Search

In this paper we prove structure results for the singular sets of stationary harmonic maps and mean curvature flows local to particular singularities. The original work is contained in Chapter 5 and Chapter 8. Chapters 1-5 are concerned with energy minimising maps and stationary harmonic maps. Chapters 6-8 are concerned with mean curvature flows and Brakke flows. In the case of stationary harmonic maps we consider a singularity at which the spine dimension is maximal, and such that the weak tangent map is homotopically non-trivial, and has minimal density amongst singularities of maximal spine dimen- sion. Local to such a singularity we show the singular set is a bi-Hölder continuous homeomorphism of the unit disk of dimension equal to the maximal spine dimension. A weak tangent map is translation invariant along a subspace, and invariant under dilations, so it completely defined by its values on a sphere. Such a map is said to be homotopically non-trivial if the mapping of a sphere into some target manifold cannot be deformed by a homotopy to a constant map. For an n-dimensional mean curvature flow we consider a singularity at which we can find a shrinking cylinder as a tangent flow, that collapses on an (n−1)-dimensional plane. Local to such a singularity we show that all singularities have such a cylindrical tangent, or else have lower Gaussian density than that of the shrinking cylinder. The subset of cylindrical singularities can be shown to be contained in a finite union of parabolic (n − 1)-dimensional Lipschitz submanifolds. In the case that the mean curvature flow arises from elliptic regularisation we can show that all singularities local to a cylindrical singularity with (n − 1)-dimensional spine are either cylindrical singularities with (n − 1)-dimensional spine, or contained in a parabolic Hausdorff (n − 2)-dimensional set.

Three different models, relating to the study of microswimmers immersed in a low Reynolds number fluid, are presented. The underlying, mathematical concepts employed in each are developed using singularity methods of Stokes flow. The first topic concerns the motility of an artificial, three-sphere microswimmer with prescribed, non-reciprocal, internal forces. The swimmer progresses through a low Reynolds number, nonlinear, viscoelastic medium. The model developed illustrates that the presence of the viscoelastic rheology, when compared to a Newtonian environment, increases both the net displacement and swimming efficiency of the microswimmer. The second area concerns biological microswimming, modelling a sperm cell with a hyperactive waveform (vigorous, asymmetric beating), bound to the epithelial walls of the female, reproductive tract. Using resistive-force theory, the model concludes that, for certain regions in parameter space, hyperactivated sperm cells can induce mechanical forces that pull the cell away from the wall binding. This appears to occur via the regulation of the beat amplitude, wavenumber and beat asymmetry. The next topic presents a novel generalisation of slender-body theory that is capable of calculating the approximate flow field around a long, thin, slender body with circular cross sections that vary arbitrarily in radius along a curvilinear centre-line. New, permissible, slender-body shapes include a tapered flagellum and those with ribbed, wave-like structures. Finally, the detailed analytics of the generalised, slender-body theory are exploited to develop a numerical implementation capable of simulating a wider range of slender-body geometries compared to previous studies in the field.

Two Poisson brackets are called compatible if any linear combination of these brackets is a Poisson bracket again. The set of non-zero linear combinations of two compatible Poisson brackets is called a Poisson pencil. A system is called bihamiltonian (with respect to a given pencil) if it is hamiltonian with respect to any bracket of the pencil. The property of being bihamiltonian is closely related to integrability. On the one hand, many integrable systems known from physics and geometry possess a bihamiltonian structure. On the other hand, if we have a bihamiltonian system, then the Casimir functions of the brackets of the pencil are commuting integrals of the system. We consider the situation when these integrals are enough for complete integrability. As it was shown by Bolsinov and Oshemkov, many properties of the system in this case can be deduced from the properties of the Poisson pencil itself, without explicit analysis of the integrals. Developing these ideas, we introduce a notion of linearization of a Poisson pencil. In terms of linearization, we give a criterion for non-degeneracy of a singular point and describe its type. These results are applied to solve the stability problem for a free multidimensional rigid body.

