During the last decade, different theories have been proposed for developing a first order analysis on metric measure spaces. The common idea underpinning some of these non-linear theories is that, for a viable theory of first order calculus in this abstract setting, one needs plenty of curves well distributed along the space. One way of making this idea precise is to assume that the space supports a p-Poincare inequality. In this talk, we will review some of the latest results which have contributed to understand the geometrical structure of metric measure spaces supporting a p-Poincare inequality and discuss purely geometric characterizations of p-Poincare inequality for different values of the exponent p.
How much a bilipschitz map can spiral? We explore two complementary aspects: how fast and how often. Quasiconformal techniques turn out to be effective to study this problem. In many ways, rotational phenomena for bilipschitz maps are dual to stretching properties of quasiconformal maps. I will contrast these two and explain what links them together. The talk is based on joint work with K. Astala, T. Iwaniec and E. Saksman.
One can easily see that the Jacobian determinant of a W^{1,n}-mapping in n dimensions is, a priori, only in L^1, and one cannot expect any better (at least on the Lebesgue scale). Likewise, from the regularity point of view, the best regularity one can expect is the vanishing mean oscillation (VMO) of the coordinate functions. It turns out, however, that if one additionally assumes a constant sign condition on the Jacobian (which is natural e.g. in the case when the mapping in question is a homeomorphism of a domain), one obtains both higher integrability results (for the Jacobian) and higher regularity results (for the mapping). In particular, a W^{1,n} mapping in $n$ dimensions with positive Jacobian is necessarily continuous. These results are classical in the Euclidean (i.e. flat) case. In a joint work (in progress) with M. R. Pakzad and P. Hajlasz we extend them to the case of mappings between Riemannian manifolds. Moreover, we address the same problem for mappings from a slightly larger, Orlicz-Sobolev class. In that case, the topology of the manifolds plays much deeper role and we observe global topological obstacles to continuity of such mappings.
There are many ways to introduce Sobolev spaces in the Euclidean setting. One of them is by defining a set of functions, which are absolutely continous on almost every line parallel to the coordinate axes. For such functions one can define gradient almost everywhere, and having it a space ACL_p of all p-integrable functions with p-integrable gradient. It turns out, that space ACL_p is the Sobolev space W^{1,p}. During my presentations I will show a way how to extend this idea to metric spaces, where lines are replaced by curves.
We continue presentation of Newtonian spaces and related topics.
We continue presentation of Newtonian spaces and related topics.
In my talk I will discuss different definitions of quasiconformality in metric measure spaces and their connections with Sobolev spaces and quasisymmetric mappings. I will start from quasisymmetric mappings in the Euclidean setting. After brief explanation of basic facts I will move to more advanced topics in metric measure spaces.
The boundary Harnack principle is a property that two positive harmonic functions in a domain vanishing on a portion of the boundary decay at the same speed toward a smaller portion of the boundary. I will start my talk with the basic facts about harmonic functions. Then I will characterize C^{1,1} domain in R^n and formulate the boundary Harnack principle on it. The main part of my talk will be dedicated to proving the principle.
In this talk, I will give an overview of the most recent advances in the theory of quasiregular mappings in metric measure spaces as well as subRiemannian manifolds (thus in particular Carnot groups). Even foundations of the theory of quasiregular mappings are essentially done after several recent works, the full picture is still not yet very complete and so I will indicate these natural open problems. The talk is based on several joint works with Marshall Williams, and also some joint work with Tony Liimatainen, and with Katrin Fassler, respectively.
Harmonic functions on a metric measure space are usually defined as minima of the Dirichlet energy. In my talk I will present another approach. We say that a function is harmonic on an open set U if it satisfies the mean value property for all balls contained in U. If the mean value property holds only for some admissible radii at every point of U we call a function weakly harmonic. I will start with presenting some regularity results. After that, I will discuss some results on the Dirichlet problem and geometric properties of harmonic and weakly harmonic functions. The talk is based on a joint work with Tomasz Adamowicz and Przemyslaw Gorka.
We look at prime ends in the Heisenberg group with an emphasis on the Caratheodory extension theorem and the Lendelof theorem.
One of the classical theorems in complex analysis is the Picard's theorem stating that a non-constant entire holomorphic map from the complex plane to the Riemann sphere omits at most two points. In the late 1960's and early 1970's, results of Reshetnyak and Martio-Rickman-Vaisala showed that mappings of bounded distortion, also called as quasiregular mappings, can be viewed as a counterpart for holomorphic mappings in quasiconformal geometry. One of the natural goals from the very beginning in this theory was obtain Picard-type results. In 1980, Rickman showed that a non-constant quasiregular mapping from the Euclidean n-space to the n-sphere omits only finitely many points, where the number depends only on the dimension and distortion. The sharpness of Rickman's theorem was not as simple issue as in the classical Picard theorem. In 1984, Rickman showed by a surprising and elaborate construction that given any finite set in the 3-sphere there exists a quasiregular from the Euclidean 3-space into the 3-sphere omitting exactly that set. In this talk, I will discuss joint work with David Drasin on the sharpness of Rickman's Picard theorem in all dimensions. Especially, I will discuss the role of bilipschitz geometry in the proof which leads to a stronger statement on the metric properties of the map and is a crucial ingredient in dimensions n > 3.
