested in Algebraic Geometry.
Martin Lotz can supervise projects in effective, computational and numerical algebraic geometry.
Miles Reid has a variety of possible projects on offer related to modern research in algebraic geometry, and to MA4A5 Algebraic Geometry or MA426 Elliptic Curves. He can also propose problems related to advanced topics in Galois theory, commutative algebra and algebraic number theory for suitably motivated students.
Finite subgroups of SL(2, CC) and SL(3, CC)
Explicit generators, invariant theory. Work of Klein around 1870, but recently developed in many directions. Start from my preliminary chapter on cyclic quotient singularities. See for example my Bourbaki seminar or the references given on my McKay correspondence website.
Quaternions, octonions, special geometric structures and exceptional Lie groups
See any introductory text on quaternions (for example, Balazs Szendroi and Miles Reid, Geometry and Topology, Chapter 8), followed by John C Baez, The Octonions, 56 pages, preprint available from uk.arXiv.org as math.RA/0105155, and John H. Conway and Derek A. Smith, On quaternions and octonions: their geometry, arithmetic, and symmetry, A K Peters, Ltd. 2003 IBSN 1568811349
Topology, projective geometry and enumerative geometry of Grassmann varieties and projective homogeneous spaces.
Applications of Riemann Roch on curves and surfaces
Graded rings, computer algebra calculations. Computer enumeration of graded rings of interest in algebraic geometry. Work of mine and my students. It contains lots of fairly simple minded combinatorial problems that lend themselves to computing. Start from my preliminary chapter on graded rings, then look at Gavin Brown's GRDB website.
Damiano Testa is interested in Algebraic geometry and Number Theory. Prospective students would have to see him to discuss further details.
Analysis/Applied Analysis/Partial Differential Equations
Keith Ball offers a project on large deviations in geometry and functional analysis.
David Bate is interested in geometric measure theory, real analysis and functional analysis.
Bertram Düring's research interests are in applied and computational partial differential equations, spanning modelling, analysis, numerical analysis and optimal control, with a particular interest in applications from finance and socio-economics.
Charles Elliott’s research is centred around the analysis of nonlinear partial differential equations and computational mathematics with applications (mathematical biology, material science, continuum mechanics, phase transitions etc) including numerical analysis and applied analysis. In particular, finite element methods, free boundary problems, geometric evolution equations and surface growth, two phase flow, cell motility, biomembranes and PDE optimisation. In the year 2020/21 he will be giving a PhD module Variational Analysis and Evolution Equations. This module is a natural companion to Theory of PDEs, Advanced PDEs, Numerical Analysis of PDEs and Advanced Real Analysis as well as other modules in Analysis and Applied Mathematics. Lecture notes will b e available at the beginning of the academic year. Projects may be in any of applied analysis, numerical analysis, computation and modelling and topics include:- (A) Variational problems with point constraints involving inverse, control or cell biology applications (B) PDE methods applied to networks (C) Functions of bounded variation: applications and computation. (D) Time dependent function spaces and PDEs in evolving domains (E) Information propagation on graphs. Prospective students would have to contact him to discuss further details. Please consult his web page for more details of his research interests.
Josephine Evans is interested in analysis of PDEs, stochastic processes or functional inequalities related to many body systems (kinetic theory) coming from both physics and biology.
Susana Gomes is interested in partial differential equations, in particular modelling, control and inverse problems in applications such as pedestrian dynamics or fluid dynamics
András Máthé is interested in geometric measure theory, fractal geometry and real analysis (of combinatorial nature).
Filip Rindler research concerns singularities in nonlinear PDEs and the modern theory of the calculus of variations. In particular, he is interested in oscillation and concentration phenomena and what can be rigorously proved about their "shape". Applications include elasticity and elasto-plasticity theory.
Jose Rodrigo is interested primarily interested in nonlinear partial differential equations (fluid mechanics, reaction-diffusion systems with mass conservation, fractional diffusion, ...) but is also happy to supervise projects on some areas of harmonic analysis (following on from MA433 or MA4J0). Prospective students would have to see him to discuss further details.