We investigate in this thesis the generic properties of curves in the Minkowski plane R2 1 and of smooth Lorentzian surfaces. The generic properties of curves in R2 1 are obtained by studying the contacts of curves in R2 1 with lines and pseudo-circles. These contacts are captured by the singularities of the families of height and distancesquared functions on the curves. On the other hand, the generic properties of smooth Lorentzian surfaces are obtained by studying certain Binary Differential Equations defined on the surfaces.

Morihiko Saito’s theory of Hodge modules have made an incredible impact in the study of singularities. So far, the strongest results have been obtained in the case of hypersurface singularities, using the strong properties of the V -filtration along hypersurfaces which is built into the theory of Hodge modules. This thesis extends two important tools from the case of hypersurfaces. The first is a compatibility property between the Hodge filtration of a mixed Hodge module and the V - filtration along a higher codimension subvariety. The second is a formula explaining how to restrict to a smooth subvariety of higher codimension using the V -filtration along that subvariety. The main tool at work in proving these theorems is the blow-up along the smooth subvariety. There are two main applications of these theorems: the first is to analyze the Hodge and weight filtration on the local cohomology module along a singular locally complete intersection subvariety. We define the minimal exponent of a locally complete intersection variety and show that its value dictates when the Hodge filtration on local cohomology is equal to the pole order filtration. This shows that the minimal exponent understands information about k-du Bois singularities, and it turns out that the minimal exponent also understands k-rational singularities, by its relation to the weight filtration on local cohomology. The second application is to the study of the Fourier-Laplace transform of monodromic mixed Hodge modules. These modules naturally arise through Verdier’s specialization construction. We explicitly write out the Hodge and weight filtrations for such modules.

In this thesis we study singularities of Hilbert schemes and show that there are many (components) of Hilbert schemes that are smooth or mildly singular and use them to explore phenomena in birational geometry and commutative algebra. Specifically, we study the Hilbert scheme compactification of a pair of linear spaces, describe all the sub- schemes parameterized by this component and show that it is a smooth Mori dream space. We study Hilbert schemes with two Borel-fixed points and prove that they are reduced, and that their irreducible components have normal and Cohen-Macaulay singularities. We study the Hilbert scheme of points on a threefold and extend results on the Hilbert scheme of points of a surface to this case; we also provide bounds on the dimension of this Hilbert scheme. Finally, we generalize the Hilbert and Quot schemes to construct the fiber-full scheme, which is a fine moduli space that controls all the cohomological data of a variety instead of just the Hilbert polynomial.

In low-dimensional topology one often decomposes a space into pieces that can be individually understood then put back together, yielding a combinatorial description, and such descriptions can be used for diagrammatic reasoning, computations of algebraic invariants, constructions of new objects, and so on. A powerful tool for this is singularity theory, where a classification of the types of singularities that generically might be present (for example, Morse critical points) leads to systematic approaches to decompositions, and in the ideal case the classification reduces to analyzing singularities of polynomial functions. We develop singularity theory relevant to "n-categorical" decompositions of smooth manifolds, in which pieces each have a recursive decomposition of their boundaries. In particular, we study smooth functions from manifolds to flag-foliated R^n, which serves as a model for the composition laws of an n-category. We use a refinement of the jet transversality techniques for the Thom--Boardman singularity types to define dense sets of functions that are suitably generic with respect to the foliations. We show how to compute codimensions of the submanifolds that correspond to a singularity type. For a few situations, for instance surfaces with embedded curves, we carry out a classification of the germs that appear in this dense set up to structure-preserving diffeomorphism. To this end, we prove stability results that aid in proving equivalence of germs and in finding polynomial representatives.We explain how to use this classification of singularities to give a presentation of the symmetric monoidal 2-category of curves with embedded curves. As an application, we give a way to compute the Krushkal polynomial, which is a combinatorial invariant of graphs in surfaces, using an extended TQFT on the symmetric monoidal 2-category of surfaces that are decomposed into black and white regions.

