# Matematiska kollokviet

Matematiska kollokviet är en seminarieserie vid Matematiska institutionen som vänder sig till en bred matematisk publik. Alla intresserade är välkomna.

## Organisatörer

Kollokviet organiseras av Anders Björn, Milagros Izquierdo, Vladimir Kozlov och Hans Lundmark vid avdelningen Matematik och tillämpad matematik.

## Tid och lokal

Seminarietiden är vanligtvis varje onsdag kl 13:15-14:15 i Hopningspunkten som ligger i B-huset, ingång 23, plan 2, Campus Valla i Linköping.

## Kommande seminarier

### Fredag 14 december 2018, Antonio F. Costa, UNED, Madrid, Spanien

Seminariet arrangeras av Pedagogiska klubben på Matematiska institutionen i samarbete med Matematiska kollokviet.

Titel: Mathematics and E-learning

Tid och plats: Fredag 14 december, Kompakta rummet, 13.15-15.00

Sammanfattning: TBA

### Onsdag 30 januari 2019, Pavel Exner, Czech Academy of Sciences, Řež, Tjeckien, och Institut Mittag-Leffler, Djursholm

Titel: TBA

Tid och plats: Onsdag 30 januari 2019, Hopningspunkten, 13.15-14.15

Sammanfattning: TBA

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## Tidigare seminarier 2018

### Onsdag 12 december 2018, Juha Lehrbäck, Jyväskylä University, Finland

Titel: Assouad-type dimensions of inhomogeneous self-similar sets

Sammanfattning: There are several possible definitions of dimension for subsets of the Euclidean space (or a more general metric space). Assouad-type dimensions reflect, in a sense, the extreme local behavior of sets. While the upper and lower Assouad dimensions of a self-similar set always agree with its Hausdorff dimension, this no longer holds for inhomogeneous self-similar sets, which are obtained from a self-similar set E by adding to E a compact condensation set C and all the iterates of C under the iterated function system defining the self-similar set E. In addition, rather delicate separation conditions are needed in order to obtain nice formulas for the Assouad dimensions of inhomogeneous self-similar sets. After a general introduction recalling some of the basic definitions and facts from fractal geometry, I will discuss the above issues together with some examples and applications. The main results of this talk are based on my joint work with Antti Käenmäki.

### Onsdag 5 december 2018, Ugur Abdulla, Florida Institute of Technology, Melbourne, Florida, USA

Titel: The Wiener Criterion at $\infty$ for the Elliptic and Parabolic PDEs, and its Measure-Theoretical, Topological and Probabilistic Consequences

Sammanfattning: Norbert Wiener's celebrated result on the boundary regularity of harmonic functions is one of the most beautiful and delicate results in XX century mathematics. It has shaped the boundary regularity theory for elliptic and parabolic PDEs, and has become a central result in the development of potential theory. In this lecture I will describe my research developments which precisely characterize the regularity of the point at $\infty$ for second order elliptic and parabolic PDEs and broadly extend the role of the Wiener test in classical analysis. The Wiener test at $\infty$ arises as a global characterization of uniqueness in boundary value problems for arbitrary unbounded open sets. From a topological point of view, the Wiener test at $\infty$ arises as a thinness criteria at $\infty$ in ﬁne topology. In a probabilistic context, the Wiener test at $\infty$ characterizes asymptotic laws for the characteristic Markov processes whose generator is the given diﬀerential operator. The counterpart of the new Wiener test at the minimal Martin boundary point leads to uniqueness in the Dirichlet problem for a class of unbounded functions growing at a certain rate near the boundary point; a criteria for the removability of singularities, asymptotic laws for conditional Markov processes and for unique continuation at the ﬁnite boundary point.

### Onsdag 28 november 2018, Cyril Tintarev, Uppsala

Titel: Functional-analytic theory of defect of compactness

Sammanfattning: There are many important  embeddings of functional spaces that are not compact, but, instead, every bounded sequence has a subsequence with a well-structured defect of compactness (a difference between the sequence and its limit). The primary example is the Sobolev embeddings on Euclidean space. The structure of the defect of compactness is defined relatively to a group G of linear isometries on the space. If G is rich enough, then the defect of compactness is a countable sum of "elementary concentrations" of the form $g_kw$, $g_k\in G$, with the "blowup" sequences $g_k$  acting in a decoupled manner, $g_k^{-1}\tilde g_k\rightharpoonup 0$, which corresponds in applications to terms differently scaled or with asymptotically disjoint supports. In general, such structure exists if the embedding is co-compact relative to the group G - a non-trivial property similar to, but weaker than compactness, satisfied in particular, by embeddings of Besov and Triebel-Lizorkin spaces relative to the group of translations and dilations. Other examples include Strichartz embeddings, Moser-Trudinger-(-Yudovich-Peetre) embeddings, and embeddings on Sobolev type on Riemannian and sub-Riemannian manifolds. This functional-analytic approach generalizes the concentration-compactness method developed in the 1980's in the context of calculus of variations.

