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Matematiska Kollokviets Tidigare Seminarier LiU - MAI - Seminarier - Matematiska kollokviet - Arkiv för Matematiska kollokviet - 2017 Organized by Anders Björn, Milagros Izquierdo Barrios, Vladimir Kozlov, and Hans Lundmark. Onsdag 11 januari 2017, Visa Latvala, University of Eastern Finland, Joensuu, Finland Weak Harnack estimates for quasisub- and quasisuperminimizers with non-standard growth Sammanfattning: Weak Harnack estimates can be regarded as generalizations of submeanvalue and supermeanvalue properties of classical subharmonic and superharmonic functions. These estimates are important tools in the nonlinear potential theory. For the -Laplace type equations, weak Harnack estimates can be obtained either by Moser's iteration method or by De Giorgi's method. DiBenedetto and Trudinger extended the latter method in 1980's to a very general class of quasisub- and quasisuperminimizers. The purpose of the talk is to give an overview on results related to quasiminimizers of the variational integrals with non-standard growth. We focus on the variable exponent growth but also comment on the recent development related to generalized Orlicz spaces. Onsdag 18 januari 2017, Alexander I. Nazarov, Saint Petersburg, Ryssland The influence of geometry of a manifold on attainability of the norm of the Sobolev trace embedding operator Sammanfattning: Let be a smooth compact Riemannian -dimensional manifold with smooth boundary, and let . We consider the trace Sobolev embedding (here ). Similarly to the case of conventional Sobolev embedding (see the survey [3] and references therein), the attainability of the infimum in (1) heavily depends on the geometry of . This talk is partially based on the following joint papers with Alexander Reznikov. 1. A.I. Nazarov, A.B. Reznikov, On the existence of an extremal function in critical Sobolev trace embedding theorem, J. Funct. Anal. 258 (2010), N11, 3906--3921. 2. A.I. Nazarov, A.B. Reznikov, Attainability of infima in the critical Sobolev trace embedding theorem on manifolds, AMS Transl. Series 229 (2010), 197--210. 3. A.I. Nazarov, Dirichlet and Neumann problems to critical Emden-Fowler type equations, J. Global Optim. 40 (2008), 289--303. Onsdag 25 januari 2017, Adrian Muntean, Karlstads universitet Reaction-diffusion systems with distributed microstructures: well-posedness and homogenization asymptotics Sammanfattning: The discussion will address the multiscale modeling, analysis and approximation of a class of reaction-diffusion systems posed simultaneously on both macroscopic and microscopic space scales. The coupling between the scales is done via micro-macro transmission conditions. Our target system has a typical structure for reaction-diffusion-flow problems in media with distributed microstructures (also called, double porosity materials), reminding the work of G. Barenblatt on flows through fractured media. The talk will focus on an elementary derivation of the two-scale structure based on formal homogenization asymptotics. Then I will give a hint on how can one handle the weak solvability of the system and will point out a route to justify the homogenization asymptotics rigorously. A numerical example will illustrate how the weak solution "communicates" between the space scales. Onsdag 1 februari 2017, Clas Rydergren, ITN, Norrköping New sources of input data for travel demand estimation models Sammanfattning: Forecasts of how journeys are made, from where and to where, is usually done using simple mathematical models. The results from the models are used as estimates of the load on the transport system. Estimates are made for the load today, in the future, and to estimate the change in travel demand when changes in the infrastructure are made. The model result is critical input to traffic planners. Different types of models are used depending on, among other things, if the analysis requires the results to include all modes of transport, or not, and the length of the forecast horizon. Models for long-term forecasts (several years) often contain components to describe the travelers' values and perceptions whereas models for the current situation or with a very short forecast horizon, often is based exclusively a network model and data from dedicated traffic measurements. During this seminar, I will present an example of a traditional demand model, and exemplify how this model is used. The ongoing digitization have led to new sources of input data for this type of models. I will exemplify this by presenting inputs used in a couple of research projects ongoing at the Division of Communications and Transport System (KTS) at ITN. Onsdag 8 februari 2017, Viktor Kolyada, Karlstads universitet On Gagliardo-Nirenberg type inequalities Sammanfattning: We prove a Gagliardo-Nirenberg type multiplicative