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These are the notes of my lectures at the 1996 European Congress of Mathematicians. {} Polynomials appear in mathematics frequently, and we all know from experience that low degree polynomials are easier to deal with than high degree ones. It is, however, not clear that there is a well defined class of "low degree" polynomials. For many questions, polynomials behave well if their degree is low enough, but the precise bound on the degree depends on the concrete problem. {} It turns out that there is a collection of basic questions in arithmetic, algebraic geometry and topology all of which give the same class of "low degree" polynomials. The aim of this lecture is to explain these properties and to provide a survey of the known results.

After introducing some motivations for this survey, we describe a formalism to parametrize a wide class of algebraic structures occurring naturally in various problems of topology, geometry and mathematical physics. This allows us to define an "up to homotopy version" of algebraic structures which is coherent (in the sense of $\infty$-category theory) at a high level of generality. To understand the classification and deformation theory of these structures on a given object, a relevant idea inspired by geometry is to gather them in a moduli space with nice homotopical and geometric properties. Derived geometry provides the appropriate framework to describe moduli spaces classifying objects up to weak equivalences and encoding in a geometrically meaningful way their deformation and obstruction theory. As an instance of the power of such methods, I will describe several results of a joint work with Gregory Ginot related to longstanding conjectures in deformation theory of bialgebras, $E_n$-algebras and quantum group theory.

In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple modules of left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A fairly complete picture of the representation theory for CW left regular bands is obtained.

In this thesis I present my research on the exotic configurations of antiferromagnetic systems characterised by a topological invariant. The research presented outlines the construction of novel local antiferromagnetic degrees of freedom for low dimensional antiferromagnetic lattices. This new construction reproduces the real division algebra hierarchy satisfied by the nonlinear sigma-model and reveals the presence of a novel local gauge field. I have also studied elastic magnetic vesicles of spherical and toric genus in the presence of a magnetic soliton. My studies reveal a global shrinking, with local swellings in the regions where the soliton presents a spin-flip. The geometrical origin of this novel phenomena led me to interpret the geometric frustration of magnetic vesicles as the competition between the two topological orders present. The microscopic mechanism and topological competition suggested above go beyond the scope of this thesis, providing a microscopic explanation for the Fermi-Bose transmutation and a way to deal with this type of exotic physics.

In this thesis I present my research on the exotic configurations of antiferromagnetic systems characterised by a topological invariant. The research presented outlines the construction of novel local antiferromagnetic degrees of freedom for low dimensional antiferromagnetic lattices. This new construction reproduces the real division algebra hierarchy satisfied by the nonlinear sigma-model and reveals the presence of a novel local gauge field. I have also studied elastic magnetic vesicles of spherical and toric genus in the presence of a magnetic soliton. My studies reveal a global shrinking, with local swellings in the regions where the soliton presents a spin-flip. The geometrical origin of this novel phenomena led me to interpret the geometric frustration of magnetic vesicles as the competition between the two topological orders present. The microscopic mechanism and topological competition suggested above go beyond the scope of this thesis, providing a microscopic explanation for the Fermi-Bose transmutation and a way to deal with this type of exotic physics.

A course on Differential Geometry And Topology

A course on Differential Geometry And Topology

A course on Differential Geometry And Topology

The overall objective of this work is to introduce a drastically innovative technology for repair and automatic extraction of high-order geometric data from platform models developed with CAD (Computer Aided Design) software tools. The results will have applications to large-scale CEM analysis of low observable aircraft by high-order fast solvers as well as to a variety of other computational problems requiring high accuracy geometry representation. The effort carried out under this project was planned as an assessment of the current capabilities of GeomFix, a software system developed recently by Matis for efficient and reliable repair of geometric deficiencies in CAD models intended for use with computational electromagnetics methods.