### Onsdag 14 november 2018, André Raspaud, Université de Bordeaux, Frankrike

Titel: Strong edge-coloring and star edge-coloring of graphs

Sammanfattning: A proper edge-coloring of a graph $G$ is a coloring of the edges of $G$ such that every two adjacent edges receive two distinct colors.

In this talk we will give a short survey of the following different close notions of edge-coloring of graphs.

• strong edge-coloring of a graph $G$ is a proper edge-coloring of $G$  such that every  two edges adjacent to a same edge receive two distinct colors.

The strong chromatic index of $G$, denoted by $\chi_s'(G)$, is the smallest integer $k$ such that $G$ admits a strong edge-coloring with $k$ colors.
• An acyclic edge-coloring is a proper edge-coloring of $G$ with the property that every cycle contains edges of at least three distinct colors.

The acyclic chromatic index of $G$, denoted by $\chi_a'(G)$, is the minimum number $k$ such that $G$ admits an acyclic edge-coloring with $k$ colors.
• star edge-coloring of a graph $G$ is a proper edge coloring such that every $2$-colored connected subgraph of $G$ is a path of length at most $3$.

The star chromatic index of $G$, denoted by $\chi_{st}'(G)$, is the minimum number of colors needed for a star edge coloring of $G$.

We have the following easy inequality:

$$\boxed{\chi_{a}'(G)\leq \chi_{st}'(G)\leq \chi_s'(G)}$$

We will also present in this talk our results concerning the strong edge coloring and the star edge-coloring.

### Onsdag 24 oktober 2018, Vladimir Kozlov, Matematiska institutionen, Linköpings universitet

Titel: Dynamical behaviour of SIR model with co-infection of two viruses

Sammanfattning: Co-infection with multiple strains  in a single host is very common.  Multiple viruses are widely studied because of their negative effect on the health of host as well as on whole population.  Many mathematical models have been developed and analyzed with multiple strains.  In this talk, we formulate a SIR model with co-infection and density dependence which represents a $4\times 4$ Lotka-Volterra system.

The global dynamics of the corresponding dynamical system  will be described and a special attention will be given to the dependence of the dynamics on system's parameters. Changing the parameters you can switch the system from one stable dynamics to another one.  In particular it will be shown that the dynamics becomes more and more complicated when the carrying capacity of population increases. This supports the enrichment paradox for this system.

The mathematical analysis of this system is based on an interplay between the theory of linear complementarity problem from optimization theory and a global stability analysis, which uses a generalized  Volterra function.

This is a joint work with Samia Ghersheen, Vladimir Tkachev and Uno Wennergren (Linköping University).

### Onsdag 17 oktober 2018, Sergey Vakulenko, St Petersburg, Ryssland

Titel: How evolution can create complex phenotypes?

Sammanfattning: We consider evolution of a population, where fitness of each organism is defined by many phenotypical traits. These traits result from expression of N genes, where N >>1. Well adapted organism should satisfy Nc >> N environmental constraints. The fitness is defined by a random Boolean circuits, for example, K-SAT model. The well known estimates (obtained first by E. Friedgut) show then that the probability to satisfy Nc constraints is exponentially small thus the evolution rate is exponentially small in N (the same fact follows from the classical Fisher geometric model and the Valiant approach).

We show that this fundamental obstacle can be overcome if the evolution goes, in certain sense, step by step and it is canalized in Waddington sense, i.e., in the end of each evolution rounds, the phenotype is stabilized with respect to mutations as a result of special gene regulation mechanisms. Moreover, we show that in such evolution process the number of mutations necessary for adaptation is sharply reduced. The most of mutations are neutral, and this neutralism increases during evolution.

