5 November 2014 Speaker: Dr Ben Torsney (Glasgow),. Title: Optimal Design, Lagrangian and Linear Model Theories: Further Developments on a Fusion. Abstract: TBC.

20 November 2014 Speaker: Dr Vadim Lozin (Warwick). Please note change of room for this - M/2.01. Title: Combinatorics and algorithms for augmenting graphs. Abstract: The notion of augmenting graphs generalizes the Berge's idea of augmenting chains that has been used by Edmonds in his celebrated solution of the maximum matching problem. This problem is a special case of the more general problem of finding a maximum independent set in a graph. Recently, the augmenting graph approach has been successfully applied to solve the maximum independent set problem in various other special cases. However, our knowledge of augmenting graphs is still very limited, and we do not even know what the minimal *infinite* classes of augmenting graphs are. In this talk, we give an answer to this question and apply it to extend the area of polynomial-time solvability of the maximum independent set problem.

28 January 2015 This will take place in room M/2.06. Speaker: Prof. Robert John (Nottingham). Title: Type-2 Fuzzy Logic in Decision Support Abstract: This talk will provide an overview of Bob's research in type-2 fuzzy logic and its application in Decision Support. Type-2 fuzzy sets are fuzzy-fuzzy sets - that is, where the fuzzy set has membership grades that are themselves fuzzy sets, rather than numbers in [0,1]. Fuzzy sets (type-1) have had significant success in control applications but by their very definition are not particularly 'fuzzy' and struggle in applications that attempt to mimic human reasoning in decision support systems. Introduced in 1975, type-2 fuzzy logic really started to grow in the late '90s led by Bob and Jerry Mendel. In the intervening period the number of type-2 papers and researchers has grown considerably. This talk will introduce the audience to type-2 fuzzy logic and provide a brief history

Bob will describe practical application of his work in decision support, such as the aggregation of uncertain information, supply chain modelling and medical diagnosis. Bob John has a BSc in Mathematics, a MSc in Statistics and a PhD in Fuzzy Logic. He worked in industry for 10 years as a mathematician and knowledge engineer developing knowledge based systems for British Gas and the financial services industry. Bob spent 24 years at De Montfort University in various roles including Head of Department, Head of School and Deputy Dean. He led the Centre for Computational Intelligence research group from 2001 until 2012. Bob joined Nottingham in 2013 where he led on the LANCS initiative and Heads up the research group Automated Scheduling, Optimisation and Planning (ASAP) in the School of Computer Science. The LANCS Initiative is built on a collaboration between four U.K. Universities: Lancaster, Nottingham, Cardiff and Southampton. The research group carries out multi-disciplinary research into mathematical models and algorithms for a variety of real world optimisation problems. It has 8 academic staff, 9 researchers and over 30 PhD students.

11 February 2015 Speaker: Dr Fabricio Oliveira (Rio de Janeiro). Title: Optimising under uncertainty: an introduction and applications in healthcare related problems. Abstract: In this talk we will discuss the importance, as well as some of the available tools, to consider the stochastic nature of input parameters in optimisation problems. It is well known that the static nature of optimisation problems makes it difficult to support decision making when the input data is subject to uncertainty. We will present how one can incorporate such uncertainty by means of stochastic and robust optimisation, using examples of current applications in healthcare related problems. Some previous background in optimisation is of good value, but not necessarily mandatory. Hopefully, at the end of the talk, the audience will be able to understand how it is possible to include uncertainty in optimization-based decision support tools to improve the decision-making process. 18 February 2015 Speaker: Christian Henning (UCL). Title: TBC. Abstract: TBC. 4 March 2015 Speaker: Dr Dmitrii Pasechnik (Oxford). Title: TBC. Abstract: TBC. 18 March 2015 at 15:00 Speaker: Haeran Cho (Bristol). Title: TBC. Abstract: TBC.