estimates for Lorentz norms of a function. These estimates are expressed in terms of norms of derivatives of a function and its homogeneous Besov norms of negative order. One of our main results is a refinement of the well-known strong version of the Sobolev embedding theorem involving Lorentz norms. Our methods are based on estimates of nonincreasing rearrangements of functions in terms of heat kernels. These methods enable us to cover also the case of Sobolev norms with . This is joint work with F.J. Pérez Lázaro (Universidad de La Rioja, Spain). Onsdag 15 februari 2017, Maria Przybylska, Zielona Góra, Polen Analytical method of spectra calculations for quantum optics systems in the Bargmann representation Sammanfattning: The fundamental problem of quantum mechanics is solving the eigenvalue problem for a given Hamilton operator, i.e. determination of eigenfunction called a wave function and corresponding eigenvalue called the energy spectrum. We show that for a certain class of quantum mechanics problems one can use the so-called Bargmann representation which allows one to rewrite the eigenvalue equation as a system of linear equations with one independent complex variable. Using this representation we distinguish three types of conditions determining the spectrum: local conditions around each singularity, compatibility condition joining local solutions around different singularities and normalization condition related to the proper growth order of solutions. A few examples of quantum optics systems describing the interaction of one mode of electromagnetic radiation with two-level atom will be considered and obstructions on spectrum given by these conditions will be presented. In some cases, one can find closed form expression on spectrum formulated by means of transcendental functions of parameters of the systems such as confluent Heun functions or generalised Heun functions. Onsdag 1 mars 2017, Milagros Izquierdo, MAI On the Connectivity of Branch Loci of Spaces of Curves Sammanfattning: Since the 19th century the theory of Riemann surfaces has a central place in mathematics putting together complex analysis, algebraic and hyperbolic geometry, group theory and combinatorial methods. Since Riemann, Klein and Poincaré among others, we know that a compact Riemann surface is a complex curve, and also the quotient of the hyperbolic plane by a Fuchsian group. In this talk we study the connectivity of the moduli spaces of Riemann surfaces (i.e in spaces of Fuchsian groups). Spaces of Fuchsian groups are orbifolds where the singular locus is formed by Riemann surfaces with automorphisms: the branch loci: With a few exceptions the branch loci is disconnected and consists of several connected components. This talk is a survey of the different methods and topics playing together in the theory of Riemann surfaces. Onsdag 8 mars 2017, Natan Kruglyak, MAI Theory of Interpolation (review) Sammanfattning: A year ago I gave a talk during which I have discussed what was done in interpolation theory before 1980. Now I plan to remind (shortly) what was discussed last year and will focus on some results which were obtained after 1980. Onsdag 15 mars 2017, Håkan Hedenmalm, KTH Bloch functions, asymptotic variance, and geometric zero packing Sammanfattning: In connection with the study of the universal integral means spectrum for quasiconformal mapping, it turns out that the main term for small exponents and small Beltrami coefficients is governed by the asymptotic variance introduced by McMullen for a dynamical situation. This follows from work of Oleg Ivrii. The fact that this universal asymptotic variance is less than 1 is shown here. This then leads to the unexpected result that the quasiconformal universal variance is not of the form assumed so far. To obtain the result, we use duality to turn the problem into a problem of analyzing an improvement in the Cauchy-Schwarz inequality. The resulting dual problem has geometric interpretation in terms of "zero packing". In the planar case this is related with Abrikosov´s analysis of superconductivity for which he obtained the Nobel prize. Onsdag 22 mars 2017, Thomas Geisser, Rikkyo University, Tokyo, Japan, och Institut Mittag-Leffler Special values of zeta-functions Sammanfattning: One can associate to an algebraic variety over Z, i.e. the solution set of multi-variable polynomial equations with integer coefficients, the so-called zeta-function, which encodes the number of solutions of the equations with entries in finite fields. This generalizes, for example, the Riemann zeta function. Expressing the value of the zeta-function at integers in terms of other invariants often gives deep arithmetic formulas. We give an introduction to zeta-functions and examples of such formulae. Onsdag 29 mars 2017, Panu
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