These results are consistent with experimental data. They show that with a few number of genes one can obtain a complex organism, and that phenotypic stability is not an obstacle to evolution. These results also explain QTL data: evolution can involve genetic changes of relatively large effects and often the total number of changes are surprisingly small.

This is a joint work with John Reinitz, USA, Dmitry Grigoriev, France, Andreas Weber, Germany, Ovidiu Radulescu, France, and Dominik Michels, Saudi Arabia.

### Fredag 12 oktober 2018, Alexandre Karassev, Nipissing University, Kanada

Titel: Dimension and decomposition complexity

Sammanfattning: In attempts to capture asymptotic properties of finitely generated groups, manifolds, and general metric spaces, various dimension- like properties have been introduced recently, including asymptotic dimension, asymptotic dimension growth, asymptotic property C and asymptotic property D. We prove that if X is a tree-graded space (as introduced by C. Drutu and M. Sapir) and the family of all pieces of X satisfies one of the dimension-like properties, then X satisfies the same property, with explicit control over the parameters used in the property. In particular, the free product of finitely generated groups G*H satisfies a dimension-like property if the property holds for each group G and H. This is a joint with Nikolay Brodskiy

### Onsdag 10 oktober 2018, Evgeniy Lokharu, Lunds universitet

Titel: Three-dimensional steady water waves with vorticity

Sammanfattning: We will consider the nonlinear problem of steady gravity-driven waves on the free surface of a three-dimensional flow of an incompressible fluid. In the talk we will discuss a recent progress on three-dimensional waves with vorticity, which is a relatively new subject. The rotational nature of the flow is modeled by the assumption on the velocity field, that is proportional to its curl. Such vector fields are known in magnetohydrodynamics as Beltrami fields. We plan to give a necessary background on the topic and prove the existence of a three-dimensional doubly periodic waves with vorticity.

The talked is based on a joint work with Erik Wahlén and Douglas Svensson Seth from Lund University.

### Onsdag 3 oktober 2018, Håkan Lennerstad, Blekinge tekniska högskola, Karlskrona

Titel: Distance-consistent graph labelings, the ampleness of a graph, and graph functionals

Sammanfattning: A natural labeling of a simple connected graph $G=(V,E)$ is a labeling $c$ of the nodes with natural numbers $1,2,...,|V|$. Such a labeling induces a labeling distance $c(u,v)=|c(u)-c(v)|$ alongside the usual graph distance $d(u,v)$. A natural labeling that realizes the minimum $$l(G)=\min_{c}\sum_{u,v\in V}(c(u,v)-d(u,v))^2$$ is a distance-consistent labeling, and $l(G)$ is the ampleness of $G$. It trivial that $l(G)=0$ iff $G$ is a path graph, and I'll give the proof that $l(G)≤l(K_{n})$ for all $G$ with $n=|V$|. The normalized ampleness $L(G)=l(G)/(Kn), \, 0 \le L(G) \le1$ is studied for different graph classes such as the bipartite graph $K_{n,n}$, the star graph $S_{n}$, the cycle graph $C_{n}$ and a few other types, particularly for $n \to \infty$.
The quantity $$\min_{c}\sum_{u,v\in V}(c(u,v)-d(u,v))^2$$ is a graph functional; mapping graphs to non-negative integers. It can be thought of as the "inverse listness" of a graph - being zero for lists only (path graphs). The quantity $c(u,v)$ can be replaced by other quantities defining the "inverse cycleness" or "inverse starness" of any graph, in which case the corresponding functional is zero if an only if the graph is $C_{n}$ or $S_{n}$, respectively.

### Onsdag 26 september 2018, Johan Öinert, Blekinge tekniska högskola, Karlskrona

Titel: Epsilon-strongly group graded rings, Leavitt path algebras and crossed products by twisted partial actions

Sammanfattning: Epsilon-strongly group graded rings constitute a class of rings which contains all strongly group graded rings and all crossed products associated with unital twisted partial group actions. A result of Năstăsescu, Van den Bergh and Van Oystaeyen (1989) gives a characterization of strongly group graded rings which are separable over their canonical 'degree zero' subrings. A more recent result of Bagio, Lazzarin and Paques (2010) gives a characterization of crossed products, associated with unital twisted partial group actions, which are separable over their coefficient subrings. We are able to simultaneously generalize both of these results by giving a characterization of separable epsilon-strongly group graded rings. We also provide examples of separable epsilon-strongly group graded rings (not strongly graded!) and thereby answer a question of Le Bruyn, Van den Bergh and Van Oystaeyen (1988).