We study singularities of algebraic varieties, in particular those arising in birational geometry, from several points of view. The first is that of arc schemes: arc schemes parametrize “infinitesimal curves” on a variety, and their geometry reflects properties of singularities. We show that morphisms of arc schemes (more precisely, of “local” arc schemes) can detect local isomorphisms of varieties. More precisely, we use the triviality of a certain ideal-closure operation to show that if a morphism induces an isomorphism of local arc schemes then it must be an isomorphism on local rings. We then use arc schemes, in conjunction with the theory of determinantal rings, to verify the semicontinuity conjecture for the behavior of the minimal log discrepancy (a subtle invariant of singularities) in the case of determinantal varieties. In particular, we calculate the Nash ideal of a generic square determinantal variety, which then allows us to give an explicit formula for the minimal log discrepancies of pairs of determinantal varieties and determinantal subvarieties. This allows us to verify the semicontinuity conjecture for such pairs. We then take another point of view, via the study of differential operators on singular rings. At least since [Levasseur and Stafford 1989], the question had been asked of whether one can characterize singularities of rings via certain properties of their rings of differential operators. In particular, one question is whether a ring with mild singularities is a simple module under the action of its ring of differential operators. While an answer in characteristic p had been provided by [Smith 1995], no answer had been forthcoming in characteristic 0. We provide a counterexample showing that the expected connection does not exist, through the study of the global geometry of Fano varieties. More specifically, we show that certain del Pezzo surfaces do not have big tangent bundles, and thus their homogeneous coordinate rings are not simple under the action of their rings of differential operators, despite having “mild” singularities.

This dissertation concerns the Inverse Mean Curvature Flow of closed hypersurfaces in Euclidean Space, and its relationship with minimal surfaces. Inverse Mean Curvature Flow is an extrinsic geometric flow which has become prominent in differential geometry because of its applications to geometric inequalities and general relativity, but deep questions persist about its analytic and geometric structure. The first four chapters of this dissertation focus on singularity formation in the flow, the flow behavior near singularities, and the dynamical stability of round spheres under mean-convex perturbations.On the topic of singularities, I establish the formation of a singularity for all embedded flow solutions which do not have spherical topology within a prescribed time interval. I later show that mean-convex, rotationally symmetric tori undergo a flow singularity wherein the flow surfaces converge to a limit surface without rescaling, contrasting sharply with the singularities of other extrinsic geometric flows. On the topic of long-time behavior, I show that all flow solutions whichexist and remain embedded for some minimal time depending only on initial data must exist for all time and asymptotically converge to round spheres at large times. In the fourth chapter, I utilize this characterization to establish dynamical stability of the round sphere under certain mean-convex, axially symmetric perturbations that are not necessarily star-shaped. In the last chapter, I relate questions of singularities and dynamical stability for the InverseMean Curvature Flow to the mathematics of soap films. Specifically, I show that certain families of solutions to Plateau’s problem do not self-intersect and remain contained within a given region of Euclidean space. I accomplish this using a barrier method arising from global embedded solutions of Inverse Mean Curvature Flow. Conversely, I also use minimal disks to establish that a singularity likely forms in the flow of a specific mean-convex embedded sphere.

Differential games provide a framework for solving dynamic systems of two competing interests.One family of differential games focuses on two competing agents.Another family of differential games has competing teams of agents known as teaming games.Solutions to teaming games have inherit challenges due to the number of agents.By increasing the number of agents on each team, the dimension of the state space increases and additional termination cases are created.Singularities are a challenge in the majority of differential games and increasing the number of agents compounds that difficulty.This study aimed to overcome the challenges presented in solving teaming differential games by solving corresponding optimal control problems.The teaming game in question is a single pursuer, dual evader pursuit-evasion differential game.By fixing the control strategy of the pursuer, the teaming differential game is transformed into an optimal control problem for the team of evaders.Conversely, fixing the control strategies of the evaders results in an optimal control problem for the pursuer.For both problems, the optimal control strategy for the team in question is determined along with any singularities present within the control strategies.The teaming game is then reconsidered.Similarities between the optimal control problems and differential game allow for a simplified development of the solution to teaming game.The work concludes by demonstrating how solving the corresponding optimal control problems helps to overcome the inherit challenges of solving a teaming differential game.