15 April 2015 Speaker: Dr Vladimir Deineko (Warwick). Title: Special structures in polynomially solvable cases: Is there much in common? Abstract: In our talk we present a survey of polynomially solvable cases of NP-hard problems with an emphasis on common structures in these cases. We concentrate on the cases where specially structured matrices are involved. It most considered cases permuting rows and columns of specially structured matrices destroy the properties needed. We discuss the arising recognition problems and pose quite a few open questions from this exciting area of research. 29 April 2015 Speaker: Prof. Mark Kelbert (Moscow). Title: Shannon's entropy power inequality and weighted differential entropies. Abstract: We establish a number of new inequalities for weighted differential entropies and analyze in details a Bayesian problem of estimating probability of success in a series of trials with binary outcomes. In particular, the weighted Rao-Cram\'er inequality is presented.

6 May 2015 Speaker: Prof. Mark Girolami (Warwick). Title: Differential Geometric Markov chain Monte Carlo Methods Abstract: Monte Carlo methods are the dominant approach to perform inference over increasingly sophisticated statistical models used to describe complex phenomena. This presents a major challenge as issues surrounding correct and efficient MCMC -based statistical inference over such models are of growing importance. This talk will argue that differential geometry provides the tools required to develop MCMC sampling methods suitable for challenging statistical models. By defining appropriate Riemannian metric tensors and corresponding Levi-Civita manifold connections MCMC methods based on Langevin diffusions across the model manifold are developed. Furthermore proposal mechanisms which follow geodesic flows across the manifold will be presented. The optimality of these methods in terms of mixing time shall be discussed and the strengths (and weaknesses) of such methods will be experimentally assessed on a range of statistical models will also be considered. This talk is based on work that was presented as a Discussion Paper to the Royal Statistical Society and it remains the most downloaded article from the journal website. Details Here.

27 May 2015 Speaker: Dr Timm Oertel (Zurich). Title: A polyhedral Frobenius theorem with applications to integer optimization. Abstract: We prove a representation theorem of projections of sets of integer points by an integer matrix W. Our result can be seen as a polyhedral analogue of several classical and recent results related to the Frobenius problem. Our result is motivated by a large class of non-linear integer optimization problems in variable dimension. Concretely, we aim to optimize f(Wx) over the set of integers in P, where f is a non-linear function, P is a n-dimensional polyhedron and W is a d x n matrix. As a consequence of our representation theorem, we obtain a general efficient transformation from the latter class of problems to integer linear programming. Our bounds depends polynomially on various important parameters of the input data leading, among others, to first polynomial time algorithms for several classes of non-linear optimization problems.

Reliable adaptive algorithms are based on a posteriori error estimates, i.e. estimates of the error in terms of values obtained in the computation process: computed solution and current mesh. Such a posteriori error estimates for parabolic partial differential equations will be the subject of this talk. For classical and singularly perturbed semilinear parabolic equations, we give computable a posteriori error estimates in the maximum norm, which, in the singularly perturbed regime, hold uniformly in the small perturbation parameter. The parabolic equations are discretized in time using the backward Euler, Crank-Nicolson and discontinuous Galerkin methods. Both semidiscrete (no spatial discretization) and fully discrete cases will be considered. The analysis invokes certain bounds for the Green's function of the parabolic operator. When dealing with the full discretizations, we also employ the elliptic reconstruction technique. Although parts of our analysis are quite technical, it will be demonstrated (using a first-order ODE example as a trivial case of a parabolic PDE) that some main ideas are quite elementary. 18 November 2014 Speaker: Robert Style (University of Oxford). Title: The mechanics of soft solids - breaking classical laws. Abstract: Soft solids make up the bulk of biological material, and are increasingly being used for new technology like wearable electronics, and soft robotics. However, despite their importance, experiments show that many classical laws fail to describe them. For example, I will show how classical theories of wetting, composite mechanics and contact mechanics significantly break down at a critical `elastocapillary' lengthscale -- because of solid surface tension. Furthermore, I will show how these phenomena highlight the existence of a swathe of new, small-scale behaviour in soft materials. Co-Host: Dr. Maurice Blount.