Given an arbitrary group G, we will explain how to equip any Leavitt path algebra over a finite (directed) graph with an epsilon-strong G-gradation.

This talk is based on recent joint work with Patrik Nystedt (University West, Sweden) and Héctor Pinedo (Industrial University of Santander, Colombia).

### Fredag 21 september 2018, Lashi Bandara, University of Potsdam, Tyskland

Titel: When functional calculus, harmonic analysis, and geometry party together ...

Sammanfattning: Functional calculus emerged in the latter half of last century as a convenient tool particularly in the analysis of partial differential equations. In the last thirty years, harmonic analysis has entered the picture to interact with functional calculus in an extraordinarily fruitful way. More recently, geometry has crashed the scene, with an abundance of interesting and important problems, which can be effectively dealt with using the tools coming from functional calculus and harmonic analysis. Moreover, there are fascinating geometric interpretations associated with the latter tools, although these investigations are still in their infancy.

The goal of this talk will be to flesh out a brief narrative of the journey of functional calculus, how it came to interact with harmonic analysis, and the party they've been recently having together with geometry. It will culminate with state-of-the-art results, but the beginnings will be humble, starting with the Fourier series! For the majority of the talk, no background will be assumed beyond Hilbert spaces, self-adjoint operators, and the spectrum of an operator.

### Onsdag 12 september 2018, Nikolay Kuznetsov, Russian Academy of Sciences, St. Petersburg, Ryssland

Titel: Studies of water waves: An early history and some up-to-date developments

Sammanfattning: Main results from 1687 (Newton's Principia) to 1895 (Lamb's Hydrodynamics) will be outlined with emphasis on the work of Scott Russell on solitary waves and of Stokes on the waves named after him. The influence of the early work on achievements of the last four decades will be traced.

### Onsdag 5 september 2018, Tilman Bauer, KTH

Titel: Realizing Cohomology

Sammanfattning: To compute the cohomology of a given space is generally considered a tractable problem. The inverse problem, namely to construct a space with prescribed cohomology, is a much subtler problem. For the case of cohomology with coefficients modulo an (even) prime, I will discuss the classical problem posed by Steenrod in 1960 of which polynomial rings can be cohomology rings of spaces, starting from obstructions imposed by Adams’s solution to the Hopf invariant one problem to the discovery and classification of p-compact groups by Dwyer-Wilkerson, Andersen-Grodal, and others. I will then address the problem of realizing certain very small cohomology rings by combining exotic p-compact groups with exotic new cohomology theories. The last part is joint work in progress with A. Baker.

### Onsdag 13 juni 2018, German Zavorokhin, Steklov Math. Institute, St. Petersburg, Ryssland

Titel: Pressure drop matrix for a bifurcation of an artery with defects

Sammanfattning: We consider a bifurcation of an artery. The influence of defects of the vessel's wall near the bifurcation point on the pressure drop matrix is analyzed. The elements of this matrix are included in the modified Kirchhoff transmission conditions, which were introduced earlier in the works of V. Kozlov and S. Nazarov, and which give better approximation of 3D flow by 1D flow near the bifurcation point in comparison with the classical Kirchhoff conditions.
This is joint work with V. Kozlov and S. Nazarov.

### Tisdag 5 juni 2018, Vlad Bally, Université Paris-Est Marne-la-Vallée, Frankrike

Seminariet är ett samarrangemang med Seminarier i Matematisk statistik.

Titel: Asymptotic integration by parts formula and regularity of probability laws

Sammanfattning: We consider a sequence of random variables $F_{n}\sim p_{n}(x)dx$ which converge to a random variable $F.$ If we know that $p_{n}\rightarrow p$ in some sweated sense, then we obtain $F\sim p(x)dx.$ But in many interesting situations $p_{n}$ blows up as $n\rightarrow \infty .$ Our aim is to give a criterion which says that, if there is a "good equilibrium" between $\left\Vert F-F_{n}\right\Vert _{1}\rightarrow 0$ and $\left\Vert p_{n}\right\Vert \uparrow \infty$ then we are still able to obtain the absolute continuity of the law of $F$ and to study the regularity of the density $p.$ Moreover we get some upper bounds for $p.$ The blow up of $p_{n}$ is characterized in terms of integration by parts formulae.