The overall objective of this work is to introduce a drastically innovative technology for repair and automatic extraction of high-order geometric data from platform models developed with CAD (Computer Aided Design) software tools. The results will have applications to large-scale CEM analysis of low observable aircraft by high-order fast solvers as well as to a variety of other computational problems requiring high accuracy geometry representation. The effort carried out under this project was planned as an assessment of the current capabilities of GeomFix, a software system developed recently by Matis for efficient and reliable repair of geometric deficiencies in CAD models intended for use with computational electromagnetics methods.

The overall objective of this work is to introduce a drastically innovative technology for repair and automatic extraction of high-order geometric data from platform models developed with CAD (Computer Aided Design) software tools. The results will have applications to large-scale CEM analysis of low observable aircraft by high-order fast solvers as well as to a variety of other computational problems requiring high accuracy geometry representation. The effort carried out under this project was planned as an assessment of the current capabilities of GeomFix, a software system developed recently by Matis for efficient and reliable repair of geometric deficiencies in CAD models intended for use with computational electromagnetics methods.

We present a new ansatz which can successfully be used to determine the morphological properties of the supercluster-void network. The ansatz is based on a surface modelling scheme SURFGEN, which generates a triangulated surface from a discrete data set representing (say) the distribution of galaxies in real (or redshift) space. Four Minkowski functionals -- surface area, volume, extrinsic curvature and genus -- describe the geometry and topology of the supercluster-void network. Ratio's of Minkowski functionals -- Shapefinders -- provide us with an excellent diagnostic of three dimensional shapes of clusters, superclusters and voids. Minkowski functionals and Shapefinders are determined for a triangulated iso-density surface using SURFGEN. SURFGEN is tested against both simply and multiply connected eikonal surfaces such as triaxial ellipsoids and tori. Remarkably, the first three Minkowski functionals are computed to better than 1% accuracy while the fourth (genus) is known exactly. SURFGEN also gives excellent results when applied to Gaussian random fields. Our results indicate that the surface modelling scheme SURFGEN is accurate and robust and can successfully be used to quantify the topology and morphology of the supercluster-void network in the universe. We apply SURFGEN to three cosmological models, $\L$CDM, $\T$CDM and SCDM and obtain interesting new results pertaining to the geometry, morphology and topology of large scale structure.

We study knotted polymers in equilibrium with an array of obstacles which models confinement in a gel or immersion in a melt. We find a crossover in both the geometrical and the topological behavior of the polymer. When the polymers' radius of gyration, $R_G$, and that of the region containing the knot, $R_{G,k}$, are small compared to the distance b between the obstacles, the knot is weakly localised and $R_G$ scales as in a good solvent with an amplitude that depends on knot type. In an intermediate regime where $R_G > b > R_{G,k}$, the geometry of the polymer becomes branched. When $R_{G,k}$ exceeds b, the knot delocalises and becomes also branched. In this regime, $R_G$ is independent of knot type. We discuss the implications of this behavior for gel electrophoresis experiments on knotted DNA in weak fields.

We study knotted polymers in equilibrium with an array of obstacles which models confinement in a gel or immersion in a melt. We find a crossover in both the geometrical and the topological behavior of the polymer. When the polymers' radius of gyration, $R_G$, and that of the region containing the knot, $R_{G,k}$, are small compared to the distance b between the obstacles, the knot is weakly localised and $R_G$ scales as in a good solvent with an amplitude that depends on knot type. In an intermediate regime where $R_G > b > R_{G,k}$, the geometry of the polymer becomes branched. When $R_{G,k}$ exceeds b, the knot delocalises and becomes also branched. In this regime, $R_G$ is independent of knot type. We discuss the implications of this behavior for gel electrophoresis experiments on knotted DNA in weak fields.

We study knotted polymers in equilibrium with an array of obstacles which models confinement in a gel or immersion in a melt. We find a crossover in both the geometrical and the topological behavior of the polymer. When the polymers' radius of gyration, $R_G$, and that of the region containing the knot, $R_{G,k}$, are small compared to the distance b between the obstacles, the knot is weakly localised and $R_G$ scales as in a good solvent with an amplitude that depends on knot type. In an intermediate regime where $R_G > b > R_{G,k}$, the geometry of the polymer becomes branched. When $R_{G,k}$ exceeds b, the knot delocalises and becomes also branched. In this regime, $R_G$ is independent of knot type. We discuss the implications of this behavior for gel electrophoresis experiments on knotted DNA in weak fields.