25 November 2014 Speaker: Xuesong Wu (Imperial College London). Title: Nonlinear development of disturbances in transitional and turbulent free shear flows. Abstract: Free shear flows, such as mixing layers, jets and wakes, are inviscidly unstable due to their inflectional velocity profiles. Instability modes, which are usually excited by external perturbations, amplify on the shear floow, leading to vortex roll-up and randomization in the nonlinear stage. Interestingly, in turbulent state free shear flows exhibit a high degree of order, characterised by the prevalent presence of so-called coherent structures, the most striking of which are Brown-Roshko rollers . Both instability waves and coherent structures are known to be dynamically significant for entrainment and mixing, noise generation as well as for turbulence modelling. In this talk, I will present a nonlinear theory to describe first the development of instability modes on laminar free shear layers. The theory predicts vortex roll-up and randomisation through a generalized side-band instability mechanism. The theory will then be modified to describe formation and evolution of Brown-Roshko rollers on turbulent mixing layers. Co-Host: Dr. Chris Davies

2 December 2014 Speaker: Matthias Heil (University of Manchester). Title: Wrinkly fingers: fluid-structure interaction in elastic-walled Hele-Shaw cells. Abstract: Viscous fingering in Hele-Shaw cells is a classical and widely studied fluid-mechanical instability: When air is injected into the narrow, liquid-filled gap between parallel rigid plates, the axisymmetrically expanding air-liquid interface tends to be unstable to non-axisymmetric perturbations when the capillary number -- the ratio of (destabilising) viscous to (stabilising) capillary forces acting at the air-liquid interface -- becomes sufficiently large. The introduction of wall elasticity (via the replacement of one of the bounding plates by an elastic membrane) can weaken or even suppress the fingering instability, but it also makes the system susceptible to additional solid-mechanical instabilities.

We show that in elastic-walled Hele-Shaw cells that are bounded by sufficiently thin elastic sheets, the (fluid-based) viscous fingering instability can arise concurrently with a (solid-based) wrinkling instability. We study the interaction between these distinct instabilities, using a theoretical model that couples the depth-averaged lubrication equations for the fluid flow to the Föppl--von Kármán equations, which describe the deformation of the thin elastic sheet. We employ a linear stability analysis to determine the growth rate of non-axisymmetric perturbations to the axisymmetrically expanding bubble, and perform direct numerical simulations to study the nonlinear interactions between the instabilities. We show that the system's behaviour may be characterised by a non-dimensional parameter that indicates the strength of the fluid-structure interaction. For small [large] values of this parameter the system's behaviour is dominated by viscous fingering [wrinkling], with strong interactions between the two instabilities arising in an intermediate regime. [Joint work with Draga Pihler-Puzovic and Anne Juel]. Co-Host: Dr. Chris Davies

3 February 2015 Speaker: Helen Wilson (University College London). Title: Instabilities in viscoelastic fluids. Abstract: Non-Newtonian and viscoelastic fluids show many fascinating properties. One of the most surprising (and irritating, for those who process them in a manufacturing context) is their susceptibility to instabilities at flow rates where an "equivalent" Newtonian fluid would flow stably. I will use linear stability theory, and some asymptotic expansions, to discuss two distinct instability mechanisms which are unique to viscoelastic fluids. Co-Host: Professor Tim Phillips. 10 February 2015 Speaker: Dmitri Tseluiko (University of Loughborough). Title: Wave dynamics on a liquid film sheared by a turbulent gas