We give two examples. The first one is about diffusion processes with Hölder coefficients. The second one concerns the solution $f_{t}(dv)$ of the two dimensional homogeneous Boltzmann equation. We prove that, under some conditions on the parameters of the equation, we have $f_{t}(dv)=f_{t}(v)dv.$ The initial distribution $f_{0}(dv)$ is a general measure (except a Dirac mass) so our result says that a regularization effect is at work; moreover, if the initial distribution has exponential moments $\int e^{\left\vert v\right\vert ^{\lambda }}f_{0}(dv)<\infty ,$ then we prove that $f_{t}(v)\leq Ct^{-\eta }e^{-\left\vert v\right\vert ^{\lambda ^{\prime }}}$ for every $\lambda ^{\prime }<\lambda .$ So we have exponential upper bounds in space and at most polynomial blow up in time.

### Onsdag 30 maj 2018, Nathan Reading, North Carolina State University, USA

Titel:  To scatter or to cluster?

Sammanfattning: Scattering diagrams arose in the algebraic-geometric theory of mirror symmetry.  Recently, Gross, Hacking, Keel, and Kontsevich applied scattering diagrams to prove many longstanding conjectures about cluster algebras.  Scattering diagrams are certain collections of codimension-1 cones, each weighted with a formal power series.  In this talk, I will introduce cluster scattering diagrams and cluster algebras, and the relationship between them, focusing on rank-2 (i.e. 2-dimensional) examples.  Even 2-dimensional cluster scattering diagrams are not well-understood in general.  I will show how the two-dimensional "affine-type" cases can be constructed using cluster algebras and describe a surprising appearance of the Narayana numbers in the two-dimensional affine case.

### Onsdag 23 maj 2018, Xining Li, Sun Yat-Sen University, Guangzhou, Kina

Titel:  Characterization of Hp spaces in quasiconformal mappings

### Onsdag 16 maj 2018, Nageswari Shanmugalingam, MAI och University of Cincinnati, USA

Titel:  Geometric and analytic aspects of infinity-Poincaré inequalities

Sammanfattning: The study of absolute minimizing Lipschitz extensions and infinity-harmonic functions in the Euclidean setting was initiated by Aronsson, Crandall and Evans, and is of great interest now, with optimal regularity of solutions yet open. In the metric setting, and indeed even in the weighted Euclidean setting, studies of such solutions are possible under certain conditions on the metric space. One condition is the existence of infinity-Poincaré inequality. In this talk we will discuss this inequality, and a geometric and analytic characterizations of this inequality.

### Onsdag 2 maj 2018, Armen Asratian, MAI

Titel: A localization method in Hamiltonian graph theory

Sammanfattning: A finite graph G is called Hamiltonian if it has a cycle containing every vertex of G. Almost all of the existing sufficient conditions for a finite graph G to be Hamiltonian contain some global parameters of G (such as the number of vertices) and only apply to graphs with large edge density  and/or small diameter.

In a series of papers we have shown that some classical sufficient conditions for Hamiltonicity of graphs that contain global parameters  can be reformulated in such a way that every global parameter in those conditions is replaced by a parameter of a ball with small radius. Such results are called localization theorems and give a possibility to find new classes of Hamiltonian graphs with large diameter and small edge density.

I shall give a review of this topic and present some new results obtained with J. Granholm and N. Khachatryan. In particular,  we formulate a general method for finding localization theorems and apply this method for formulating local analogues of four well-known criteria for Hamiltonicity of finite graphs. Finally we extend some of our results to infinite locally finite graphs.

### Onsdag 18 april 2018, Sergey Nazarov, MAI och St Petersburg, Ryssland

Titel: Sharpening and smoothing near-threshold Wood anomalies in cylindrical waveguides

Sammanfattning: Gently sloped perturbation of the wall of an acoustic or elastic waveguide can lead to Wood’s anomalies which realizes as disproportionately rapid changes of the diffraction pattern near thresholds of the continuous spectrum. By means of an asymptotic analysis certain restrictions on the profile of the wall perturbations are found that provide the appearance of the anomaly, its sharpening or extinction. Secveral ways are found out to avoid the anomaly, namely either to keep the threshold resonance which itself provokes the anomaly, or to provide an embedded eigenvalue, both require a fine tuning of the profile of the perturbed wall. At the same time, violation of the fine tuning procedure usually leads to the anomaly.