James Robinson's interests are in partial differential equations and fluid mechanics; infinite-dimensional dynamical systems; embeddings of finite-dimensional sets into Euclidean spaces. Prospective students would have to see him to discuss further details.
Vedran Sohinger is interested in nonlinear dispersive PDEs and their connections with mathematical physics and probability theory. He is happy to offer projects based on the module MA4J0 (Advanced Real Analysis) and on related modules.
Marie-Therese Wolfram is interested in partial differential equations, mathematical modeling in socio-economic applications and the life sciences, numerical analysis.
(See also Joel Moreira)
Agelos Georgakopoulos offers projects combining probability and graph theory, sometimes involving computer simulations.
Vadim Lozin offers projects in graph theory, combinatorics, discrete mathematics
Oleg Pikhurko’s main research interests lie in combinatorics and graph theory, including their connections to analysis, descriptive set theory, measured group theory, etc. This is a deep and thriving area, with many possible projects that can help you to prepare for PhD studies. Prospective students should contact Prof Pikhurko by email.
Complexity Science and Mathematical Modelling and Materials
(see also Applied Analysis, Computational mathematics and numerical analysis, Fluid Dynamics, Mathematical Biology)
Colm Connaughton offers project in non-equilibrium statistical mechanics, fluid dynamics and turbulence, nonlinear waves, interacting particle systems
Thomas Hudson is interested in connecting models of physical systems at different scales through asymptotic limits. To do so, he uses mathematical tools from the analysis of PDEs and the Calculus of Variations, such as homogenization, Gamma-convergence and Large Deviations. Some examples of project ideas include modelling and simulating point defect diffusion using atomistic data, fitting material models from surface wave data, coarse-graining stochastic dynamical systems from data, studying mean-field models for dislocations, and studying nonlinear elasto-plasticity models.
Markus Kirkilionis's interests are in complex systems, mathematical biology, dynamic network models, numerical analysis, pattern formation, physiologically structured Population models, (monotone) dynamical systems. Prospective students would have to see him to discuss further details.
Robert MacKay offers projects in a range of applications of mathematics: Hierarchical Aggregation of Markov processes, Metrics on probability distributions for probabilistic cellular automata, Global Positioning Systems, Triple Linkage coding, A mixing volume-preserving vector field in a figure-eight knot complement, Optimal design of assessment graphs, Stroke treatment, Pledge game, Stellarator design, Integrable optics.
See here for further details.
Computational Mathematics and Numerical Analysis
Dwight Barkley offers projects on modern approaches to turbulence and on nonlinear waves. The projects involve numerical simulations and concepts from dynamical systems and complexity science. Students must be comfortable with numerical computations and must able to program in a high-level language C/C++/Fortran. Please see the following to get a flavour of the work: here, here, here, here, here, and here. Several past projects have resulted in publications in scientific journals.
Andreas Dedner's interests are in numerical analysis and scientific computing, with particular emphasis in high-order methods for non-linear equations and applications in geophysical flows, radiation magnetohydrodynamics, and reaction-diffusion equations. Prospective students would have to see him to discuss further details.
Martin Lotz can supervise projects on numerical linear and non-linear algebra, compressed sensing, and deep learning (theory and applications).
Björn Stinner works in the analysis and numerical analysis of partial differential equations. Specifically, he is interested in free-boundary problems, partial differential equations on manifolds, finite element methods, and phase field modelling in materials science, fluids, and cell biology. Projects may involve modelling, analysis, and computing or any combination of these.
Differential Geometry and PDE
(See Applied Analysis above and areas such a Fluid Dynamics and Computational Mathematics below for further topics in PDEs).
Mario Micallef’s research interest lie in the area of partial differential equations and differential geometry. Prospective students would have to see him to discuss further details.
Felix Schulze’s research interests lie in Geometric Analysis, more specifically geometric flows, minimal surfaces and their applications to geometric problems such as for example isoperimetric inequalities. Interested students should have a strong background in Differential Geometry and PDE as well as interest in Geometric Measure Theory. Interested students should contact Prof. Schulze by the end of week 11 of term 3. Interested MASt students are welcome to contact Prof. Schulze also after this date.