Abstract: The dynamics of a thin laminar liquid film flowing under gravity down the lower wall of an inclined channel when turbulent gas flows above the film will be discussed. The solution of the full system of equations describing the gas-liquid flow faces serious technical difficulties. However, a number of assumptions allow isolating the gas problem and solving it independently by treating the interface as a solid wall. This permits finding the perturbations to normal and tangential stresses at the interface imposed by the turbulent gas in closed form. Then the liquid film flow under the influence of these perturbations can be analysed by deriving and analysing a hierarchy of model equations describing the dynamics of the interface, i.e. boundary-layer equations, a long-wave model and a weakly nonlinear model, which turns out to be the Kuramoto-Sivashinsky equation with an additional term due to the presence of the turbulent gas. Also, by combining the long-wave approximation with a weighted-residual technique, an integral-boundary-layer approximation that is valid for moderately large values of the Reynolds number can be obtained. This model is then used for a systematic investigation of the flooding phenomenon observed in various experiments: as the gas flow rate is increased, the initially downward-falling film starts to travel upwards while just before the wave reversal the amplitude of the waves grows rapidly. Co-Host: Dr Nikos Savva.

17 February 2015 Speaker: David Needham (University of Birmingham). Title: The evolution to localized and front solutions in a non-Lipschitz reaction-diffusion Cauchy problem with trivial initial data. Abstract: This talk addresses the Cauchy Problem for a Non-Lipschitz Semi-Linear parabolic PDE with trivial initial data. The question of uniqueness is considered, in particular in relation to the existence of classes of spatially inhomogeneous solutions, and their relation to maximal and minimal solutions. Co-Host: Professor Tim Phillips. 17 March 2015 Speaker: Paul Milewski (University of Bath). Title: Modelling and computation of pilot wave-bouncing droplet dynamics in a Faraday problem. Abstract: Recent experiments by two groups, Yves Couder (Paris) and John Bush (MIT) have shown experimentally that droplets will bounce on the surface of a vertically vibrated bath (instead of coalescing with it), generating a damped Faraday wavefield at every bounce. As the forcing is increased, a pitchfork symmetry breaking bifurcation leads to a "walking" state whereby the bouncing droplet is guided by the self-generated wavefield: the droplet s pilot wave. Once this state is achieved a large array of interesting dynamics ensues with surprising analogies to quantum mechanical behaviour. The system appears to show that probabilistic quantum behaviour can arise from a physical deterministic system. We present a coupled particle-fluid model that can can be used simulate the fascinating dynamics of this problem. This is joint work with John Bush, Andre Nachbin (IMPA) and Carlos Galeano-Rios (IMPA). Co-Host: Dr Nikos Savva.

12 May 2015 Speaker: Pierre Colinet (TIPs laboratory, Université Libre de Bruxelles) Title: Evaporation of droplets with moving contact lines Abstract: Despite many years of intensive research, the modeling of contact lines moving by spreading and/or evaporation still remains a subject of debate nowadays, even for the simplest case of a pure liquid on a smooth and homogeneous horizontal substrate. In addition to the inherent complexity of the topic (singularities, micro-macro matching, intricate coupling of many physical effects, ), this also stems from the relatively limited number of studies directly comparing theoretical and experimental results, with as few fitting parameters as possible. In this presentation, I will address various related questions, focusing on the physics invoked to regularize singularities at the microscale, and discussing the impact this has at the macroscale. Two opposite minimalist theories will be detailed: i) a classical paradigm, based on the disjoining pressure in combination with the spreading coefficient; ii) a new approach, invoking evaporation/condensation in combination with the Kelvin effect (dependence of saturation conditions upon interfacial curvature). Most notably, the latter effect enables resolving both viscous and thermal singularities altogether, without needing any other regularizing effects such as disjoining pressure, precursor films or slip length. Experimental results are also presented about evaporation-induced contact angles, to partly validate the first approach, although it is argued that reality might often lie in between these two extreme cases. Co-Host: Dr Nikos Savva

27 October 2014 Speaker: Aleksander Pushnitski  (King's College). Title: Spectral asymptotics for compact Hankel operators. Abstract: : I will give a short introduction into spectral analysis of Hankel operators. After this, I will describe a class of Hankel operators with a power asymptotics of eigenvalues. I will discuss the similarity with the Weyl law for differential operators. The talk is based on my joint work with Dmitri Yafaev (University of Rennes 1).  3 November 2014 Speaker: Christian Kühn (Graz University). Title: Schrödinger operators with delta-potentials on manifolds. Abstract: We will present an approach for the definition and investigation of Schrödinger operators with delta-potentials on manifolds. In particular we will consider the case when the manifold is a closed curve in R^3.