Singularity formation is an inherent feature of equations in nonlinear physics, in many situations such as in self-focusing of light nonlinearity is essential part of the model and physical events cannot be captured by linearized equations. There are nonlinear systems, such as $1$D NLSE where singularities in analytic continuation would follow a soliton solution at fixed distances and while it is true that soliton determines the position of all the singularities, it is also true that evolution of singularities determines the solution on the real axis. Before we go further to discuss $2$D problems, we want to be more specific about analytic continuation in a $2$D problem: it is well-known that collapse in $2$D NLSE is radially symmetric and introducing radial variable $r = \\sqrt{x^2 + y^2}$ the problem becomes effectively one-dimensional. If we expand the interval spanned by $r$ from $[0,+\\infty)$ to $(-\\infty,+\\infty)$ and continue all the functions evenly across the origin, it starts to make sense to further expand $r$ to complex plane $\\mathbb{C}$ and talk about analytic continuation of functions in $r\\in\\mathbb{C}$. In $2$D nonlinear Schr\\"odinger equation (NLSE) it is common to think that a singularity appears in finite time, but one can also say that a singularity already exists in the analytic continuation of initial data and at critical time $t_c$, the singularity touches the real axis and solution reaches its maximal interval of existence. The latter point of view captures evolution in more detail, in particular it allows to ask many questions that would seem quite meaningless if you follow the "philosophy'' of a singularity just appearing at a finite time. In particular, one can ask question what is the trajectory of singularity in complex plane and how does the type of singularity change as $t\ o t_c$. For $2$D focusing NLSE and Keller-Segel model (KSE) of chemotactic bacteria, the singularities evolve towards the real axis if sufficient conditions are met by initial distribution of laser intensity (NLSE) and bacteria density (KSE) respectively. Well-established conditions are included in the text and are cited upon in corresponding sections. The central subject of this work is the study of onset of singularity towards the real axis in radially symmetric $2$D NLSE and $2$D reduced KSE models (RKSE) combining two approaches: direct numerical simulations of collapse and asymptotic analysis in the limit $t\ o t_c$. The benefit of this two-sided approach is evident when comparing results of classic theory of critical collapse in $2$D NLSE to numerical simulations: the collapse exhibits dependence on initial data even when intensity reaches enormous magnitudes and as a result is inconsistent with classical theory (e.g loglog law). An intervention of numeric approach allowed us to perform sanity checks of many assumptions and estimate regions of applicability of approximations that were used in asymptotic approach and resulted in a new corrected theory that is able to consistently describe the onset of singularity even for moderate-amplitude, developed collapse while still recovering classic theory in the limit $t\ o t_c$. The problem of $2$D potential flow of ideal fluid in free surface hydrodynamics is another example of nonlinear system containing solutions with singularities. The focus of our investigation lies in fully nonlinear travelling waves on the surface of fluid also known as Stokes waves and in particular we are interested in singularities that are present in the analytic continuation of Stokes waves. These waves computed as a part of this dissertation range from linear waves to the limit of extremely nonlinear waves that were never observed before, in addition a predicted phenomenon of parameter oscillation was confirmed for strongly nonlinear waves.

Jet schemes and arc spaces play important roles in the study of singularities of higher dimensional algebraic varieties. In this dissertation, we prove results in two directions. The first one is concerned with the study of singularities of Brill-Noether loci, in particular, the theta divisor, in terms of their jet schemes. The other concerns the generalization of the correspondence between divisorial valuations and closed irreducible cylinders in the arc space to arbitrary characteristics.