I will discuss different maximal operators and results for their boundedness. The Hardy-Littlewood maximal operator for f(x) is the supremum of integral averages of f over balls centered at x. The Stein or spherical maximal operator is the supremum over spheres. The boundedness of these two operators in L^p(R^n), p>1, is classical (for the second, Stein, 1975, p > n/(n-1)), and the boundedness of the second follows from the boundedness of the first. In the variable exponent case, boundedness was first proven by Diening (2004) for the first operator, and by Fiorenza, Gogatishvili and Kopaliani (2013) for the second one.
There are many ways to define harmonic functions on a metric measure space. During my talk I will present approach based on Cheeger differentiable structure in the setting of geodesic spaces supporting Poincare inequality. More precisely I am going to introduce the Dirichlet form, and from that the Laplace operator. Then I will discuss a few basic properties of measure-valued Laplace operator and show existence of solution of the inhomogeneous Dirichlet problem.
Presentation is based on a work by Marola, Miranda and Shanmugalingam, "Boundary measures, generalized Gauss-Green formulas, and mean value property in metric measure spaces", Rev. Mat. Iberoam. 31 (2015), no. 2, 497-530.
Lie algebroids are vector bundles with a little extra structure that allows us to think of them as a generalisation of the tangent bundle of a manifold. A key example of a Lie algebroid is the cotangent bundle of a Poisson manifold. The mantra is 'whatever you can to on a tangent bundle you can do on a Lie algebroid'. Interestingly, this mantra includes Riemannian geometry. The notion of a metric on a vector bundle is standard, but it is the structure of a Lie algebroid that allows the theory to develop parallel to the classical case. There are also some links with sub-Riemannian geometry. I will discuss some of these notions including Killing sections*. This opens up the possibility of sigma models with Riemannian Lie algebroids as their target spaces. Solution to the equations of motion of this class of sigma models represent generalised harmonic maps. Although this talk will primarily be about mathematical ideas, I will also suggest possible physical applications towards the end of the talk and especially quasi-classical quantum gravity!
* Andrew James Bruce, Killing sections and sigma models with Lie algebroid targets, to appear in Reports on Mathematical Physics.
In the latest decades it has been shown that much of the linear theory for Beltrami equations extends to the nonlinear situation under basic assumptions on the structure function to guarantee the uniform ellipticity. However, the general theory concerning the well-posedness of the equation does not take into consideration the special structure of the equation. Recent studies have revealed that knowing some structure of the nonlinearity provides interesting new information on the properties of the solutions.
We contribute to this type of questions by discussing the relation between the topological properties of the solutions and the range of the differential. The talk is based on the joint work with Kari Astala, Daniel Faraco and Albert Clop.
The nonlinear Beltrami equation is a planar nonlinear PDE that both describes all first-order elliptic nonlinear equations and has important relations to the theory of Quasiconformal mappings. We show that if the equation has Holder continuous coefficients, then any homeomorphic solution must have a positive Jacobian. The proof is based on new Schauder-type estimates for nonlinear Beltrami equations, as well as the key observation that the inverse of a homeomorphic solution also solves a similar equation. This is a joint work with Kari Astala, Albert Clop, Daniel Faraco and Jarmo Jaaskelainen.
We show that the interplay between the planar Beltrami equation governing quasiconformal and quasiregular mappings and Calderon's conductivity equation in impedance tomography admits a counterpart in the setting of the first Heisenberg group equipped with its canonical sub-Riemannian structure. This is a talk for general math audience. No deep knowledge of quasiregular maps, conductivity or sub-Riemannian geometry is required, and we will revise nontrivial results in these subjects as needed in the exposition.
Joint work with Jeremy Tyson (University of Illinois).
Discrete curvatures κ, that we consider, are functions of m+2 points in Rⁿ and measure the "flatness" of simplexes spanned by the parameters. These kind of functions have proved to be useful for solving variational problems with topological constraints and, in particular, finding ideal shapes of knots. In dimension one, a special instance called the Menger curvature, was also used in proving that 1-rectifiable sets are removable for bounded analytic functions. We shall review the regularity properties of measures μ resulting from finiteness of ∫ κ dμᵐ⁺² and show how discrete curvatures are related to the so called β-numbers of Peter Jones (a.k.a. height-excess).
In my talk I will introduce the Schwartz-Bruhat class of rapidly decreasing functions on LCA groups and show how can it be used in the theory of Sobolev spaces on metric groups.
We introduce some classes of nonsmooth domains, in particular John domains. We define harmonic functions on a metric space using the mean value property and show a few basic properties of such functions. The main part of my talk will be dedicated to proving a Carleson-type estimate for a positive harmonic function on a John domain locally connected at the boundary. This estimate says that values of a positive harmonic function vanishing on a portion of the boundary are bounded by the value at a fixed reference point and a multiplicative constant independent of the function.