10 November 2014 Speaker: Daniel Grieser  (University of Oldenburg). Title: Eigenvalues of the Laplacian on triangles. Abstract: We study the spectrum of the Laplace operator with Dirichlet boundary conditions on Euclidean triangles. I will discuss two results: The first result, joint with S. Maronna, is a new proof of the fact that a triangle is (among the set of all triangles) uniquely determined by the spectrum. The only previously known proof of this uses wave invariants. The study of these is technically difficult. Our new proof uses heat invariants and is technically simpler, and also involves a curious and interesting – and apparently new – geometric fact about triangles. The second result, joint with R. Melrose, that I will discuss is a description of the full asymptotic behavior of the eigenvalues when the triangle degenerates into a line. This may happen in various ways. More precisely, there are two parameters describing the degeneration, and we give a complete asymptotic expansion in terms of both parameters. This involves a rather intricate and unexpected blow-up of the parameter space, which will be explained in the talk.

17 November 2014 Speaker: Christoph Fischbacher (Kent). Title: On the spectrum of the XXZ spin chain. Abstract: We consider the Heisenberg XXZ spin chain in the Ising phase, which means that the anisotropy parameter $1/\Delta$ is strictly less than $1$. After having discussed some of its properties in the finite case, we extend our considerations to the infinite case. Using its conservation of total magnetization, we restrict the operator to subspaces of fixed total magnetization. After having shown that these restrictions are equivalent to fermionic many-particle Schrödinger operators with attractive interaction, we compute the lowest energy band, which is called droplet band. An HVZ type theorem allows us to determine higher band contributions to the spectrum. After a brief discussion of the structure of these higher band contributions, we show the existence of a gap above the droplet band uniformly in the particle number under the assumption that $1/\Delta < 1/2$. This work was done with Prof. G. Stolz, UAB.​

24 November 2014 Speaker: Lauri Oksanen (UCL). Title: Local reconstruction of a first order perturbation from a restricted hyperbolic Dirichlet-to-Neumann map. Abstract: We consider a wave equation on a smooth compact Riemannian manifold with boundary and show that acoustic measurements with sources and receivers on disjoints sets on the boundary determine the lower order terms in the wave equation near the set of receivers assuming that the wave equation is exactly controllable from the set of sources and that the set of receivers is strictly convex. 1 December 2014 Speaker: Ian Wood (Kent). Title: Some spectral results for waveguides.

Abstract: We study a spectral problem for the Laplacian in a weighted space which is related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic medium. The defect is infinitely extended and aligned with one of the coordinate axes. The perturbation  introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. We prove that guided mode spectrum can be created by arbitrarily small perturbations. After performing a Floquet decomposition in the axial direction of the waveguide, we study the spectrum created by the perturbation for any fixed value of the quasi-momentum. We will also briefly discuss extending the results to a similar problem for divergence form elliptic operators.​ 8 December 2014 Speaker: Beatrice Pelloni (Reading). Title: TBC. Abstract: TBC. 26 January 2015 Speaker: Claudia Wulff (Surrey) Title: Relative Lyapunov centre bifurcations. Abstract: Relative equilibria and relative periodic orbits (RPOs) are ubiquitous in symmetric Hamiltonian systems and occur for example in celestial mechanics, molecular dynamics and rigid body motion. Relative equilibria are equilibria and RPOs are periodic orbits of the symmetry reduced system. Relative Lyapunov centre bifurcations are bifurcations of relative periodic orbits from relative equilibria corresponding to Lyapunov centre bifurcations of the symmetry reduced dynamics. In this talk we prove a relative Lyapunov centre theorem by combining recent results on persistence of RPOs in Hamiltonian systems with a symmetric Lyapunov centre theorem of Montaldi et al. We then develop numerical methods for the detection of relative Lyapunov centre bifurcations along branches of RPOs and for their computation. We apply our methods to Lagrangian relative equilibria of the $N$-body problem.

