In the fall we will award the best speaker with the traditional Truly Awesome Robust Honorary Award. The T.A.R.H.A. prize is awarded for the fourth time. The awarded person is chosen by a public vote.
If you are willing to give a talk or have any questions you can contact
Riku Anttila (riku.t.anttila[at]jyu.fi) or
Onni Hinkkanen (onni.u.i.hinkkanen[at]jyu.fi).
Abstract: For \(p \in (1,\infty)\), an infinite connected graph \(G\) is \(p\)-parabolic if the capacity from some vertex to infinity is zero, and otherwise it is \(p\)-hyperbolic. These notions generalize the well-known dichotomy between recurrence and transience of random walks in the sense that simple random walk in \(G\) is recurrent if and only if \(G\) is 2-parabolic. To further understand these notions, Yamasaki introduced and studied the notion of parabolic index defined as \[ p_*(G) := \inf \{ p \in (1,\infty) : G \text{ is \(p\)-parabolic} \}. \] In this talk I will give an elementary introduction to these topics. I will show that if \(G\) is \(Q\)-Ahlfors regular for \(Q \in (1,\infty)\), meaning \(|B(x,R)| \asymp r^Q\) for any vertex \(x\) and \(R \in [1,\infty)\), then \(G\) is \(Q\)-parabolic. In particular \(p_*(G) \leq Q\). I will briefly explain why this inequality is an equality for the lattice graphs \(\mathbb{Z}^n\), and why the equality fails for some fractal-like graphs. I also present a potential candidate for the value of the parabolic index in the fractal setting.
The talk is partially based on the following paper: https://projecteuclid.org/journals/hiroshima-mathematical-journal/volume-7/issue-1/Parabolic-and-hyperbolic-infinite-networks/10.32917/hmj/1206135953.full
Abstract: In this talk, I will discuss the volume of the boundary of a Sobolev \((p,q)\) extension domain. More specifically, I will show that the volume of the boundary of a Sobolev \((p,q)\) extension domain is zero when \( 1\leq q< p <\frac{nq}{n-q}\).
This is a joint work with Pekka Koskela.
Abstract: Let x be a real number with binary expansion x=0.01101... which is "less complex than usual", for example in the sense that the pattern "111" never appears. A difficult question in number theory asks whether it is possible for also the ternary expansion of the same number to be "less complex than usual". Maybe surprisingly, the answer is expected to be negative: "Simple" binary expansion should always lead to a "complex" ternary expansion and vice versa. Maybe even more surprisingly, this phenomenon has an equivalent formulation in the language of fractal geometry: The tails of either the binary or the ternary expansion of a real number should form a set of large fractal dimension. I will attempt to illustrate this connection between the two different fields of mathematics.
Abstract: We review a technique that can be used to prove Hölder or Lipschitz estimates for PDEs that may be in a non-divergence form. The method applies to fairly general equations, but in this talk we restrict to a special case for simplicity.
Abstract: In this talk I introduce a class of mean field stochastic differential equations with a special type of discontinuity in the measure component of the diffusion coefficient: when the probability that the unknown process is below some reference value gets too small or large, we switch to a different diffusion coefficient, that is, change the magnitude of the random movement. Since not every SDE possesses a nice distribution function, I show that under suitable assumption one can transform the original equation into a new SDE with deterministic coefficients, in which case the distribution is well known. I give sufficient conditions for the existence of a unique solution, and show that if we violate these assumptions, then we might lose both existence and uniqueness of a solution.
Abstract: It is well known that every \(u \in W^{1,p}(\mathbb{R}^n)\) has a representative \(\tilde{u}\) such that \[ \lim_{t \to \infty} \tilde{u}(t\xi) = 0, \] for \(\mathcal{H}^{n-1}\)-a.e. \( \xi \in \mathbb{S}^{n-1} \) for any \( 1 \leq p < \infty\), and, for \(p > n, u(x) \to 0\) uniformly as \( x \to \infty\). In this talk, we will discuss the pointwise behavior of functions in homogenous fractional Sobolev spaces, near infinity. Moreover, we will talk about those functions in the space for which there is no limit at infinity.