The theory of self-adjoint extensions of first and second order elliptic differential operators on manifolds with boundary is studied via its most representative instances: Dirac and Laplace operators. The theory is developed by exploiting the geometrical structures attached to them and, by using an adapted Cayley transform on each case, the space $\mathcal{M}$ of such extensions is shown to have a canonical group composition law structure. The obtained results are compared with von Neumann's Theorem characterising the self-adjoint extensions of densely defined symmetric operators on Hilbert spaces. The 1D case is thoroughly investigated. The geometry of the submanifold of elliptic self-adjoint extensions $\mathcal{M}_\mathrm{ellip}$ is studied and it is shown that it is a Lagrangian submanifold of the universal Grassmannian $\mathbf{Gr}$. The topology of $\mathcal{M}_\mathrm{ellip}$ is also explored and it is shown that there is a canonical cycle whose dual is the Maslov class of the manifold. Such cycle, called the Cayley surface, plays a relevant role in the study of the phenomena of topology change. Self-adjoint extensions of Laplace operators are discussed in the path integral formalism, identifying a class of them for which both treatments leads to the same results. A theory of dissipative quantum systems is proposed based on this theory and a unitarization theorem for such class of dissipative systems is proved. The theory of self-adjoint extensions with symmetry of Dirac operators is also discussed and a reduction theorem for the self-adjoint elliptic Grasmmannian is obtained. Finally, an interpretation of spontaneous symmetry breaking is offered from the point of view of the theory of self-adjoint extensions.

Orbits of coadjoint representations of classical compact Lie groups have a lot of applications. They appear in representation theory, geometrical quantization, theory of magnetism, quantum optics etc. As geometric objects the orbits were the subject of much study. However, they remain hard for calculation and application. We propose simple solutions for the following problems: an explicit parameterization of the orbit by means of a generalized stereographic projection, obtaining a K\"{a}hlerian structure on the orbit, introducing basis two-forms for the cohomology group of the orbit.

We study the behaviour of a nonrelativistic quantum particle interacting with different potentials in the spacetimes of topological defects. We find the energy spectra and show how they differ from their free-space values.

For equivariant stable homotopy theory, equivariant KK-theory and equivariant derived categories, we show how restriction to a subgroup of finite index yields a finite commutative separable extension, analogous to finite \'etale extensions in algebraic geometry.

For equivariant stable homotopy theory, equivariant KK-theory and equivariant derived categories, we show how restriction to a subgroup of finite index yields a finite commutative separable extension, analogous to finite \'etale extensions in algebraic geometry.

For equivariant stable homotopy theory, equivariant KK-theory and equivariant derived categories, we show how restriction to a subgroup of finite index yields a finite commutative separable extension, analogous to finite \'etale extensions in algebraic geometry.

As a ramification of a motivational discussion for previous joint work, in which equations of motion for the finite spectral action of the Standard Model were derived, we provide a new analysis of the results of the calculations herein, switching from the perspective of Spectral triple to that of Fredholm module and thus from the analogy with Riemannian geometry to the pre-metrical structure of the Noncommutative geometry. Using a suggested Noncommutative version of Morse theory together with algebraic $K$-theory to analyse the vacuum solutions, the first two summands of the algebra for the finite triple of the Standard Model arise up to Morita equivalence. We also demonstrate a new vacuum solution whose features are compatible with the physical mass matrix.

Lecture notes of a course on A. Smirnov's approach to the ABC-conjecture via F1-geometry and an attempt to relate this to Jim Borger's F1-geometry based on lambda-rings.