2 February 2015 Speaker: Mariapia Palombaro (Sussex) Title: Higher gradient integrability for s-harmonic maps in dimension two Abstract: I will present some recent results concerning the higher gradient integrability of ... Read more 9 February 2015 Speaker: Serena Dipierro (Edinburgh) Title: Dislocation dynamics in crystals: nonlocal effects and a macroscopic theory in a fractional Laplace setting. Abstract: We consider an evolution equation arising in the Peierls-Nabarro model for crystal dislocation. We study the evolution of such dislocation function and show that, at a macroscopic scale, the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. These dislocation points evolve according to the external stress and an interior repulsive potential. Though the problem seems of local nature, the leading order of the diffusion turns out to be a nonlocal integrodifferential operator. 16 February 2015 Speaker: Jim Wright (Edinburgh) Title: Higher gradient integrability for s-harmonic maps in dimension two Abstract:  Affine-invariant Harmonic AnalysisAbstract: We will discuss two basic problems from euclidean harmonic analysis (the Fourier Restriction problem and L^p improving of averaging operators) and develop a new affine-invariant perspective on these problems.

23 February 2015 Speaker: Sergey Morozov (Munich) Title: Complete high energy asymptotics of the integrated density of states for a wide class of periodic and almost periodic models Abstract:  The existence of  complete asymptotic expansion for the integrated density of states in the high energy regime was long conjectured for periodic Schrödinger operators. I will discuss the history of the subject and present an eventual solution in the multidimensional situation. It turns out that the result generalises to a big class of almost periodic pseudodifferential operators with smooth symbols. The talk is based on a joint work with L. Parnovski and R. Shterenberg. 2 March 2015 Speaker: Yuri Netrusov (Bristol) Title: TBC Abstract:  TBC 9 March 2015 Speaker: Roger Moser (Bath) Title: A reduced energy for Neel walls Abstract:  Neel walls are transition layers for the magnetisation vector field in thin ferromagnetic films. We analyse a model that shows strong similarities to the theory of Ginzburg-Landau vortices. In particular, there is a "reduced energy" functional that helps to understand the interaction between Neel walls, and we can compute it explicitly. This is joint work with Radu Ignat (Toulouse).​

16 March 2015 Speaker: Nikos Katzourakis (Reading) Title: Generalised solutions for fully nonlinear PDE systems and Vectorial Calculus of Variations in $L^\infty$ Abstract: Calculus of Variations in $L^\infty$ has a long history, initiated by Aronsson in the 1960s and is under active research ever since. Mathematically, minimising the supremum is very challenging (the equations are non-divergence and highly degenerate systems) but it provides more realistic models, as opposed to the classical case of the average (integral). However, due to fundamental difficulties, until the early 2010s the field was restricted to the scalar case. In this talk I will discuss the vectorial case, which has recently been initiated by the speaker. The analysis of the $L^\infty$-equations is based on a recently proposed general duality-free PDE theory of generalised solutions for fully nonlinear systems.  23 March 2015 Speaker: Michela Ottobre (Heriot Watt) Title: Analysis of irreversible Markov Semigroups Abstract:  We will present a series of results regarding the analysis of hypoelliptic/hypocoercive linear Markov semigroups. We will focus on a technique to find sharp pointwise estimates on the time-behaviour of  the derivatives (of any order and in any direction) of the semigroup. Applications to sampling/numerical problems will be discussed.