Peter Topping offers various options in geometric analysis. There will be many different challenging projects possible in this active area. The suitable student will have interest both in differential geometry and in PDE theory (and should normally be taking Advanced PDE). It might be some effort to understand enough to start the project, but it should leave you in a great position to start PhD research in the area. Interested students must contact Prof. Topping by the end of week 9 of term 3.
Ergodic Theory and Dynamical Systems
Adam Epstein offers projects in Complex and Arithmetic Dynamical Systems, for example:
Arithmetic Questions in Holomorphic Dynamics: Consider the polynomials F_n(c) = p_c o ... o p_c(0) (n-fold self-composition) where p_c(z) = z^2 + c. It is known that all roots of F_n(c) are simple. The polynomial F_n splits into factors, some of which arise as F_m for smaller m dividing n: when such factors are divided out, the resulting polynomials are conjectured to be irreducible. Questions of this nature arise for other interesting families of rational maps, and little is known in general. Well-organised computer experimentation would be a good start. This would be an appropriate project for a student who has taken, or will be taking Algebraic Number Theory (or Galois Theory). Familiarity with basics from Complex Analysis and Dynamical Systems would also be useful.
He is also willing to supervise appropriate mutually agreed projects in set theory and logic.
Vassili Gelfreich’s research interests lie in the area of dynamical systems. Prospective students would have to see him to discuss further details.
Oleg Kozlovski is interested in Dynamical systems, ergodic theory, mathematical physics, financial mathematics. Prospective students would have to see him to discuss further details.
Ian Melbourne is interested in Ergodic theory and Dynamical systems, including probabilistic or stochastic aspects of deterministic dynamical systems. Prospective students would have to see him to discuss further details.
Joel Moreira is interested in Ergodic theory and applications to combinatorics and number theory and is willing to supervise projects in these areas.
Richard Sharp is interested in Ergodic Theory and its applications to other areas of pure mathematics. Prospective students would have to see him to discuss further details.
John Smillie is interested in translation surfaces and complex dynamics in higher dimensions.
Polina Vytnova is interested in dimension theory and dynamical systems. She is offering a project "Dimension of dynamically generated sets". Dimension is a way to quantify the size of sets with complex structure. Typically it is very difficult to compute accurately. This project relates to sets which have a certain dynamical flavour to their construction (e.g. Cantor sets, Snowflakes, Julia sets, etc.) and to the techniques available in this context for more accurate estimation of the dimension; or even explicit values in some cases. The project will involve numerical computations with Matlab/Fortran/C.
Radu Cimpeanu supervises projects at the intersection between mathematical modelling, asymptotic analysis, solutions for ordinary/partial differential equations and high performance computing (including computational linear algebra and large scale system solvers). Specific topics in fluid mechanics include interfacial flows (modelling, analysis, simulation and applications of drops, bubbles and liquid films), rheological flows (from chocolate to de-icing fluids) and novel mathematical models for the cultivated meat industry. Some projects are likely to include multi-physics aspects such as acoustics, heat transfer and electromagnetism. From a mathematical standpoint, there are exciting opportunities to combine classical (continuum) modelling with discrete and data scientific streams, as well as integrating approaches such as control theory or hybrid modelling into areas previously unexplored with such tools. More generally, work on any of the above will likely involve a mixture of analytical and computational techniques (which can be tailored depending on your own interests and what skillset you wish to develop), as well as the interplay between them.
James Sprittles offers projects on the mathematical modelling and computational simulation of fluid mechanical phenomena, particularly those driven by complex interfacial effects which are prevalent in the emerging field of micro/nanofluidics. These microflows often require new modelling approaches, involving both continuum and particle-based methods, coupled to efficient computational techniques and are currently an area of intensive research interest. Possible projects include (but are not limited to):
Gas Dynamics in Free-Surface Flows: Thin films of gas often have a huge influence on the dynamics of liquid volumes, e.g. when drops impact solids, but at present are lacking an accurate theoretical description (in fact, often their influence is ignored). The problem is that their dimension is such that classical continuum mechanics fails and kinetic theory governed by the Boltzmann equation is required. This project will involve developing new models for this class of flows, with particular attention applied to the coupling of kinetic theory with continuum mechanics, and exploiting scale-separation to make these models computationally tractable.
