BesøksadresseNiels Henrik Abels hus (kart) Moltke Moes vei 35
Abstract: If A×C ≃ B×C, is A ≃ B? We encounter this questions in many contexts, and the answer depends in often very subtle ways on the context, or in fancier language, on the category for which the question is posed. I will discuss some examples from algebra, for instance: does a polynomial ring uniquely determine its ring of coefficients, or in algebraic geometry, for instance: if X, Y, Z are algebraic varieties and X × Z ≃ Y ×Z, is X ≃ Y. Here, even when X,Y,Z are affine spaces, the answer (both yes and no occur) is known only in a few cases. In general, a lot of nice mathematics is involved both in proving positive results and in constructing counterexamples.
Coffee/Tea/Biscuits from 14.00
Modular symmetries have been central to many developments in mathematics since their discovery more than a century ago, including the proof of Fermat´s last theorem. In view of Nature´s love for order and symmetry, it is surprising that modular symmetry has not been found in the real world – until now.
Coffee/Tea/Biscuits from 14.00
Abstract: In this talk I briefly review the connection between these two areas and describe basic structures and problems underlying both fields.
Coffee/tea and cookies will be served from 14.00 to 14.30.
Abstract: Soliton equations are an important class of nonlinear partial differential equations, which contain physically relevant equations like the Korteweg-de Vries (KdV) equation, the sine-Gordon equation, and the Nonlinear Schrödinger equation. While these equations govern very different physical phenomena, they have striking common structural properties like the existence of particle-like solutions (solitons) interacting in elastic collisions. The struggle to find a mathematical explanation has led to substantial progress in mathematical physics. Today it is known that the theory of soliton equations is linked to several major branches of mathematics. Our talk will be an introduction to an operator theoretic approach to soliton equations, which may be traced back to work of Marchenkov and enables us to apply Banach geometry in the study of solution families. As a motivation, we will carefully explain this in the most accessible case of the KdV equation. Then we will discuss further developments of the method in the study of matrix equations and hierarchies. In the applications part, we will talk on the asymptotic description of multiple pole solutions, the construction of matrix solitons and countable nonlinear superposition. We will illustrate our results by Mathematica plots.
Coffee/tea and cookies will be served from 13.30 to 13.55 in the rest area, 7th floor, NHA.
Abstract: I wish to give some insight into how a generalization of a question related to the limiting behaviour of sequences of large graphs allows for a fruitful interaction between ideas and methods from combinatorics, probability theory and functional analysis. Joint work with L. Lovász and B. Szegedy.
Abstract: The idea of this talk is to introduce the concept of o-minimal theories and give a sense in how it can be useful in different areas of mathematics. O-minimality is a branch of model theory which is concerned with understanding "tame" analytic expansions of the real closed field. I will give definitions and state some of the main properties of o-minimality. I will also state and sketch applications to statistical learning theory and state some of the new applications to algebraic geometry and Diophantine number theory.
Coffee/tea and cookies from 14.00.
Abstract: In this talk, my goal is to give an introduction to some of the mathematics behind quasicrystals. Quasicrystals were discovered in 1982, when Dan Schechtmann observed a material which produced a diffraction pattern made of sharp peaks, but with a 10-fold rotational symmetry. This indicated that the material was highly ordered, but the atoms were nevertheless arranged in a non-periodic way.These quasicrystals can be defined by certain aperiodic tilings, amongst which the famous Penrose tiling. What makes aperiodic tilings so interesting--besides their aesthetic appeal--is that they can be studied using tools from many areas of mathematics: combinatorics, topology, dynamics, operator algebras... While the study of tilings borrows from various areas of mathematics, it doesn't go just one way: tiling techniques were used by Giordano, Matui, Putnam and Skau to prove a purely dynamical statement: any Z^d free minimal action on a Cantor set is orbit equivalent to an action of Z.
NB: Coffee/tea and cookies will served from 14.00-14.25 in the rest area of the 7th floor in NH Abel's House, as usual. Note that the colloquium talk will be held in Aud 2 VB from 14.30.
