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• February 8, 2021: Nate Fisher (Tufts University)

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Sub-Finsler metrics, random walks, and the Heisenberg group at infinity

Abstract: In this talk, we will discuss the real Heisenberg group and a class of sub-Finsler metrics which arise as the asymptotic cones of word metrics on the integer Heisenberg group. We will describe new results on the boundaries of these polygonal sub-Finsler metrics and explore applications to the study of random walks. Parts of this work are joint with Sebastiano Nicolussi Golo.

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• February 15, 2021: Henri Martikainen (University of Helsinki)

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Genuinely multilinear weighted estimates in product spaces​

Abstract: We prove genuinely multilinear weighted estimates for singular integrals in product spaces. The estimates complete the qualitative weighted theory in this setting. We motivate the theory by looking at unweighted applications and the key role of extrapolation in these. Examples include a free access to fractional Leibniz rules of Muscalu, Pipher, Tao and Thiele and their improvements.

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• March 22, 2021: Andrea Merlo (Université Paris-Saclay)

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Preiss rectifiability theorem in the first Heisenberg group

Abstract: In 1987 D. Preiss proved that if the upper and lower h-dimensional densities coincide almost everywhere for a Radon measure \phi, that measure is absolutely continuous with respect to the h-dimensional Hausdorff measure and it is supported on a countable union of Lipschitz graphs of dimension h.

In this talk I present a suitable extension of this rectifiability result to the first Heisenberg group, where the correct notion of rectifiability changes with the dimension.
 

• April 19, 2021: Ivan Violo (SISSA)

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Monotonicity formula for harmonic functions in RCD(0,N) spaces

Abstract: A classical result on Riemannian manifolds satisfying a lower bound on the Ricci curvature is the monotonicity of the Bishop-Gromov volume ratio. Colding and Minicozzi ('12-'14) realized that for non-negative Ricci curvature there exist analogous monotone quantities involving the Green function. Recently this has been generalized by Agostiniani, Fogagnolo and Mazzieri ('18) from the Green function to the case of an electrostatic potential and has proven to be fruitful in proving geometric inequalities. We will see that the same monotonicity formulas can be proven also in the setting of synthetic lower Ricci curvature bounds. This allows to prove some almost-rigidity results which are new also in the smooth case. This is a joint work with professor Nicola Gigli.

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• April 26, 2021: David Bate (University of Warwick)

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Characterising rectifiable metric spaces using tangent measures

Abstract: A classical result of Marstrand and Mattila states that a $S\subset \mathbb{R}^m$ (satisfying mild dimension assumptions) is n-rectifiable if and only if, for $\mathcal{H}^n$-a.e. $x\in S$, all tangent measures of $\mathcal{H}^n|_S$ at x are n-dimensional subspaces. This talk will present a generalisation of this result that replaces the ambient $\mathbb{R}^m$ with an arbitrary metric space.

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• May 3, 2021: Damian Dąbrowski (University of Jyväskylä)

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On measures with L^2 bounded Riesz transform: to AD regularity and beyond!

Abstract: The measures which define an L^2 bounded n-dimensional Riesz transform have been intensely studied in the last 50 years. Especially in the case of n-AD-regular measures (that is, measures that are "quantitatively n-dimensional") the situation is very well understood by now. In this talk I will describe how some of the recent advances in the field can be pushed beyond the AD-regular setting. Based on joint work with Xavier Tolsa.

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• May 10, 2021: Behnam Esmayli (University of Pittsburgh) 

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The Co-area Inequality

Abstract: The coarea inequality, a.k.a. Eilenberg's inequality, establishes an upper bound on the average size of the level sets of a Lipschitz map between metric spaces, in terms of the Lipschitz constant and the measure of the domain.The proof of this theorem is due to Federer but without any restrictions on spaces it was completed only after a difficult result of Davies. In this talk I will describe our new proof that avoids the result of Davies and only uses metric geometry techniques. I will then discuss our motivation for revisiting this fundamental result and the work that followed.