20 April 2015 Speaker: Filippo Cagnetti (Sussex) Title: The rigidity problem for symmetrization inequalities Abstract: Steiner symmetrization is a very useful tool in the study of isoperimetric inequality. This is also due to the fact that the perimeter of a set is less or equal than the perimeter of its Steiner symmetral. In the same way, in the Gaussian setting, it is well known that Ehrhard symmetrization does not increase the Gaussian perimeter. We will show characterization results for equality cases in both Steiner and Ehrhard perimeter inequalities. We will also characterize rigidity of equality cases. By rigidity, we mean the situation when all equality cases are trivially obtained by a translation of the Steiner symmetral (or, in the Gaussian setting, by a reflection of the Ehrhard symmetral). We will achieve this through the introduction of a suitable measure-theoretic notion of connectedness, and through a fine analysis of the barycenter function for a special class of sets. These results are obtained in collaboration with Maria Colombo, Guido De Philippis, and Francesco Maggi. 27 April 2015 Speaker: Shu Nakamura (University of Tokyo) Title: Microlocal analysis of scattering matrix, and related topics.  Abstract:  We discuss scattering theory for a class of self-adjoint operators  which include Schrodinger operators on R^n as well as discrete Schrodinger operators  on Z^d. We show that the scattering matrix is a pseudodifferential operator and  we can compute the symbol. For the perturbed differential operators on R^n,  we also consider high energy asymptotics, and compute the high energy asymptotics  of the spectrum on the scattering matrix (joint work with A. Pushnitski). 

2 June 2015 Speaker: Doaa Filali (Cardiff University) Title: Introduction on sub-Riemannian geometry.  Abstract:  TBC 15 June 2015 Speaker: Lennie Friedlander (University of Arizona) Title: Parametric Dirichlet-to-Neumann operator.  Abstract:  I will discuss spectral asymptotics for a parametric family of Dirichlet-to-Neumann operators. 

Abstract: Broadly speaking, geometric representation theory is a framework in which symmetries of geometric objects act on invariants of these objects such as cohomology theories and, more generally, derived categories associated to them. We then apply geometric machinery to study the structure of these invariants. Often the representation theoretic results obtained in this way are substantial and beyond the reach of purely algebraic methods. More specifically, in an algebro-geometric setting we can consider an algebraic group G with a subgroup H. The geometry of the space H\G/H gives rise to a number of interesting algebras and their representations, both classical and categorical. In this talk I will give several examples of this: 1) Historically, geometric representation theory was developed by Kazhdan, Lusztig, Kashiwara, Beilinson and Bernstein to prove Kazhdan-Lusztig conjectures.  Let G be a reductive  algebraic group e.g. GL(n) and let B be a Borel subgroup in G. I will discuss the Grotehndieck group of B x B-equivariant perverse sheaves on G, with the multiplication given by convolution, and its relation to Kazhdan-Lusztig theory for the finite Hecke algebra.  2) In the geometric setting of 1) Kostant and Kumar considered the Grothendieck group of B x B-equivariant coherent sheaves on G. The convolution product gives rise to an algebra structure on the group called the degenerate affine Hecke algebra. I will explain the recent work of Harada, Landveber and Sjamaar which relates this algebra to Demazure operators and its categorical version due to Arkhipov and Kanstrup. 3) If time permits, I will also discuss the geometric affine Hecke category of of Bezrukavnikov, Riche, Ben-Zvi and Nadler and its natural place in the framework of the geometric Langlands correspondence. Notes from Talk

22nd April 2015 Speaker: Prof. Alessio Corti (Imperial) Title: Lattice polygons, mirror symmetry and classification problems in algebraic geometry. Abstract: I state some elementary questions in the combinatorics of lattice polygons and explain some answers by Kasprzyk and others. Then I sketch how these questions have far-reaching implications in mirror symmetry and classification problems in algebraic geometry. If time permits I speculate about possible higher dimensional generalisations. 6th May 2015 Speaker: Prof. Mark Girolami (Warwick) Title: Differential Geometric Markov Chain Monte Carlo Methods Abstract: Monte Carlo methods are the dominant approach to perform inference over increasingly sophisticated statistical models used to describe complex phenomena. This presents a major challenge as issues surrounding correct and efficient MCMC -based statistical inference over such models are of growing importance. This talk will argue that differential geometry provides the tools required to develop MCMC sampling methods suitable for challenging statistical models. By defining appropriate Riemannian metric tensors and corresponding Levi-Civita manifold connections MCMC methods based on Langevin diffusions across the model manifold are developed. Furthermore proposal mechanisms which follow geodesic flows across the manifold will be presented. The optimality of these methods in terms of mixing time shall be discussed and the strengths (and weaknesses) of such methods will be experimentally assessed on a range of statistical models will also be considered. This talk is based on work that was presented as a Discussion Paper to the Royal Statistical Society and it remains the most downloaded article from the journal website. Details here.

