The Geometry and Analysis seminar is the main seminar of the Leeds Geometry and Analysis Group. Unless otherwise stated, seminars take place on Wednesdays at 15:00. The seminar is organised by Ben Lambert and Francesca Tripaldi.
In this talk, I will discuss inverse source problems for nonlinear evolution equations, particularly for parabolic and pseudoparabolic equations perturbed by p-Laplacian diffusion and power-law reaction terms. We also consider inverse problems for these equations with time-fractional derivatives. Such parabolic and pseudoparabolic equations, modified by p-Laplacian diffusion and power-law reaction terms, have various applications—for instance, in modeling non-Newtonian fluids, filtration processes, and more.
For all these problems, we establish sufficient conditions on the data that guarantee the existence and uniqueness of weak solutions to the given problems. In addition, we obtain some qualitative properties of the solutions, such as extinction in finite time, large time behavior, etc.
Skyrmions, topological solitons arising in nonlinear sigma models, have long been proposed as effective models for atomic nuclei. While exact solutions remain unknown, a variety of approximations, often inspired by better understood solitons such as monopoles and instantons, have been developed. In particular, Sutcliffe recently introduced a novel approximation using an ultra-discrete computation of the holonomy of a JNR instanton. In this talk, I will outline the mathematical framework behind this construction, before focusing particularly on the constituent locations of the resulting Skyrmions. I will present new theoretical and computational tools for analysing these locations, reveal their connections to classical geometry and monopole theory, and discuss their potential implications for future research. Based up joint work with Josh Cork.
On the space of matrices with rational (trigonometric/elliptic) entries there is a well-known Lie-Poisson structure, the ``r-matrix structure’’. It is an essential structure underlying the Hamiltonian dynamics of the vast majority of integrable systems, isospectral and isomonodromic evolution equations. The known r-matrices depend on parameter in rational way (trig/elliptic, respectively) and hence we think of them on the Riemann sphere (cylinder/torus).
In a relatively abstract Hamiltonian framework the isospectral evolution equations are generalized to higher genus Riemann surfaces as the “Hitchin systems”, an evolutionary integrable system on the moduli space of vector bundles. On the isomonodromic side main progress is attributable to Krichever who used a quite explicit coordinatization of vector bundles on Riemann surfaces that we can call “Tyurin parametrization”.
In this talk I report on the fully explicit generalization of the r-matrix structure to an arbitrary genus Riemann surface merging the Tyurin-Krichever approach with the general framework of Hitchin’s. The key tool is a (fully explicit) matrix-valued kernel that plays crucial role also in setting up integral equations in related area of the "non-abelian steepest descent” method.
In this talk I will focus on simply-connected holomorphic symplectic manifolds - both Kähler (hyperkähler manifolds) and non-Kähler. The only known example of the latter was constructed by Bogomolov-Guan. First, I will remind the construction of known families which are mostly based on the same idea. Then I will outline some geometry of some particular examples with its connection to G_2-manifolds and birational geometry. Later, I will speak about the subvarieties of such manifolds with the main ideas rely on the deformation theory, calibrated manifolds and Lagrangian fibrations. In particular, I will show the nonexistence of some tori in hyperkahler manifolds.
The classical Minkowski problem asks for the existence and uniqueness of a convex body with prescribed Gauss curvature, while the family of Christoffel-Minkowski problems generalize this question to find convex bodies with prescribed elementary symmetric polynomial of the principal radii. The full resolution of the Minkowski problem was given by works of Minkowski, Aleksandrov, Pogorelov, Nirenberg, Cheng-Yau, while sufficient conditions for the resolution of the Christoffel-Minkowski problem were given by Guan-Ma and Sheng-Trudinger-Wang. In this talk we discuss recent work with Yingxiang Hu and Mohammad Ivaki, which gives an analogous set of sufficient conditions to solve the Christoffel-Minkowski problem in the class of capillary surfaces in a half spaces with angle less than 90 degrees.
Thin homotopy is an equivalence relation on paths which consist of two basic equivalences: reparametrizations and cancellation of retracings. This naturally arises in geometry when studying the invariances inherent in parallel transport on principal bundles. The path signature is the parallel transport map with respect to a “universal” translation-invariant connection on $R^n$, which has recently been used to develop the theory of rough paths. In fact, the path signature characterizes thin homotopy classes of paths.
In this talk, we discuss the generalization of this story to piecewise linear surfaces. We develop an algebraic model of piecewise linear paths and surfaces akin to the free group construction, and use this to study the relationship between thin homotopy of surfaces, and the notion of a “surface signature.” Based on joint work with Francis Bischoff.
NOTES: Online seminar, also streamed in RSLT11.
Coassociatives are four-dimensional calibrated submanifolds of $G_2$ manifolds, seven dimensional manifolds with holonomy $G_2$. There is especially rich geometry to be studied when two coassociatives intersect transversely inside the ambient $G_2$ manifold. Non-compact examples of this phenomenon involving the Harvey-Lawson $Sp(1)$ invariant coassociatives in $R^7$ have been studied by Lotay and Kapouleas, who use the $U(1)$ symmetry in these examples to show the transverse intersection can be resolved by gluing in a family of Lawlor necks. The topology of the resulting coassociative is that of a generalized connected sum: the connected sum of the original two coassociatives along their intersection circles. In this talk, I will report on progress towards a generalization of this gluing construction for compact coassociative submanifolds intersecting transversely in an arbitrary $G_2$ manifold. If time permits, I will also describe an invariant for certain types of such transverse coassociative pairs.
Gregory-Laflamme type instabilities seem to plague black holes in dimensions greater than 4. Gregory-Laflamme instabilities are exponentially growing solutions to the linearised Einstein vacuum equation and have been identified heuristically and numerically for black holes that have an event horizon that has one direction that is large compared to all others. In this talk, I will discuss a direct rigorous mathematical proof of the Gregory-Laflamme instability for the 5D Schwarzschild black string. The proof relies upon reducing the linearised vacuum Einstein equation to a Schrödinger equation to which direct variational methods can be applied.
The entropy of a continuous random variable behaves, in some ways, similarly to the logarithm of the volume of a set. In particular, the Entropy Power Inequality (EPI) is widely considered as an information-theoretic analogue of the Brunn-Minkowski inequality. In fact, a common proof of the two inequalities exists via a sharp form of Young's inequality.
After briefly reviewing this connection between convex geometry and information theory, we will present a new inequality for entropy, which improves the EPI under assumptions on the marginals, in the same way that Bonnesen's inequality improves the Brunn-Minkowski inequality under assumptions on the volume along some projection. We will characterize the equality case in the latter inequality. Furthermore, we will show how this inequality follows from a more general entropy inequality, which reduces to Bergström's inequality for determinants in the Gaussian case. We will also discuss a related inequality for the Fisher information.
This talk is based on joint work with Matthieu Fradelizi and Martin Rapaport.
We introduce an area formula for computing the spherical measure of an intrinsic graph of any codimension in an arbitrary homogeneous group. Our approach only assumes that the map generating the intrinsic graph is continuously intrinsically differentiable. The important novelty lies in the notion of Jacobian, which is built by the auxiliary Euclidean distance. The introduction of this Jacobian allows the spherical factor to appear in the area formula and enables explicit computations. This is joint work with Francesca Corni (University of Bologna).