### Onsdag 11 april 2018, Lucia Lopez de Medrano, UNAM, Mexico City, Mexiko, och Institut Mittag-Leffler

Titel: Tropical Geometry

Sammanfattning: In this talk we will review basic aspects of tropical geometry and discuss some of its applications in classical algebraic geometry.

### Måndag 9 april 2018, Agnieszka Kałamajaska, University of Warsaw, Polen

Titel: Dirichlet's problem for critical Hamilton-Jacobi fractional equation

Sammanfattning: Using  an extended approach of Dan Henry, we study solvability of the Dirichlet problem on a bounded smooth  domain for the Hamilton-Jacobi equation with critical nonlinearity posed in Sobolev spaces:
$\begin{cases} u_t + (-\Delta)^{1/2}u + H(u, \nabla u) =0, &t>0, x\in \Omega,\\ u(t,x) =0, &t>0, x \in \partial\Omega,\\ u(0,x)=u_0, &x\in \Omega. \end{cases}$
We will also discuss the additional regularity and uniqueness of the limiting weak solution.  The talk will be based on joint work with Tomasz Dlotko.

### Onsdag 4 april 2018, Panu Lahti, Jyväskylä University, Finland

Titel:  A new approximation of BV functions on metric spaces

Sammanfattning: I will discuss a new way of approximating BV functions in the so-called strict sense, and pointwise uniformly, by SBV functions, which are BV functions whose variation measure has no Cantor part. This is based on a careful analysis of capacities. I will consider this in the setting of metric spaces with a doubling measure and Poincaré inequality but the result may be new even in Euclidean spaces. Lastly I will discuss possible applications for variational problems.

### Tisdag 27 mars 2018, Lars-Erik Persson, Luleå tekniska universitet

Sammanfattning: First I describe shortly the dramatic around 10 years period until G.H. Hardy formulated and proved his famous inequality in 1925. After that I describe some selected steps in what today is referred to as Hardy-type inequalities (see e.g. the book [1] and references therein). Finally, I turn to shortly describe some remarkable examples of developments mostly from the really last years. See e.g. Chapter 7 of the second edition of our book [1] and also my Lecture Notes [2] from P.L. Lions seminar. In particular, some open questions are presented .
[1] A.Kufner, L.E. Persson and N. Samko, Weighted Inequalities of Hardy type, World Scientific, Second edition, New Jersey-London-etc., 2017.
[2] L.E. Persson, Lecture Notes, Collège de France, Pierre-Louis Lions Seminar, November 2015.

### Onsdag 21 mars 2018, Dag Nilsson, Lunds universitet

Titel:  Existence of solitary waves: A spatial dynamics approach

Sammanfattning: In 1982 Kirchgässner studied a class of semilinear elliptic boundary value problems in an infinite strip. By treating the unbounded coordinate $x$ as time he formulated his problem as a dynamical system of the form $$\label{system} u_x=Lu+F(u),$$ where $L$ is a linear operator and $F=\mathcal{O}(|u|^2)$. Using methods from dynamical systems theory he was then able to prove existence of solutions of the system above. Today this procedure is called spatial dynamics. In 1988 Kirchgässner applied this technique to the two dimensional irrotational water wave problem and was able to prove existence of solitary wave solutions in the presence of strong surface tension. This work was later expanded upon by other researchers who considered different parameter regimes, for example weak surface tension. The method was extended to the three dimensional setting by Groves and Mielke, under the extra assumption that the waves are periodic in one spatial direction.

In my talk I will present results from three of my papers where the method of spatial dynamics is used to prove existence of solitary waves for three different physical situations: two-dimensional internal waves, waves on a cylindrical ferrofluid jet and three dimensional internal waves. In particular I will compare my findings with known results for surface waves. Parts of the talk are based on a collaboration with Mark Groves of Saarland University.

