Schedule for: 22w5080 - Smooth Functions on Rough Spaces and Fractals with Connections to Curvature Functional Inequalities

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We provide a comprehensive survey on Dirichlet forms on metric measure spaces. In particular, we discuss how to pass from mm-spaces to Dirichlet forms and vice versa, and under which conditions these transitions commute. Moreover, we outline the basic transformations of the respective data: measure change, metric change, time change, conformal change.

We will review some of the recent developments in the theory of Korevaar-Schoen-Sobolev spaces. While this theory is equivalent to that of Cheeger and Shanmugalingam if the space supports a Poincare inequality, it offers new perspectives in situations, like fractals, where such inequalities are not available.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

The conformal Assouad dimension is the infimum of all possible values of Assouad dimension after a quasisymmetric change of metric. We show that the conformal Assouad dimension equals a critical exponent associated to the combinatorial modulus for any compact doubling metric space. This generalizes a similar result obtained by Carrasco Piaggio for the Ahlfors regular conformal dimension to a larger family of spaces. We also show that the value of conformal Assouad dimension is unaffected if we replace quasisymmetry with power quasisymmetry in its definition.

It is an established result in the field of analysis of diffusion processes on fractals, that the transition density of the diffusion typically satisfies analogs of Gaussian bounds which involve a space-time scaling exponent $\beta$ greater than two and thereby are called SUB-Gaussian bounds. The exponent $\beta$, called the walk dimension of the diffusion, could be considered as representing ``how close the geometry of the fractal is to being smooth''. It has been observed by Kigami in [Math.\ Ann.\ \textbf{340} (2008), 781--804] that, in the case of the standard two-dimensional Sierpi\'{n}ski gasket, one can decrease this exponent to two (so that Gaussian bounds hold) by suitable changes of the metric and the measure while keeping the associated Dirichlet form (the quadratic energy functional) the same. Then it is natural to ask how general this phenomenon is for diffusions. This talk is aimed at presenting (partial) answers to this question. More specifically, the talk will present the following results: (1) For any symmetric diffusion on a metric measure space in which any bounded closed set is compact, the infimum over all possible values of the exponent $\beta$ after ``suitable'' changes of the metric and the measure is ALWAYS two unless it is infinite. (We call this infimum the conformal walk dimension of the diffusion.) (2) The infimum as in (1) above is NOT attained, in the case of the Brownian motion on the standard (two-dimensional) Sierpi\'{n}ski carpet, as well as on the standard three- and higher-dimensional Sierpi\'{n}ski gaskets. Some related open problems will also be discussed. For (1), it is not known whether the changes of the metric can be provided by geodesic metrics, or in other words, whether we can require the sub-Gaussian bounds to hold in the full off-diagonal regime. For (2), some (slight) new knowledge about local and global behavior of harmonic functions on the fractal is the key, and for better understanding of related phenomena it would be very important to analyze behavior of harmonic functions on Sierpi\'{n}ski carpets more deeply. This talk is based on joint works with Mathav Murugan (University of British Columbia). The results are given in https://link.springer.com/article/10.1007/s00222-022-01148-3 (Invent.\ math., in press), except for the non-attainment result for the Sierpi\'{n}ski carpet in (2) above, which is in progress.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk we review some Lipschitz-type results in connection with geometric properties and differentiable structures of metric measure spaces.

In the setting of nonsmooth spaces verifying synthetic lower Ricci curvature bounds (the so-called RCD metric measure spaces), a very refined differential calculus is available by now. By combining these calculus tools with the compactness and stability properties of the class of RCD spaces, it was possible to obtain several results on the isoperimetric problem that are new even in the case of non-compact Riemannian manifolds. Among other things, I will discuss the second-order differential behaviour of the isoperimetric profile, as well as some of its consequences, such as the existence of isoperimetric sets for large volumes and the sharp Lévy-Gromov isoperimetric inequality with the rigidity case.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

An opportunity to further discuss questions and problems.

An opportunity to further discuss questions and problems.

An opportunity to further discuss questions and problems.

An opportunity to further discuss questions and problems.

An opportunity to further discuss questions and problems.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

We construct symmetric self-similar Dirichlet forms on unconstrained Sierpinski carpets, which are natural extension of planar Sierpinski carpets by allowing the small cells to live off the 1/k grids. The intersection of two cells can be a line segment of irrational length, and we also drop the non-diagonal assumption in this recurrent setting. A uniqueness theorem is also provided. Moreover, the additional freedom of unconstrained Sierpinski carpets allows us to slide the cells around. In this situation, we view unconstrained Sierpinski carpets as moving fractals. We prove that the self-similar Dirichlet forms will vary continuously in a Γ-convergence sense, and the generated diffusion processes, viewed as processes in ℝ2, will converge in distribution. This is a joint work with Hua Qiu.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The counterpart of “Sobolev spaces” on metric spaces has been intensively studied for the last 20 years after the pioneering works by Cheeger, Hajlasz, and Shanmugalingam. The mainstream of the ideas is to use the local Lipschitz constant of a function as a suitable substitute for its gradient. However, a recent study by Kajino and Murugan on the conformal walk dimension revealed that the Dirichlet form associated with the Brownian motion on the Sierpinski carpet can not be a Sobolev space in this sense. In this talk, we will propose a new way of constructing “Sobolev spaces” on compact metric spaces including the Sierpinski carpet.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Dyadic cubes are ubiquitous in analysis in Euclidean spaces. First constructions preserving some of their key features in much more general spaces have been given by David and Christ. I have explored further elaborations in my works with Martikainen, Kairema, Auscher, and Tapiola; in particular, metric versions of random dyadic cubes (inspired by several works of Nazarov, Treil and Volberg on Euclidean spaces), the "1/3 trick" of adjacent/shifted dyadic cubes, and constructions of Hölder-regular "splines" and "wavelets" adapted to these dyadic structures.

