Jyväskylä Geometric Analysis Spring Seminar 2022

The seminar usually starts on Mondays at 14:15 and talks take place on campus and/or via Zoom depending on the pandemic situation.

• February 7, 2022: Vesa Julin (University of Jyväskylä)

The motion of charged liquid drop

Abstract: We study the motion of charged liquid drop in three dimensions where the equations of motions are given by the Euler equations with free boundary with an electric field. This is a well-known problem in physics going back to the famous work by Rayleigh. Due to experiments and numerical simulations one expects the charged drop to form conical singularities called Taylor cones, which we interpret as singularities of the flow. In my talk I will discuss about our recent results on the well-posedness, regularity and the formation of singularities of the solution. This is a joint work with Domenico La Manna.

• February 28, 2022: Emanuele Caputo (University of Jyväskylä)

Parallel transport on non-collapsed RCD(K,N) spaces

Abstract: In this seminar, we address the construction of parallel transport in the setting of non-collapsed RCD(K,N) spaces, which are metric measure spaces with an intrinsic notion of lower bound on the Ricci curvature and upper bound on the dimension, equipped with the N-dimensional Hausdorff measure as reference measure. In a joint work with N. Gigli and E. Pasqualetto, we obtain existence and uniqueness results of parallel transport in this setting. To be more specific, we establish a Leibniz formula granting uniqueness and I would like to emphasize the functional analytic tools developed for the proof of this formula.

Our theory covers the case of curves obtained via the flow of sufficiently regular time dependent Sobolev vector fields: in this generality, we cannot study parallel transport along a single such curve, but along a well distributed family of flow lines, in a sense related to the notion of regular Lagrangian flows in the nonsmooth setting. The construction relies on the regularity theory of such flows in RCD spaces.

• March 7, 2022: Francesco Nobili (University of Jyväskylä)

Progress on the independence on p of p weak gradients

Abstract: The independence on p of p-weak gradients is a classical problem in the Sobolev space metric theory. However, if no regularity assumptions are enforced to the underlying space, the dependence of weak gradients on the integrable exponent is typically expected.

In this seminar, we propose a new strategy based on optimal transportation techniques to achieve a strong kind of independence of p-weak gradients on spaces satisfying certain Wasserstein interpolation $L^{\infty}$-estimates. This improves previously available results in settings relying on Doubling & Poincarè assumptions. We push then this analysis to deduce fundamental information concerning the Sobolev and BV calculus on metric measure spaces. This is based on joint works with N. Gigli and E. Pasqualetto, T. Schultz.

• March 28, 2022: Teemu Saksala (North Carolina State University)

Reconstruction of a manifold from travel time data

Abstract: In this talk I will introduce several geometric data sets related to the distance function either on Riemannian or Finsler manifolds with boundary. I consider differences of these data sets, and for each of them I will provide geometric conditions that are sufficient to determine the isometry class of the manifold producing the data. This talk is based on joint works with Maarten V. de Hoop, Joonas Ilmavirta, Matti Lassas and Ella Pavlechko.

• April 4, 2022: Tommaso Rossi (University of Bonn)

Almost-Riemannian manifolds do not satisfy the curvature-dimension condition

Abstract: The Lott-Sturm-Villani curvature-dimension condition CD(K, N) provides a synthetic notion for a metric-measure space to have curvature bounded from below by K and dimension bounded from above by N. We present a new strategy to disprove the CD condition for almost-Riemannian manifolds. This is a particular class of sub-Riemannian manifolds, which can be thought as non-complete Riemannian manifolds. In particular, we prove that 2-dimensional almost-Riemannian manifolds and strongly regular almost-Riemannian manifolds do not satisfy the CD(K, N) condition for any K in R and N>1. This is a joint work with M. Magnabosco.

• May 9, 2022 (starting at 15:00):

Emil Airta (University of Helsinki)

Multi-parameter commutators and two-weight estimates

Abstract: This talk is on commutator estimates in the multi-parameter framework. In particular, I am focusing on the two-weight boundedness theory referred to as Bloom estimates. The Bloom-type two-weight theory consists of two distinct weights and the third function in some weighted oscillation class, where the weight is a combination of the other two. Also, a unique feature is to assume that the weights are in a reasonable class. The topic has been very active since commutator theory by itself has some practical applications. Furthermore, the two-weight theory characterizes the weighted BMO spaces class. The presentation covers major parts of my doctoral dissertation that I will defend in a few weeks.

Tuomas Oikari (University of Helsinki)

Weak factorization and commutator lower bounds

Abstract: A commutator is formed by commuting a singular integral with a function (symbol) of pointwise multiplication. A classical result states that a commutator is bounded on L^p if and only if the symbol of the commutator belongs to the space BMO. This result has a dual formulation through a factorisation of the Hardy space H^1.  In this talk I will outline this duality and discuss a recent approximate weak factorisation (awf) argument that allows to assess half of this duality on both sides, namely, prove commutator lower bounds and a factorisation of H^1. The awf argument has proved itself useful e.g. in the multilinear and parabolic settings, in which I recall some recent results of my own.