### Onsdag 14 mars 2018, Lukáš Malý, Chalmers och Göteborgs universitet

Titel:  Self-improvement of generalized Poincaré inequalities

Sammanfattning:  Many parts of the theory of first-order analysis in metric spaces rely on various types of Poincaré inequalities (PI), which are indispensable ingredients of Sobolev-type and Morrey-type embeddings of Sobolev functions. It was proven by Keith and Zhong that a $p$-Poincaré inequality is an open-ended condition. Specifically, if a complete metric space endowed with a doubling measure admits a $p$-Poincaré inequality with $p>1$, then the metric space admits a $q$-Poincaré inequality for some $q<p$. A $p$-Poincaré inequality need not be the most natural choice when constructing a refined theory of Sobolev-type spaces, where the gradients lie in an Orlicz or a Lorentz space. For instance, Tuominen applied Orlicz-type PI and Costea–Miranda applied a Lorentz-type PI to study the respective Sobolev spaces.
In my talk, I will discuss self-improvement of such more general Poincaré inequalities. I will also provide an elementary proof that Orlicz-type Poincaré inequalities are, in fact, $p$-Poincaré inequalities in disguise and undergo self-improvement by the original result of Keith and Zhong. The method serves also as an alternative proof for relating Orlicz-type Muckenhoupt weights to the standard $A_p$ weights. The situation for Lorentz-type Poincaré inequalities is however more delicate and one can construct a fairly simple metric space where the self-improvement result fails. In particular, a Lorentz-type Poincaré inequality need not be an open-ended condition.

### Onsdag 7 mars 2018, Andreas Sykora, München, Tyskland

Titel:  Fuzzy surfaces from graphs and embedding functions

### Onsdag 28 februari 2018, Rebekah Jones, University of Cincinnati, USA

Titel:  Dimension distortion of sets of finite perimeter under a quasisymmetric map in a metric space

Sammanfattning: One characterization of quasiconformal maps is that they quasi-preserve the modulus of curves, i.e. there exists $C>0$ such that for any collection of curves $\Gamma \subset \mathbb{R}^n$, $C^{-1} \text{Mod}_n(\Gamma) \le \text{Mod}_n(f(\Gamma)) \le C \text{Mod}_n (\Gamma)$. In 1973, Kelly showed that also the $\frac{n}{n-1}$-modulus of surfaces is quasi-preserved. In particular, this implies that almost every surface does not increase in dimension under such a map. We show that, under the appropriate geometric assumptions, such a result is valid in a metric space. This talk is based on joint work with Panu Lahti and Nageswari Shanmugalingam.

### Onsdag 21 februari 2018, Stefan Rauch, MAI

Titel: Understanding reversals of a rattleback

Sammanfattning: The rattleback is a rigid body having a boat-like shape (modelled as the bottom half of an 3-axial ellipsoid) having asymmetric (chiral) distribution of mass. When the rattleback is spun on its bottom in the “wrong” direction then it starts to rattle, it slows down and acquires rotation in the opposite, preferred sense of direction. This behavior defies our intuition about conservation of angular momentum as the force and the torque responsible for changing the angular momentum (and the direction of spinning) are not obvious.

The overwhelming majority of papers on the rattleback´s motion study the dependence of stability for spinning solutions on the sense of rotation, on the shape of the rattleback´s surface and on the distribution of mass. There has been no available simple, intuitive explanation of the rattleback´s behavior in terms of physical forces and torques.

In a joint paper with M. Przybylska, just published in Regular and Chaotic Dynamics (a journal of Steklov Mathematical Institute), we explain the motion of a toy rattleback by using frictionless Newton equations of motion for a rigid body rolling without sliding in a plane. It is the reaction force of the supporting surface that is the source of the torque turning the rattleback in the preferred sense of rotation.

The picture is, however, more subtle as it appears that the direction of the torque depends on the initial conditions and a frictionless, low energy rattleback admits reversals in both directions(!).

I will discuss how the rattleback´s motion depends on initial conditions and how it agrees with results of simulations of the rattleback´s equations for tapping and spinning initial conditions. Simulations show also that the long time behavior of such a rattleback is, for low energy initial conditions, quasi-periodic and there are infinitely many reversals in both directions.

### Onsdag 14 februari 2018, Maria Przybylska, Zielona Góra, Polen

Titel: Integrability properties of certain generalisations of non-holonomic Suslov problem

Sammanfattning: One of the basic examples of nonholonomic mechanics is the Suslov system. Two its generalizations will be presented. The first one is based on the classical heavy gyrostat. Its equations of motion are restricted by the non-holonomic Suslov constraint: the projection of the angular velocity of the body onto a vector constant in the body frame vanishes. Integrability of the obtained system is analysed. It appears that certain integrable cases of the Suslov problem have their integrable generalisation. Additionally it is proved that for a wide range of parameters of the problem, the system is not integrable in the Jacobi sense.