Universal Behaviour of Drop Formation: The formation of a drop from a column of liquid (as seen from a dripping tap) is a singular flow which, remarkably, is governed by a 'universal' similarity solution (i.e. the final stages of the liquid breaking always look the same). However, recent computational results have revealed, twenty years after the original theory was proposed, that the approach to this solution is not uniform, so that the liquid's surface oscillates in time. This project will involve developing theory and computations to investigate the stability of the similarity solution in order to understand these new surprising results.
Oleg Zaboronski offers a project related to
Metastability in stochastic Euler dynamics. During this project, the student will learn about:
1. Arnold's algebraic formulation of 2d Euler equation
2. Zeitlin's su(N) approximation of 2d Euler flow
3. Stochastic Euler equation on su(N)
4. Control of stochastic Euler flow on su(N) using gradient dissipation built out of integrals of motion and the associated invariant measures.
5. Dynamics of stochastic Euler flow on su(N) for multimodal invariant measures using Wentzel-Fradkin theory.
Geometry, Topology and Geometric Group Theory
(see also Marco Schlichting)
Brian Bowditch offers project in hyperbolic geometry, low-dimensional topology, geometric group theory
John Greenlees is interested in algebraic topology, homotopy theory, commutative algebra and representation theory.
Robert Kropholler would be interested in supervising projects in geometric group theory and low-dimensional topology. Students should contact him to discuss potential projects in more detail.
Martin Lotz can supervise projects in topological data analysis and persistent homology.
Saul Schleimer is interested in geometric topology, group theory, and computation. Prospective students would have to see him to discuss further details.
Karen Vogtmann is interested in geometric group theory, low-dimensional topology, cohomology of groups.
Weiyi Zhang is interested in symplectic topology, complex geometry and low dimensional topology.
Mathematical Biology, Epidemiology, Ecology and Evolution
Louise Dyson works on mathematical modelling of biological systems, especially the epidemiology of neglected tropical diseases and the analysis of biological systems in which noise plays an important role.
Matt Keeling offers projects in the areas of Epidemiology,Ecology or Evolution. Epidemiology (the study of infectious diseases and their spread in populations), Ecology (the study of animal/plant populations and their environment), Epidemiology (the study of disease spread and control) and Evolution (the study of the long-term dynamics of populations) present a wide variety of interesting problems that require a mathematical approach. There are a vast number of problems and approaches that could be studied, ranging from model-development, to computer simulation, to statistical analysis. If you've attended (or planning to attend) MA4E7: Population Dynamics, that would be a distinct advantage although not essential. I like to offer projects that show how the mathematical techniques you've learnt can be applied to real questions to obtain useful or meaningful insights. If you're interest, its probably easiest to come and talk with me, and together we can determine a specific project that matches your interests and skills.
Recent projects have included the evolution and competition of influenza strains, diffusion approximations to disease spread on networks, optimal control of spatial epidemics, Nicholson-Bailey lattice models.
Kat Rock is a mathematical epidemiologist interested in population-level disease dynamics. In particular her focus is on transmission of vector-borne (insect-transmitted) infections, infections in low- or middle-income countries, and bringing together dynamic transmission modelling and health economic evaluation. She teaches the MA4M1 “Epidemiology by Example” course and it would be recommended to have taken this and/or “Population Dynamics” (MA4E7) already or alongside an R-project.
Dave Wood would be willing to discuss possible topics given the information below. His main interests for projects include ecological type modelling, investigating systems with symmetry (including but not limited to applications to arthropod locomotion) and applications of mathematics in industry. Systems with symmetry could be theoretical or applied, but would concentrate on using methods from MA3J3 Bifurcations, Catastrophes and Symmetry, so this may interest students who have taken that. The industrial applications could cover a broad range of mathematical disciplines and be a survey of some problems that have already been studied or a look at a new problem involving original research. Ecological modelling I would be happy to consider any suitable application that a student feels passionate about (see below for a couple of past such projects).