Let $Y$ be a compact K\"ahler normal space and $\alpha \in H^{1,1}(Y,\mathbb{R})$ a K\"ahler class. We study metric properties of the space $\mathcal{H}_\alpha$ of K\"ahler metrics in $\alpha$ using Mabuchi geodesics. We extend several results by Calabi, Chen, Darvas previously established when the underlying space is smooth. As an application we analytically characterize the existence of K\"ahler-Einstein metrics on $\mathbb{Q}$-Fano varieties, generalizing a result of Tian, and illustrate these concepts in the case of toric varieties.

We study the relationship between initial data sets with horizons and the existence of metrics of positive scalar curvature. We define a Cauchy Domain of Outer Communications (CDOC) to be an asymptotically flat initial set $(M, g, K)$ such that the boundary $\partial M$ of $M$ is a collection of Marginally Outer (or Inner) Trapped Surfaces (MOTSs and/or MITSs) and such that $M\setminus \partial M$ contains no MOTSs or MITSs. This definition is meant to capture, on the level of the initial data sets, the well known notion of the domain of outer communications (DOC) as the region of spacetime outside of all the black holes (and white holes). Our main theorem establishes that in dimensions $3\leq n \leq 7$, a CDOC which satisfies the dominant energy condition and has a strictly stable boundary has a positive scalar curvature metric which smoothly compactifies the asymptotically flat end and is a Riemannian product metric near the boundary where the cross sectional metric is conformal to a small perturbation of the initial metric on the boundary $\partial M$ induced by $g$. This result may be viewed as a generalization of Galloway and Schoen's higher dimensional black hole topology theorem \cite{GS06} to the exterior of the horizon. We also show how this result leads to a number of topological restrictions on the CDOC, which allows one to also view this as an extension of the initial data topological censorship theorem, established in \cite{EGP13} in dimension $n=3$, to higher dimensions.

Hypertoric varieties are quaternionic analogues of toric varieties, important for their interaction with the combinatorics of matroids as well as for their prominent place in the rapidly expanding field of algebraic symplectic and hyperkahler geometry. The aim of this survey is to give clear definitions and statements of known results, serving both as a reference and as a point of entry to this beautiful subject.

Using $S^1$-equivariant symplectic homology, in particular its mean Euler characteristic, of the natural filling of links of Brieskorn-Pham polynomials, we prove the existence of infinitely many inequivalent contact structures on various manifolds, including in dimension 5 the k-fold connected sums of $S^2\times S^3$ and certain rational homology spheres. We then apply our result to show that on these manifolds the moduli space of classes of positive Sasakian structures has infinitely many components. We also apply our results to give lower bounds on the number of components of the moduli space of Sasaki-Einstein metrics on certain homotopy spheres. Finally a new family of Sasaki-Einstein metrics of real dimension 20 on $S^5$ is exhibited.

This is an overview of math.AG/0310186, math.AG/0309290, math.AG/0501247, math.AG/0401002 and math.AG/0504584 written for the Proceedings of the AMS Meeting on Algebraic Geometry, Seattle, 2005.

We analyze the dynamics of 2-D stationary, line-driven winds from accretion disks in cataclysmic variable stars. The driving force is that of line radiation pressure, in the formalism developed by Castor, Abbott & Klein for O stars. Our main assumption is that wind helical streamlines lie on straight cones. We find that the Euler equation for the disk wind has two eigenvalues, the mass loss rate and the flow tilt angle with the disk. Both are calculated self-consistently. The wind is characterized by two distinct regions, an outer wind launched beyond four white dwarf radii from the rotation axis, and an inner wind launched within this radius. The inner wind is very steep, up to 80 degrees with the disk plane, while the outer wind has a typical tilt of 60 degrees. In both cases the ray dispersion is small. We, therefore, confirm the bi-conical geometry of disk winds as suggested by observations and kinematical modeling. The wind collimation angle appears to be robust and depends only on the disk temperature stratification. The flow critical points lie high above the disk for the inner wind, but close to the disk photosphere for the outer wind. Comparison with existing kinematical and dynamical models is provided. Mass loss rates from the disk as well as wind velocity laws are discussed in a subsequent paper.

The goal of this article is to survey recent developments in the theory of contact structures in dimension three.