This is a joint work with Pekka Koskela and Kaushik Mohanto.
Abstract: In this presentation I will give an overview of continued fractions appearing in various analytic and geometric settings. Continued fractions are well known in number theory for their property of giving good rational approximations to a given real number. One could guess that this property allows continued fractions to show up in analysis and geometry. This guess is absolutely correct but the various answers, that do not include tons of technical details, are hard to come by. Hence I will showcase some neat and possibly less well-known results related to continued fractions in the following settings: hyperbolic geometry, quadratic operator equations, dynamical systems and complex analysis. Gauss will be mentioned several times in the presentation.
Abstract: In this talk I will introduce the notion of extended metric-topological measure spaces (e.m.t.m. spaces) and define a Sobolev space on them by using Cheeger energy. In this setting the topology does not need to be induced by metric (though there is some connection between them). This has some advantages, for example when the topology is coarser it is easier to check if your measure is Radon. One example of such setting is a dual Banach space with weak*-topology.
Abstract: Metric currents are a generalization of Euclidean currents to arbitrary metric spaces. Formally, a metric current is a multilinear functional on Lipschitz tuples that satisfies additional properties of continuity, locality and finite mass. These currents can also be viewed more geometrically, as a class of generalized surfaces.
The goal of this presentation is to give a non-technical introduction to the theory of metric currents. I will also state a decomposition result for acyclic normal currents in complete metric spaces.
Abstract: Fractal spaces arising as limits of Iterated Graph Systems (IGS) were introduced by Anttila and Eriksson-Bique, providing the first examples of self-similar, combinatorially Loewner spaces which are not quasisymmetric to a Loewner space. The attainment of Ahlfors regular conformal dimension is implied by the existence of a loewner metric in the conformal gauge of a metric space, and in a joint work with the aforementioned authors, we give an exact classification of this attainment problem among symmetric IGS-fractals.
In this talk, I will provide a brief overview of the construction of IGS-fractals and then discuss the construction of a specific class of quasisymmetric metrics on these fractals, which serves as a key tool in our classification.
Abstract: The description complexity of a finite structure — such as a binary string or a graph — in a given language L is the length of the shortest description in L that uniquely identifies the structure up to isomorphism. This talk will highlight the importance of this notion and review some recent progress on understanding it in the context of mathematical logic.
Abstract: We review Thomas Wolff's notes in 1999 on the connection between \(L^1\)-means of Fourier transforms of measures and \((s,1)\)-Furstenberg sets. Namely, for \(s\in[0,1]\), let \(\gamma(s)\) be the infimum of Hausdorff dimensions of \((s,1)\)-Furstenberg sets and let \(\sigma_1(s)\) be the supremum of numbers \(\sigma>0\) such that \[\int_{-\pi}^{\pi} |\hat{\mu}(Re^{i\theta)}| d\theta\lesssim_{\epsilon} R^{\epsilon-\sigma},\] where \(\mu\) is a positive Borel measure with compact support in \(B(1)\subset\mathbb R^2\) and \(\text{I}_s(\mu)=1\). Then we have the nice inequality \[1-s+\gamma(s)-4\sigma_1(\gamma(s))\geq 0.\]
Abstract: We will award the fourth edition of T.A.R.H.A. to the most popular speaker of the spring chosen by a public vote.
Antti Kykkänen
Janne Nurminen
Tapio Kurkinen
Timo Schultz
Ville Kivioja
Damian Dabrowski (Spring 2024)
Tapio Kurkinen (Autumn 2023)
Henri Hänninen (Spring 2023)
Fall 2022
Spring 2020
Autumn 2019
Autumn 2018
Autumn 2017
Spring 2014