The standard mathematical model for the motion of surface waves in shallow water is the Euler equations for inviscid, incompressible flow, supplemented by free surface conditions. Without known explicit solutions in a simple form, simplifying assumptions are invoked to derive weakly nonlinear models and other variants that describe wave propagation. These models have provided good results in various applications, but their region of validity is not known precisely. The goal in this thesis is to assess the asymptotic errors in the KdV model by computing solutions to Euler’s equations numerically and comparing them with the predictions from the KdV equation directly.

This dissertation studies certain invariants of singularities of complex algebraic surfaces. In the first main result, we show that all integrally closed ideals on log terminal surfaces are multiplier ideals by extending an existing proof for smooth surfaces. Next, we turn our attention to the computation of the jumping numbers of an ideal in the local ring of a surface at a rational singularity. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an algorithm for finding the jumping numbers of the ideal. This shows, in particular, how to compute the jumping numbers of any plane curve from the numerical data of its minimal resolution. In addition, the jumping numbers of the maximal ideal at the singular point in a Du Val or toric surface singularity are computed, and applications to the smooth case are explored. We end by giving a new and insightful computation of the jumping numbers of the germ of a unibranch plane curve.

Two-dimensional free surface flows may be formulated through boundary integrals which allows analytic continuation in the unphysical complex plane. For two dimensional water waves, Tanveer showed that the only form of singularity in the unphysical plane is of the square-root type. One may wonder what influence these singularities may have on the behavior of water waves. This thesis resolves some of these questions by tracking singularities in the unphysical domain and relating their close approach to the real axis with wave breaking.The main result is the direct verification of Tanveer’s singularity result. A boundaryintegral technique is used to simulate deep water wave motion. A spectral procedure is used to form-fit the Fourier spectrum of the curvature of the wave profile to a prescribed asymptotic expression. The form-fit provides information on the power and location of the closest singularity to the real axis. The power of the curvature singularities is found to be -3/2 when the curvature is expressed as a function of the Lagrangian variable. This singularity is associated with a pole singularity in the complex arclength plane, and is not an artifact of the parametrization. The singularity approaches the real axis when a plunging breaker occurs. For nonbreaking waves, the singularity wanders above some level in the unphysical plane. It is then established that this curvature singularity is theoretically equivalent to Tanveer’s one-half power singularity. When the surface elevation is viewed as a function of horizontal distance, a different type of singularity arises. It is a square root type singularity that takes the form of a breaking wave when it reaches the real axis of the horizontal coordinate.Nonlinear interactions among various wavelengths are considered important in random ocean waves. A particularly important nonlinear interaction is the Benjamin-Feir instability. For moderate initial amplitudes, the end-state of this instability is either wave breaking or the Fermi-Pasta-Ulam recurrence. Clearly, singularities in the unphysical domain will play a role. Starting with initial conditions that contain several singularities away from the real axis, their trajectories are studied. When breaking occurs, results show that it is one of the crests in the wave train that breaks like a plunging breaker while others remain moderately flat. One of the singularities moves close to the real axis while the other singularities stay far away. When recurrence occurs, evidence indicates that the closest singularity remains far away from the real axis for all time. The hope and the possibility is that a study of singularities in Benjamin-Feir instability may lead to insight into nonlinear wave interactions in general.

Solids with spatial variations in the crystalline axes naturally evolve into cells or grains separated by sharp walls. Such variations are mathematically described using the Nye dislocation density tensor. At high temperatures, polycrystalline grains form from the melt and coarsen with time; the dislocations can both climb and glide. At low temperatures under shear the dislocations (which allow only glide) form into cell structures. While both the microscopic laws of dislocation motion and the macroscopic laws of coarsening and plastic deformation are well studied, we hitherto have had no simple, continuum explanation for the evolution of dislocations into sharp walls. We present here a mesoscale theory of dislocation motion. It provides a quantitative description of deformation and rotation, grounded in a microscopic order parameter field exhibiting the topologically conserved quantities. The current of the Nye dislocation density tensor is derived from downhill motion driven by the microscopic Peach--Koehler forces between dislocations, making use of a simple closure approximation. The resulting theory is shown to form sharp dislocation walls in finite time mathematically described by Burgers equation---similar to those seen in the theory of sonic booms and traffic jams. Our finite-difference methods use special upwind and Fourier techniques for dealing with the shock formation. The outcomes of our simulations in one and two dimensions are in good agreement with experiment and other discrete dislocation simulations. These results provide fundamental insights into the basic phenomena of plastic deformation in crystals, and offers predictions for residual stress, cell-structure refinement, and the distinguish features of different hardening stages in plastic deformation.