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• May 24, 2021: Raffaele Grande (Cardiff University)

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Convergence results for horizontal mean curvature flow and related stochastic controlled dynamics

Abstract: The horizontal mean curvature flow is widely used in neurogeometry (e.g. Citti-Sarti model for the visual cortex) and in computer science. It represents the evolution of a hypersurface embedded in a particular geometrical setting, called sub-Riemannian geometry, in which not all the curves are admissible by definition. This leads the existence of some points called characteristic points in which is not possible to define the horizontal normal. In order to avoid this problem, it is possible to use the notion of Riemannian approximation applied to the horizontal mean curvature flow.

In this talk I will explain the connection between the evolution of a generic hypersurface in this setting and the associated stochastic optimal control problem (as stated e.g. by Cardaileguet, Buckdahn and Quincampoix and  Soner and Touzi in the Riemannian case and Dirr, Dragoni and von Renessee in the sub-Riemannian setting). At the end, I will show some results which I have found in collaboration of N. Dirr and F. Dragoni about asymptotic optimal controls in the Heisenberg group (in the horizontal and the approximated case) and about the convergence of approximated solutions to the horizontal solutions.

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• May 31, 2021: Miguel Garcia Bravo (University of Jyväskylä)

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Abstract: The theory of Sobolev extension domains starts with the works of Calderón and Stein. In the past years there has been an increasing interest in understanding the geometric properties of those domains for which the extension of Sobolev functions to the whole Euclidean space is possible. The goal of this talk is to show the relation between W^{1,1}- extension domains and BV-extension domains. It will be shown that these are the same if some extension of sets of finite perimeter in a ‘’strong’’ sense is possible. In the case of bounded planar BV-extension domains it can also be proved that a rectifiablity condition on some part of their boundaries characterizes W^{1,1}-extension domains.

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• June 7, 2021: Toni Ikonen (University of Jyväskylä)

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Vector fields on metric measure spaces

Abstract: In the absence of a linear or a smooth structure on a general metric (measure) space, it is not clear how to control the infinitesimal behaviour of a given Lipschitz function f. A prime candidate is given by the so-called pointwise Lipschitz constant lip(f) measuring the pointwise metric behaviour of f.  A different approach, based on pathwise control, is given by the so-called minimal upper gradient |df| of f. In the smooth framework, the two elements |df| and lip(f) coincide at almost every point, and this fact was generalized to PI spaces by Cheeger in 1999.

 

During this talk, we sketch a proof of a recent result (by the speaker and coauthors) showing that for any metric measure space for which the inequality lip(f) <= C |df| holds for every Lipschitz f, with C > 0 independent of f, the inequality improves to an equality lip(f) = |df|. The proof is based on showing that the Weaver vector fields, introduced by Nik Weaver in 1999, and Sobolev vector fields, introduced by Nicola Gigli in 2015, coincide for spaces for which lip(f) <= C |df| holds.

 

The talk is based on a recent joint work with Enrico Pasqualetto and Elefterios Soultanis.

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• June 14, 2021: Keijo Mönkkönen (University of Jyväskylä)

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Travel time tomography on Finsler manifolds

Abstract: In travel time tomography one wants to determine the speed of sound inside the Earth by measuring travel times of seismic waves on the surface of the Earth. This is an inverse problem arising in seismology and it has been studied since the 1900s. One can geometrize the problem using Riemannian geometry: the Earth is a smooth compact manifold M with boundary and the speed of sound defines a Riemannian metric on M. In this way one obtains the boundary rigidity problem: determine the Riemannian metric from the distances between boundary points. However, when the speed of sound depends on the direction of propagation we need more general Finsler geometry. Finsler geometry is a generalization of Riemannian geometry where the fiberwise inner product depends on direction and the induced distance function is not necessarily symmetric.

 

In this talk I focus on special non-reversible Finsler norms called Randers metrics which consist of a Riemannian metric and a one-form. I show that under certain assumptions the distances between boundary points determine the Randers metric uniquely up to an almost isometry. Using Zermelo's navigation problem I give an application of the main theorem to seismology where the seismic wave propagates in a moving medium. In the end I discuss the linearization of the boundary rigidity problem and how it is related to the injectivity of the geodesic ray transform on Finsler manifolds.

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