13 May 2015 Speaker: Prof. Bernard Schutz (Cardiff) Title: Data Science Challenges at Cardiff University Abstract: The new Cardiff Data Innovation Institute (DII) has been established to do research in data science, defined rather broadly. The DII is just starting to build up staff and seek collaborations. The spectrum of research at the University that involves the exploitation of large and/or complex data sets is remarkably wide. I will describe the mission and organisation of the DII and then highlight some of the areas where there are challenges that the DII could hope to address, either by itself or in the role of a “matchmaker” between data producers and existing fundamental data science researchers at the University. 24th June 2015 Speaker: Prof. Jesus De Loera (UC Davis) Title: Helly's theorem: A jewel of 20th century geometry and its new 21st century applications. Abstract: The classical theorem of Eduard Helly (1913) is a masterpiece of geometry. It states that if a finite family $\Gamma$ of convex sets in $R^n$ has the property that every $n+1$ of the sets have a non-empty intersection, then all the convex sets must intersect. This theorem has since found applications in many areas, most particularly  the study of solvability of systems of linear inequalities and the theory of optimization. My lecture will be accessible to undergraduate students, it will begin explaining the basics of convex geometry and  proceed with a selection of lovely applications of Helly's theorem. The last part of my talk will deal with some surprising new generalizations, my favorite one is our brand new version when the intersection(s) contain(s) a lattice point. It originated in the 1970's  work of Doignon, Bell, and Scarf (arising in Economics theory). Along the way I will mention the history of the subject. All new results are based on joint work with I. Aliev, R. Basset, Q. Louveaux, R. La Haye, D. Oliveros and E. Roldan-Pensado.

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Academic Staff Dr Iskander Aliev Dr Tracey England Dr Dafydd Evans Professor Jeff Griffiths Professor Paul Harper Dr Izabela Komenda Dr Vincent Knight Dr Rhyd Lewis Dr Jonathan Thompson Dr Julie Vile

Academic Staff Dr Roger Behrend Professor David E Evans Dr Gandalf Lechner Dr Timothy Logvinenko Dr Mathew Pugh G A Elliott (Honorary Professor) V F R Jones (Honorary Professor)

Academic Staff Dr Roger Behrend Professor David E Evans Dr Gandalf Lechner Dr Timothy Logvinenko Dr Mathew Pugh G A Elliott (Honorary Professor) V F R Jones (Honorary Professor)

PhD students Stephen Moore Cellan White

Academic Staff Dr Maurice Blount Dr Mikhail Cherdantsev Dr Christopher Davies Professor A Russell Davies (Honorary) Dr Claire Heaney Dr Angela Mihai Professor Tim Phillips Professor John Pryce Dr Nikos Sav

Academic Staff Dr Iskander Aliev Professor Alexander Balinsky Professor B.M. Brown (School of Computer Science) Dr Mikhail Cherdantsev Dr Kirill Cherednichenko Dr Nicolas Dirr Dr Federica Dragoni Professor David Edmunds (Honorary Professor) Dr Jonathan Eckhardt (Austrian Presidential Research Fellow - Joint with COMSC) Professor W Desmond Evans (retired) Dr Matthew Lettington Professor Marco Marletta Dr Juan Reyes (Joint with COMSC) Dr Karl Michael Schmidt Professor U Smilansky