Colloquium

Associate Professor Erik Adli from the Department of Physics at UiO will give a Colloquium Talk on 

                                 Particle Accelerators

The speedskating 1000-m is a Formula One event, for the Olympics and the annual World Sprint Championships. The skaters are running in pairs, with one starting in inner lane and the other in outer lane, and then exchanging lanes on the backstretch. With speeds up to 60 km/h, there will in particular be technical challenges when negogiating the curves. Once upon a time I managed to change the Olympics, by pinpointing the statistically significant non-zero difference between last inner and last outer lane for the 500-m; the practical solution is to have the skaters running the 500-m twice, with one start in inner and one in outer. I have now properly examined the 1000-m, and found that there is a similar and in fact more dramatic Olympic Unfairness Parameter associated with that event.  

Coffee and cakes from 12.00 in the lunch room.

Abstract:

The Hodge conjecture is a major unsolved problem in mathematics which forms a subtle bridge between algebraic geometry, differential geometry and topology. Formulated by Hodge in his 1950 ICM address, it asks for a characterisation of the subvarieties of a projective variety in terms of the topology of that variety. Today the conjecture remains widely open, and it is one of the seven `Millenium problems' for which the Clay Institute offers a prize of one million dollars. In this talk I will give an basic introduction to the circle of ideas around the conjecture, focusing on specific examples, pictures and important special cases.

NB! Coffee/Tea/Biscuits from 14.00.

Abstract:

If X is a compact metric space, then a measure on X is a linear, bounded and positive operator on the space of continuous functions on X. If f:X->Y is a continuous map between compact metric spaces, then by tautology we can pushforward any measure on X. Can we pullback a measure? No answer had been found in the literature.

In this talk I will explain why we cannot pullback a measure to a measure in general, even in the case where f is an isomorphism over a dense open subset of Y. On the other hand, I will show that if f is a finite covering over a dense open subset of Y, then we can pullback any measure on Y to a more general class of so-called positive strong submeasures. This is then applied to dynamics of meromorphic maps of compact Kahler manifolds. This general class of strong submeasures can also be used in the problem of intersection of hypersurfaces. 

NB! Coffee/Tea/Biscuits from 14.00.

Abstract

NB! Coffee/Tea/Biscuits from 14.00.

Professor Emeritus Erling Størmer will turn 80 on November 2, and he will use this occasion to give a Colloquium Talk with tittle:

        POSITIVE LINEAR MAPS ON MATRIX ALGEBRAS

Abstract: The first part of the talk will be an introduction to the theory of positive linear maps between matrix algebras, plus a few words on the operator algebra case. For the last 20 years, physicists working in quantum information theory have been active in the subject and I’ll devote the last half of the talk to a negative solution to a conjecture they had.  

Abstract: I will discuss the two main syntheses that occured duing the "wavelet revolution" in the second half of the 1980s: one concerning the many aspects of what we now call wavelets, another concerning the relations between conjugate mirror filters, multiresolution approximations and orthonormal wavelet bases.

Coffee/tea and cookies will be served from 14.00 to 14.30.

Abstract

Coffee/Tea/Biscuit from 14.00

Continuing from last week, I will talk about Andrew Wiles' proof of the Taniyama-Shimura-Weil Conjecture, and why it implies Fermat's Last Theorem.

There will be served coffee, tea and biscuits from 14.00.

In preparation for the Abel Prize week, I will talk about Fermat's Last Theorem and the Taniyama-Shimura-Weil Conjecture about the modularity of elliptic curves.

There will be served coffee, tea and biscuits from 14.00.

Abstract:

Let X=Spec(R) be a smooth affine variety and let Y be a closed subvariety corresponding to an ideal I in R. Finding a set of generators for I is in general a very hard problem, even in the case where X is an affine space over a field k. However, an easy application of Nakayama’s lemma shows that the number n of generators of I is at least the number m of generators of its conormal bundle and at most m+1. We will show how to actually determine when n=m using homotopical and cohomological methods, answering in particular a long standing conjecture of Murthy.

Coffee/Tea/Biscuit from 14.00

Abstract; Grassmannians and flag varieties play fundamental role in  modern mathematics. They show up in many different situations and have been extensively studied from the point of view of algebraic geometry, topology,representation theory and combinatorics. In the first part of my talk I will recall the main properties of the Grassmannians and flag varieties. In particular, I will describe the connection with the representation theory of the general linear group. In the second part of the talk I will describe recent results on the Poincaré–Birkhoff–Witt degenerations. The main idea to be utilized is the connection between the representation spaces and the varieties we are interested in. 

Coffee/Tea/Biscuits from 14.00

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.