The second model is a Lie-Poisson system on six-dimensional class A co-algebras generated by a quadratic Hamiltonian and restricted by a nonholonomic constraint which is a generalisation of the Suslov constraint. We obtain counterparts of classical integrable cases. Moreover, in the case without a potential conditions of meromorphicity of solutions lead to cases with additional polynomial first integrals. They are constructed by means of solutions of a third order linear differential equations that in two cases appear to be generalised hypergeometric equations defining 3F2 hypergeometric function. It appears that for all class A co-algebras there exists a generalised version the Kozlov case when the system is described by a natural Hamiltonian with two degrees of freedom. It is shown that this system is not integrable except in one case.

### Fredag 9 februari 2018, Yakov Krasnov, Bar-Ilan University, Ramat Gan, Israel

Titel:  Methods of nonassociative algebras in differential equations

Sammanfattning: Many well-known (classes of) differential equations  may be viewed as a Riccati type equation in a certain commutative nonassociative algebra. We develop further the principal idea of L. Markus for deriving algebraic properties of solutions to ODEs and PDEs directly from the equations defining them.
Our main purpose is (a) to show how the algebraic formalism can be applied with great success to a remarkably elegant description of the geometry of curves being solutions to homogeneous polynomial ODEs, and, on the other hand, (b) to motivate the recent interest in applications of nonassociative algebra methods to PDEs. More precisely, given a differential equation on an algebra A, we are interested in the following two problems:
1. Which properties of the differential equation determine certain algebraic structures on $A$ such as to be associative, unital or division algebra.
2. In the converse direction, which properties of $A$ imply certain qualitative information about the differential equation, for example topological equivalent classes, existence of a bounded, periodic, ray solutions, ellipticity etc.
We also define and discuss syzygies between Peirce numbers which provide an effective tool for our study. (Some results here are based on a recent joint work with V. Tkachev.)

### Onsdag 7 februari 2018, Salvador Rodríguez-López, Stockholms universitet

Titel:  Regularity properties for solutions of hyperbolic equations in some Function spaces

Sammanfattning: Fourier analysis methods play an important role in the study of some linear and nonlinear PDEs. In this talk, we will first give a general overview of some tools of Fourier analysis, such as Littlewood-Paley decomposition, some associated function spaces, namely Besov and Triebel-Lizorkin spaces, and certain operators such as pseudo-differential and Fourier integral operators. We will also briefly discuss some recent results on the regularity for the solutions to linear hyperbolic partial differential equations, which encompasses the wave equation. More precisely, since the solution of these equations can be written as a linear combination of the so-called Fourier integral operators, the regularity is established by obtaining some boundedness properties of these. Specifically, we will present an extension of the result of A. Seeger, C.D. Sogge and E. M. Stein on L^p spaces to the scale of Besov and Triebel-Lizorkin spaces. We will finish the talk by pointing out some ongoing research and open problems.

### Onsdag 31 januari 2018, Anders Björn, MAI

Titel:  Some parts from the history of analysis in the 19th century

Sammanfattning: While preparing for the Real Analysis, honours course, (Analys överkurs) during the autumn I tried to find out some details about who invented what and when. Some of these facts doesn't seem to be so well known, but may be of interest to others. I will discuss things like who introduced and proved uniform continuity, and who first showed that continuous functions can be integrated. I will also discuss how the Riemann zeta function, and esp. the Riemann hypothesis, influences estimates for $\pi(x)=$ the number of primes $\le x$, and the history around these results.

### Onsdag 24 januari 2018, Veronica Crispin Quiñonez, Uppsala

Titel: Hilbert series of quadratic forms in the exterior algebra

Sammanfattning (PDF)

### Onsdag 17 januari 2018, Sylvester Eriksson-Bique, UCLA, Los Angeles, USA

Titel: Poincaré inequalities and notions of connectivity

Sammanfattning: What does it mean for a space to be well-connected, and how can one quantify that? In this talk I will discuss a few notions of connectivity, and how they relate to Poincaré inequalities. This initial discussion revolves around the seemingly innocent question: If the gradient of a function is small, can I conclude that the function is almost constant? This can be made effective in several ways, leading to various inequalities, some of which are classical Poincaré inequalities, and one of the is a new quantitative notion of connectivity. These inequalities have appeared in various contexts and are related to many applications. Interestingly, all of these turn out to be equivalent, as long as equivalent is properly interpreted, to the classical notion of a Poincaré inequality. With time, I might discuss briefly some recent applications to non-self-similar carpets, self-similar spaces and self-improvement phenomena.

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