Previous titles he has supervised include: “Coupled cell networks, bifurcations and symmetry”, “The effects of tuna fishing on dolphin populations”, “Symmetry in coupled cells and neuronal networks”, “Applications of maths in industry”, “Discrete maths in industry”, “Modelling the Future of the Hawaiian Honeycreeper: An Ecological and Epidemiological problem” (the latter of which led to a successful PhD).
Nigel Burroughs applies mathematical and statistical methods to biological systems. He is primarily interested in understanding the mechanics and mechanisms of cells, in effect how they work and achieve the spectacular range of behaviours that are observed. He uses a combination of model development, mathematical analysis (dynamical systems, perturbation theory), simulation and statistical computation (Markov chain Monte Carlo methods), with both deterministic and stochastic models/systems, although most are stochastic given the fact that cells often show stochastic behaviour. Projects are in the mechanics of cell division, from the duplication and separation of the chromosomes to the mechanical separation of cells into two daughter cells, (1, 2 below), microtubule modelling (3, 4) and cancer modelling (5):
1. Modelling chromosome oscillations. Chromosomes are duplicated but then have to be divided so that each daughter cell gets one and only one copy. This is achieved by 'holding' the pairs at the cell equator until all pairs are in position. Mathematically we can think of this system as two particles in a 1D box connected by a spring and pushed/pulled separately from the ends. The surprising observation is that chromosomes oscillate from side to side across the cell equator. How this occurs is poorly understood, but this is believed to involve a tension sensor that effectively acts as a communication system between the two chromosomes. This project would involve analysis of deterministic and stochastic models of oscillation of paired chromosomes (dynamical systems techniques and simulation), examining different mechanisms of feedback/communication.
2. Statistical computation (Markov chain Monte Carlo, MCMC) analysis of chromosome oscillations. See above (1) for system description. Here you would use our large database of paired chromosome trajectories (1000s of trajectories) to understand the statistical structure of chromosome oscillations, fitting saw-tooth like oscillatory profiles. Experience with MCMC, hidden Markov chains and probability theory is essential.
3. Modelling microtubule bending. Microtubules are biological polymers that polymerise into tubes and exhibit elastic bending properties. Here you will examine models and simulations of microtubule bending in fluid flow and binding of proteins that enhance the curvature.
4. Reverse engineering microtubules. Microtubules are biological polymers; they polymerise into tubes and exhibit what is called dynamical instability, switching from periods of growth to decay. What causes this (stochastic) switching is unknown but likely an emergent property of the microtubule lattice. The idea of this project would be to construct models and an MCMC algorithm to fit those models of lattice dynamics to data to examine the degree to which such data can inform on the underlying processes. Experience with MCMC and probability theory is essential.
5. Cancer evolution. Cancer is a Darwinian process whereby the cell functional pathways are mutated to escape growth control processes. There are multiple control processes, each controlled by a large number of genes. Under Darwinian evolution one predicts that only one mutation is required in each of these groups of genes. You will construct an MCMC algorithm to test this on real data to demonstrate this mutual exclusion prediction.
Prospective students should contact him to discuss further details. [Experience with MCMC means acquaintance with Gibbs and Metropolis-Hastings algorithms and their use in simulating posterior probability distributions. Experience coding an algorithm for a simple problem would be an advantage].
David Rand. His main research interest is Systems Biology, particularly understanding the design principles of regulatory and signalling systems in cells. Prospective students would have to see him to discuss further details.
Mike Tildesley works on the development of mathematical models to simulate the spread of livestock and zoonotic diseases.
Mathematical Physics, Molecular Dynamics, and Statistical Mechanics
Stefan Adams offers project on large deviation theory, probability theory, Brownian motions, statistical mechanics, gradient models, multiscale systems.
Siri Chongchitnan offers projects on cosmology and theoretical astrophysics. Potential topics include: cosmological inflation, primordial gravitational waves, large-scale structures and Bayesian methods in cosmology. These projects will involve extensive use of Python. Background knowledge of cosmology or astrophysics, whilst ideal, is not necessary. An ideal student would be someone who is deeply curious about the nature and the origin of the Universe. Further enquiries are welcome.
Roman Kotecky's interests are in discrete mathematics, probability theory, statistical mechanics, mathematical physics. Prospective students would have to see him to discuss further details.