In recent years a resurgence of interest in wave singularities (of which optical vortices are a prominent example), light angular momentum and the relations between them has occurred. Many applications in various areas of linear and non-linear optics have been based on studying effects related to angular momentum and optical vortices. This dissertation examines the use of such wave singularities for studying the light propagation in highly inhomogeneous media and the relationship to angular momentum transfer. Angular momentum carried by light can be, in many cases, divided in two terms. The first one relates to the polarization of light and can be associated, in the quantum description, to the spin of a photon. The second is determined by the electromagnetic field distribution and, in analogy to atomic physics, is associated with the orbital angular momentum (OAM) of a photon. Under the paraxial approximation appropriate for the case of beam propagation, the two terms do not couple. However, each of them can be modified by the interaction with different media in which the light propagates through processes which involve angular momentum exchange. The decoupling of spin and orbital parts of light angular momentum can not, in general, be assumed for non paraxial propagation in turbid media, especially when backscattering is concerned. In Chapter 3 of this dissertation, scattering effects on angular momentum of light are discussed both for the single and multiple scattering processes. It is demonstrated for the first time that scattering from a spherically symmetric scattering potential, couples the spin and the OAM such that the total angular momentum flux density in conserved in every direction. Remarkably, the conservation of angular momentum occurs also for some classes of multiple scattering trajectories and this phenomenon manifests itself in ubiquitous polarization patterns observed in back-scattering from turbid media. It is newly shown in this dissertation that the polarization patterns a result of OAM carrying optical vortices which have a geometrical origin. These geometrical phase vortices are analyzed using the helicity space approach for optical geometrical phase (Berry phase). This approach, introduced in the con- text of random media, elucidates several aspects specific to propagation in helicity preserving and non-preserving scattering trajectories. Another aspect of singular waves interaction with turbid media relates to singularities embedded in the incident waves. Chapter 4 of the dissertation discusses how the phase distribution associated with an optical vortex leads to changes in the spatial correlations of the electromagnetic field. This change can be used to control the properties of the effect of enhanced backscattering in a way which allows inferring the optical properties of the medium. A detailed theoretical and experimental study of this effect is presented here for the first time for both double-pass geometries and diffusive media. It is also demonstrated that this novel experimental technique can be used to determine the optical properties of turbid media and, moreover, it permits to sense the depth of reflective inclusions in opaque media. When considering a regime of weakly inhomogeneous media, the paraxial approximation is still valid and therefore the spin and OAM do not couple. If, In addition, the medium is optically isotropic then the polarization is not affected. However, when the medium is non-axially symmetric for any specific realization, the OAM does change as a result of interaction with the medium. This effect can be studied using a newly developed method of coherent modes coupling which is presented in Chapter 5. This approach allows studying the power spread across propagating modes which carry different orbital angular momentum. The powerful concept of coherent modes coupling can be applied to fully coherent, fully polarized sources as well to partially coherent, partially polarized ones. An example of this scattering regime is atmospheric turbulence and the propagation through turbulence is thoroughly examined in Chapter 5. The results included in this dissertation are of fundamental relevance for a variety of applications which involves probing different types of random media. Such applications include remote sensing in atmospheric and maritime environments, optical techniques for biomedical diagnostics, optical characterization procedures in material sciences and others.