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.
Location: MaD302

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 wellknown 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 wellposedness, 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 noncollapsed RCD(K,N) spaces
Abstract: In this seminar, we address the construction of parallel transport in the setting of noncollapsed 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 Ndimensional 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 pweak 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 pweak 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)
AlmostRiemannian manifolds do not satisfy the curvaturedimension condition
Abstract: The LottSturmVillani curvaturedimension condition CD(K, N) provides a synthetic notion for a metricmeasure 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 almostRiemannian manifolds. This is a particular class of subRiemannian manifolds, which can be thought as noncomplete Riemannian manifolds. In particular, we prove that 2dimensional almostRiemannian manifolds and strongly regular almostRiemannian 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)
Multiparameter commutators and twoweight estimates
Abstract: This talk is on commutator estimates in the multiparameter framework. In particular, I am focusing on the twoweight boundedness theory referred to as Bloom estimates. The Bloomtype twoweight 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 twoweight 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.