Florian Theil offers project in the area of molecular dynamics which is an active and fast developing field in Applied Mathematics. The objective is to obtain insight into complex molecular systems by means of computer simulations instead of using experimental techniques. Due the high dimensionality the simulations are very costly and consequentially analytical results can provide valuable insights which potentially improve the efficiency of the numerical approaches. The proposed fourth year projects involve the application of tools from Stochastic Analysis and PDE theory to MD systems.
Daniel Ueltschi's interests are in analysis, probability theory, statistical mechanics, mathematical physics. Prospective students would have to see him to discuss further details.
Roger Tribe offers projects on determinantal point processes - randomly arranged points whose distribution is characterized via determinants. (See Terrence Tao blog on https://terrytao.wordpress.com/2009/08/23/determinantal-processes/). Other projects available: random matrices/interacting particle systems/large deviations.
Demi Allen is happy to supervise projects in Metric Number Theory/Diophantine Approximation and Fractal Geometry. Here is one possible project which neatly combines both of these topics. Please get in touch if you are interested in this project or are interested in discussing other possible projects in these broad areas.
Sam Chow supervises projects in analytic number theory, diophantine equations, and diophantine approximation.
Adam Harper is interested in analytic and probabilistic number theory, and would be willing to supervise projects in these areas. The basic goal is to use methods of analysis (real, complex, Fourier, probability) to understand the distribution of number theoretic objects, like prime numbers. A project would likely involve working through a couple of papers from the (fairly) recent research literature, and trying to understand and synthesise them.
David Loeffler would be willing to supervise projects in various areas of number theory, with possible topics including aspects of p-adic numbers, modular forms, and elliptic curves. See here for a list of previous topics.
- Minhyong Kim works on arithmetic geometry, the study of mathematical structures incorporating a mixture of arithmetic and geometric properties. He is also interested in topology and mathematical physics, especially the foundational aspects of quantum field theory. He is willing to supervise a project in any combination of these areas.
Simon Myerson offers projects in analytic number theory and harmonic analysis. You do not need to have taken any particular courses; in particular you do not have to take MA4L6. An interest in number theory and analysis is a must. There are applications to PDEs which would interest students talking MA4J0, although they are unlikely to be the focus of the project. Projects close to algebraic number theory would also be possible if you have an idea in mind.
- Samir Siksek is interested in number theory and diophantine equations. Prospective students would have to see him to discuss further details.
Stochastic Analysis and Probability
Tobias Grafke is interested in numerical methods and mathematical tools to analyse stochastic systems. Applications include fluid dynamics and turbulence, atmosphere and ocean dynamics, and biological and chemical systems.
Tim Sullivan offers projects in uncertainty quantification and inverse problems, understood as the meeting point of numerical analysis, applied probability and statistics, and scientific computation. He also has interests in data science and machine learning.
Sebastian Vollmer is interested in Monte Carlo Methods, Stochastic Gradient Methods, Stochastic Processes, Applications of Data Science.
- INTEGRABLE PROBABILITY : this is a rather new area within probability which explores remarkable connections between probability and algebraic structures (representation theory, algebraic combinatorics etc). A motivation for this development has been the understanding of the so called Kardar-Parisi-Zhang universality, which governs fluctuations of stochastic growth model (colonies of bacteria, spread of fluid in porous media etc). Remarkably, these fluctuation are not governed by the standard central limit theorem but rather from distributions related to random matrix theory. Some favour on this topic is provided by these notes
- SCALING LIMITS OF DISORDERED SYSTEMS AND SPDEs: Stochastic PDEs (SPDEs) have been proposed as continuum scaling limits of statistical mechanics models. However, both the well posedness of the SPDEs and the convergence of the discrete models to these are very challenging problems. The first problem has been recently largely settled below the so-called "critical dimension”) through the theory of Regularity Structures (by Hairer) or Paracontrollled Districutions (by Gubinelli-Imkelle-Perkowski). At the critical dimension things are much more challenging. This project can spin around the above theories, some exploration of critical dimensional statistical mechanics and/or scaling limits. Some flavour of the topic can be obtained in these notes