This is a survey of some of the work of Tom Farrell and Lowell Jones. This is the lead article of a special issue of the Pure and Applied Mathematics Quarterly. This issue is published in conjunction with the conference "Geometry,Topology, and their Interactions" held in Morelia, Mexico.

The problem of low Reynolds number turbulence in active nematic fluids is theoretically addressed. Using numerical simulations I demonstrate that an incompressible turbulent flow, in two-dimensional active nematics, consists of an ensemble of vortices whose areas are exponentially distributed within a range of scales. Building on this evidence, I construct a mean-field theory of active turbulence by which several measurable quantities, including the spectral densities and the correlation functions, can be analytically calculated. Because of the profound connection between the flow geometry and the topological properties of the nematic director, the theory sheds light on the mechanisms leading to the proliferation of topological defects in active nematics and provides a number of testable predictions. A hypothesis, inspired by Onsager's statistical hydrodynamics, is finally introduced to account for the equilibrium probability distribution of the vortex sizes.

This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It also establishes new connections --- some heuristic and others rigorous --- based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.

The face poset of the permutohedron realizes the combinatorics of linearly ordered partitions of the set $[n]=\{1,...,n\}$. Similarly, the cyclopermutohedron is a virtual polytope that realizes the combinatorics of cyclically ordered partitions of the set $[n+1]$. The cyclopermutohedron was introduced by the third author by motivations coming from configuration spaces of polygonal linkages. In the paper we prove two facts: (1) the volume of the cyclopermutohedron equals zero, and (2) the homology groups $H_k$ for $k=0,...,n-2$ of the face poset of the cyclopermutohedron are non-zero free abelian groups. We also present a short formula for their ranks.

Gaussian random fields pervade all areas of science. However, it is often the departures from Gaussianity that carry the crucial signature of the nonlinear mechanisms at the heart of diverse phenomena, ranging from structure formation in condensed matter and cosmology to biomedical imaging. The standard test of non-Gaussianity is to measure higher order correlation functions. In the present work, we take a different route. We show how geometric and topological properties of Gaussian fields, such as the statistics of extrema, are modified by the presence of a non-Gaussian perturbation. The resulting discrepancies give an independent way to detect and quantify non-Gaussianities. In our treatment, we consider both local and nonlocal mechanisms that generate non-Gaussian fields, both statically and dynamically through nonlinear diffusion.

Contents: 1. Clockwork topology. 2. From 2-complexes to groups. 3. From 2-complexes to groups: more advanced features. 4. Presentations. 5. Fundamental groups of graphs. 6. Coverings and subgroups. 7. Applications of coverings. 8. Fundamental groups of graphs of groups. 9. Word problems. 10. The topological dictionary. 11. Hyperbolic geometry. 12. Geometric properties of groups. 13. Hyperbolic groups. Lecture Notes Collection FreeScience.info ID1755 Obtained from http://www-users.york.ac.uk/~bje1/gtgroups.pdf http://www.freescience.info/go.php?pagename=books&id=1755

About the book This problem book is compiled by eminent Moscow university teachers. Based on many years of teaching experience at the mechanics-and-mathematics department, it contains problems practically for all sections of the differential geometry and topology course delivered for university students: besides classical branches of the theory of curves and surfaces, the reader win be offered problems in smooth manifold theory, Riemannian geometry, vector fields and differential forms, general topology, homotopy theory and elements of variational calculus. The structure of the volume corresponds to A Course of Differential Geometry and Topology (Moscow University Press 1980)  by Prof. A. T. Fomenko and Prof. A. S. Mishchenko Some problems however, touch upon topics outside the course lectures. The corresponding sections are provided with all necessary theoretical foundations. Preface 5 1. Application of Linear Algebra to Geometry 7 2. Systems of Coordinates 9 3. Riemannian Metric 14 4. Theory of Curves 16 5. Surfaces 34 6. Manifolds 53 7. Transformation Groups 60 8. Vector Fields 64 9. Tensor Analysis 70 10. Differential Forms, Integral Formulae, De Rham Cohomology 75 11. General Topology 81 12. Homotopy Theory 87 13. Covering Maps, Fibre Spaces, Riemann Surfaces 97 14. Degree of Mapping 105 15. Simplest Variational Problems 108 Answers and Hints 113 Bibliography 208