It is widely known that the exit time of a diffusion process from a domain reflects geometric and spectral information of the domain. In this talk we consider a diffusion on a metric measure space equipped with a local regular Dirichlet form. With suitable assumptions such as volume doubling property and heat kernel sub-Gaussian upper bound we obtain estimates on the survival probability $\mathbb{P}(\tau_D>t)$ of the diffusion, where $\tau_D$ is its first exit time from domain $D$. The applications of this estimate include a uniform upper bound for the product $\lambda(D)\sup_{x\in D} \mathbb{E}_x(\tau_D)$. and a partial answer to a conjecture of Grigor'yan, Hu and Lau. These results apply to many examples in sub-Riemannian manifolds, fractals, as well as fractal-like manifolds. This is a joint work with Phanuel Mariano.

It is well known that scale invariant boundary Harnack inequality holds for Laplacian \Delta on uniform domains and holds for fractional Laplacians \Delta^s on any open sets. It has been an open problem whether the scale-invariant boundary Harnack inequality holds on bounded Lipschitz domains for Levy processes with Gaussian components such as the independent sum of a Brownian motion and an isotropic stable process (which corresponds to \Delta + \Delta^s). In this talk, I will present a necessary and sufficient condition for the scale-invariant boundary Harnack inequality to hold for a class of non-local operators on metric measure spaces. This result will then be a[[lied to give a sufficient geometric condition for the scale-invariant boundary Harnack inequality to hold for subordinate Brownian motion on bounded Lipschitz domains in Euclidean spaces. A counter-example will be given showing that the scale-invariant BHP may fail on some bounded Lipschitz domains with small Lipschitz constants. Joint work with Jie-Ming Wang.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

We present a general criterium for the density in energy of suitable subalgebras of Lipschitz functions in the p-metric-Sobolev space associated with a Polish metric-measure space. We then apply our result to the case of the algebra of cylindrical functions in the 2-Sobolev-Wasserstein space arising from a positive Borel measure on the 2-Kantorivich-Rubinstein-Wasserstein space of probability measures on the Euclidean space. We show that such a Sobolev space is always Hilbertian, independently of the choice of the reference measure and we briefly mention how the density result can be extended to more general Sobolev-Wasserstien spaces. This talk is based on a joint work with Massimo Fornasier (TU München, Germany) and Giuseppe Savaré (Bocconi University, Milano, Italy)

We provide a review of construction of p-energy and (1,p)-Sobolev space on the Sierpinski carpet when p is strictly greater than its Ahlfors regular conformal dimension. For p = 2, our 2-energy and (1,2)-Sobolev space correspond to the canonical DIrichlet form on the Sierpinski carpet given by Barlow--Bass and Kusuoka--Zhou. We will see that the condition related to the Ahlfors regular conformal dimension plays the role of ``strongly recurrence'', which implies very good regularity of functions in our Sobolev space

There are various dimension notions that are used to distinguish different fractals. Some depend on measures (e.g. Hausdorff and packing dimension) and others depend only on the metric of the space (e.g. box-counting and Assouad dimension). Even when considering all these notions, however, they might not be enough to distinguish or classify certain fractals. In such situations, it is useful to consider a collection of dimensions instead, known as a dimension spectrum. In this talk, we will present how the Assouad spectrum of a given set changes under quasiconformal maps and use this result to quasiconformally classify polynomial spirals, which would not be possible considering only the Hausdorff, box-counting and Assouad dimension notions. This talk is based on joint work with Jeremy Tyson.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

By substituting the modulus of gradient with the notion of upper gradient, one can ask if there is a universal inequality on metric spaces that mimics the classical coarea formula. It is reasonable to assume for example that the metric space is (locally) homeomorphic to $\bbbr^n$ and with locally finite Hausdorff-n measure. Under stronger further geometric assumptions on a metric space, the affirmative answer follows from a localization of the Eilenberg’s inequality. Only under the former assumptions, we prove for the case of $n=2$, such an equality for monotone Sobolev functions. I will also discuss counterexamples showing difficulties of generalizing further.

The Vicsek set is a tree-like fractal on which neither analog of curvature nor differential structure exists, whereas the heat kernel satisfies sub-Gaussian estimates. I will talk about Sobolev spaces and scale invariant $L^p$ Poincar\'e inequalities on the Vicsek set. Several approaches will be discussed, including the metric approach of Korevaar-Schoen and the approach by limit approximation of discrete p-energies.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.