Judith Packer (University of Colorado at Boulder, USA) will give a Colloquium Talk:
Fractals, fractal wavelets, and measures on solenoids
Abstract: We discuss a construction, first due to D. Dutkay and P. Jorgensen, that can be used to define generalized wavelets on inflated fractal spaces arising from iterated function systems.
Self-similarity relations defining the fractal spaces also give rise to filter functions defined on the torus. These filter functions can be used to construct isometries, as well as probability measures on solenoids. Representations of the Baumslag-Solitar group can be obtained from the probability measures, and some properties of the representation are related to properties of the original wavelet and filter systems. Some results joint with L. Baggett, N. Larsen, K. Merrill, I. Raeburn, and A. Ramsay will be discussed.
NB: Coffee/tea and cookies will served from 14.00-14.25 in the rest area of the 7th floor in NH Abel's House, as usual. But note that the colloquium talk will be held in Aud 2 VB from 14.30.
Abstract: Rudin's classic treatise "Function Theory in Polydiscs" from 1969 studies complex analysis on polydiscs starting from the following question: How much of our extremely detailed knowledge about holomorphic functions in the unit disc can be carried over to an analogous situation in several variables, namely to polydiscs? A different perspective comes from the work of Bohr and Bohnenblust--Hille several decades earlier; here the main issue is function theory in the infinite-dimensional polydisc, which to a large extent is concerned with the asymptotics of function theory in finite-dimensional polydiscs. I will discuss some examples, old and new, of what these different viewpoints have led to, as well as connections to other areas.
There will be served coffee, tea and biscuits from 14.00.
Et av høydepunktene i Pierre Deligne, årets Abelprisvinner, sitt arbeid er beviset for den siste og avgjørende delen av Weil-formodningene. Både før og etter hans bevis har disse formodningene spilt en nøkkelrolle i studiet av forholdet mellom aritmetikk og geometri. Mitt foredrag, på engelsk, vil i stor grad bygge på notater av Runar Ile.
Abstract: In this talk we will discuss some of the foundations of computational mathematics. In particular, I will ask the rather fundamental questions: "Can everything be computed?" and: "In what way?" and discuss possible ways to answer them. This is done by linking some new developments in computational mathematics (in particular the Solvability Complexity Index and towers of algorithms) to some of the fundamental works of Godel, McMullen and Smale
In 1975 this year's Abel Prize recipient, the Hungarian mathematician Endre Szemerédi, proved a long-standing conjecture of Erdøs: if A is a subset of the natural numbers of positive upper density, then A contains arbitrarily long arithmetic progressions. His proof is purely of combinatorial nature and extremely complicated--only a handful people have read and understood it. In 1977 Furstenberg gave a new proof using ergodic theory. Specifically, he showed that any measure-preserving system has a multiple recurrence property, which is a vast generalization of a classical result of Poincaré. What is amazing is that Szemerédi's theorem and Furstenberg's theorem actually are equivalent! We will indicate how the Szemeredi theorem -- and Furstenberg's proof of it -- plays a key role in the proof of the now famous Green/Tao result from 2004: within the prime numbers there exist arbitrarily long arithmetic progressions.
The 2012 Abel Prize is awarded to Endre Szemerédi (Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, and Department of Computer Science, Rutgers) "for his fundamental contributions to discrete mathematics and theoretical computer science, and in recognition of the profound and lasting impact of these contributions on additive number theory and ergodic theory." The purpose of this talk is to present some of Szemerédi's work in combinatorics, in particular extremal combinatorics. A central result is Szemerédi’s Regularity Lemma which is a fundamental tool in graph theory. It says something about the structure of large graphs and their relation to random graphs. We also mention some applications/connections to other areas.
In recent years, a number of finite element spaces compatible with the differential operators grad, curl and div, have been given a unified presentation in the language of differential forms, enabling a likewise unified analysis of discretizations of partial differential equations related to the Hodge Laplacian. I will present a general framework for the construction of finite element spaces of differential forms, allowing for polyhedral meshes and non-polynomial basis functions. We apply it to get a variant of finite element exterior calculus, incorporating a form of upwinding adapted to convection dominated flows.