We derive some more results on the nature of the singularities arising in the collapse of inhomogeneous dust spheres. (i) It is shown that there are future-pointing radial and non-radial time-like geodesics emerging from the singularity if and only if there are future-pointing radial null geodesics emerging from the singularity. (ii) Limits of various space-time invariants and other useful quantities (relating to Thorne's point-cigar-barrel-pancake classification and to isotropy/entropy measures) are studied in the approach to the singularity. (iii) The topology of the singularity is studied from the point of view of ideal boundary structure. In each case, the different nature of the visible and censored region of the singularity is emphasized.

We derive some more results on the nature of the singularities arising in the collapse of inhomogeneous dust spheres. (i) It is shown that there are future-pointing radial and non-radial time-like geodesics emerging from the singularity if and only if there are future-pointing radial null geodesics emerging from the singularity. (ii) Limits of various space-time invariants and other useful quantities (relating to Thorne's point-cigar-barrel-pancake classification and to isotropy/entropy measures) are studied in the approach to the singularity. (iii) The topology of the singularity is studied from the point of view of ideal boundary structure. In each case, the different nature of the visible and censored region of the singularity is emphasized.

Let M be the blow-up of a complex manifold along a submanifold. We determine the integral cohomology ring and obtain a formula for the Chern classes of M. As applications we determine the cohomology rings for the varieties of complete conics and complete quadrices in 3-space, and justify two enumerative results due to Schubert [S1].

Let M be the blow-up of a complex manifold along a submanifold. We determine the integral cohomology ring and obtain a formula for the Chern classes of M. As applications we determine the cohomology rings for the varieties of complete conics and complete quadrices in 3-space, and justify two enumerative results due to Schubert [S1].

Neurodegeneration affects cortical gray matter leading to loss of cortical mantle volume. As a result of such volume loss, the geometrical arrangement of the regions on the cortical surface is expected to be altered in comparison to healthy brains. Here we present a novel method to study the alterations in brain cortical surface geometry in Parkinson's disease (PD) subjects with a \emph{Geometry Networks (GN)} framework. The local geometrical arrangement of the cortical surface is captured as the 3D coordinates of the centroids of anatomically defined parcels on the surface. The inter-regional distance between cortical patches is the signal of interest and is captured as a geometry network. We study its topology by computing the dimensionality of simplicial complexes induced on a filtration of binary undirected networks for each geometry network. In a permutation statistics test, a statistically significant ($p

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This paper is a revised version of arXiv:0805.0529 and Phys.Rev. D78, 064038 (2008), taking into account the erratum published in Phys.Rev. D85, 069903(E) (2012). Geometrical and topological properties of the quasi-plane Szekeres model and of the plane symmetric dust model are discussed. Some related comments on the quasi-hyperbolical model are made. These properties include: (1) The pattern of expansion in the plane symmetric case, and the Newtonian model that imitates it; (2) The possibility of toroidal topology of the $t =$ const sections in the plane symmetric model; (3) The absence of apparent horizons in the quasi-plane and quasi-hyperbolic models (they are globally trapped); (4) Description of the toroidal topology in the Szekeres coordinates; (5) Interpretation of the mass function in the quasi-plane model.

This is a survey article on the recent development of "stringy geometry and topology of orbifolds", a new subject of mathematics motivated by orbifold string theory.

This is a survey article on the recent development of "stringy geometry and topology of orbifolds", a new subject of mathematics motivated by orbifold string theory.

If you imagine a three dimensional maze from which there is no escape, how can you map it? Is there a way to describe what all possible mazes look like, and how do mathematicians set about investigating them? Subscribe for regular science videos: http://bit.ly/RiSubscRibe Caroline Series describes how hyperbolic geometry is playing a crucial role in answering such questions, illustrating her talk with pictures that have inspired some striking examples of digital art. Caroline Series is Professor of Mathematics at the Warwick Mathematics Institute, University of Warwick Her academic interests are in hyperbolic geometry, Kleinian groups and dynamical systems. Her research has been on the theory of dynamical systems and geometric patterns in three-dimensional hyperbolic spaces. Her work has been recognised through awards and prizes, most recently with the Senior Anne Bennett Prize from the London Mathematical Society. She is also a founding member of the European Women in Mathematics (EWM), which aims to support and encourage female mathematicians across Europe. This was filmed at a Friday Evening Discourse at the Royal Institution on 28 November 2014. Subscribe for regular science videos: http://bit.ly/RiSubscRibe The Ri is on Twitter: http://twitter.com/ri_science and Facebook: http://www.facebook.com/royalinstitution and Tumblr: http://ri-science.tumblr.com/ Our editorial policy: http://www.rigb.org/home/editorial-policy Subscribe for the latest science videos: http://bit.ly/RiNewsletter

If you imagine a three dimensional maze from which there is no escape, how can you map it? Is there a way to describe what all possible mazes look like, and how do mathematicians set about investigating them? Subscribe for regular science videos: http://bit.ly/RiSubscRibe Caroline Series describes how hyperbolic geometry is playing a crucial role in answering such questions, illustrating her talk with pictures that have inspired some striking examples of digital art. Caroline Series is Professor of Mathematics at the Warwick Mathematics Institute, University of Warwick Her academic interests are in hyperbolic geometry, Kleinian groups and dynamical systems. Her research has been on the theory of dynamical systems and geometric patterns in three-dimensional hyperbolic spaces. Her work has been recognised through awards and prizes, most recently with the Senior Anne Bennett Prize from the London Mathematical Society. She is also a founding member of the European Women in Mathematics (EWM), which aims to support and encourage female mathematicians across Europe. This was filmed at a Friday Evening Discourse at the Royal Institution on 28 November 2014. Subscribe for regular science videos: http://bit.ly/RiSubscRibe The Ri is on Twitter: http://twitter.com/ri_science and Facebook: http://www.facebook.com/royalinstitution and Tumblr: http://ri-science.tumblr.com/ Our editorial policy: http://www.rigb.org/home/editorial-policy Subscribe for the latest science videos: http://bit.ly/RiNewsletter

In a granular solid, mechanical equilibrium requires a delicate balance of forces at the disordered grain scale. To understand how macroscopic rigidity can emerge in this amorphous solid, it is crucial that we understand how Newton's laws pass from the disordered grain scale to the laboratory scale. In this work, we introduce an exact discrete calculus, in which Newton's laws appear as differential relations at the scale of a single grain. Using this calculus, we introduce gauge variables which describe identically force- and torque-balanced configurations. In a first, intrinsic formulation, we use the topology of the contact network, but not its geometry. In a second, extrinsic formulation, we introduce geometry with the Delaunay triangulation. These formulations show, with exact methods, how topology and geometry in a disordered medium are related by constraints. In particular, we derive Airy's expression for a divergence-free, symmetric stress tensor in two and three dimensions.

Given a nonconstant holomorphic map f: X -> Y between compact Riemann surfaces, one of the first objects we learn to construct is its ramification divisor R_f, which describes the locus at which f fails to be locally injective. The divisor R_f is a finite formal linear combination of points of X that is combinatorially constrained by the Hurwitz formula. Now let k be an algebraically closed field that is complete with respect to a nontrivial non-Archimedean absolute value. For example, k = C_p. Here the role of a Riemann surface is played by a projective Berkovich analytic curve. As these curves have many points that are not algebraic over k, some new (non-algebraic) ramification behavior appears for maps between them. For example, the ramification locus is no longer a divisor, but rather a closed analytic subspace. This article initiates a detailed study of the ramification locus for self-maps f: P^1 -> P^1. This simplest first case has the benefit of being approachable by concrete (and often combinatorial) techniques.