Colloquium

Abstract:

A well-studied problem in computer vision is "structure from motion", where 3D structures and camera poses are reconstructed from given 2D images taken by the unknown cameras. The most classical instance is the 5-point problem: given 2 images of 5 points, the 3D coordinates of the points and the 2 camera poses can be reconstructed. In fact, given 2 generic images of 5 points, this problem has 20 solutions (i.e., 3D coordinates + 2 camera poses) over the complex numbers. Reconstruction problems which have a finite positive number of solutions given generic input images, such as the 5-point problem, are called "minimal". These are the most relevant problem instances for practical algorithms, in particular those with a small generic number of solutions. 

We formally define minimal problems from the point of view of algebraic geometry. Our algebraic techniques lead to a classification of all minimal problems for point-line arrangements and any number of cameras. We compute their generic number of solutions with symbolic and numerical methods.

This is joint work with Timothy Duff, Anton Leykin, and Tomas Pajdla.​

NB! Coffee/Tea/Biscuits from 14.00.

Abstract:

Transmitting a quantum state from one location to another is a key task within a quantum computer. This task can be realised through the use of a spin network, which can be modelled by an undirected graph G: vertices in G represent spins, and two edges are adjacent in G whenever the corresponding spins interact. 

As the system evolves over time, the quality of the state transfer is measured by a function called the fidelity, which can be expressed in terms of the Hamiltonian for the system; that Hamiltonian is the adjacency matrix of G. The fidelity at time t, f(t), is a number between 0 and 1, and if it happens that f(t_0)=1 for some t_0>0, then we say that there is perfect state transfer (PST). Similarly if f(t) can be made arbitrarily close to 1 via suitable choice of t, then there is pretty good state transfer (PGST). 

In this talk we will give an introduction to perfect and pretty good state transfer on graphs, with a focus on identifying some families of graphs that possess PST/PGST, and other families that don't. If time permits, there will also be a discussion of the effect of errors on the fidelity. Tools from matrix analysis, spectral graph theory and number theory will each play a role.

NB! Coffee/Tea/Biscuits from 14.00.

Abstract:

I will begin with a non-technical introduction to curvature of surfaces and then move on to the Uniformization Theorem which says that any Riemann surface can be deformed in a conformal way (i.e. preserving angles) into a Riemann surface of constant curvature. This theorem, which was proved in 1907, can be seen as the one-dimensional case of a problem in complex geometry going back to the 1930’s to determine which complex manifolds admits Kähler-Einstein metrics. Using the one-dimensional case as illustration, I will talk about an important milestone reached in 2013 when a conjecture (the Yau-Tian-Donaldson conjecture) relating existence of Kähler-Einstein metrics on projective manifolds to algebraic properties of the manifold was verified. I will end by talking about some more recent developments and some open problems in the field. 

NB! Coffee/Tea/Biscuits from 14.00.

Abstract:

The Clay Mathematics Institute awards $1 million to anyone who proves (or disproves) the existence of smooth solutions of the Navier-Stokes equations [1]. In this lecture we will give a non-technical introduction to the Navier-Stokes equations and some related partial differential equations. I will give a brief overview of some of the partial results available, and try to explain why this is a difficult problem. 

The talk should be accessible to math students in their third year and up.

[1] http://www.claymath.org/millennium-problems/navier–stokes-equation

NB! Coffee/Tea/Biscuits from 14.00.

Abstract: 

This year the Center for Advanced Study is holding a special year on Homotopy Type Theory and Univalent Foundations, organized by Marc Bezem and Bjørn Dundas.

Homotopy type theory, with the partition of types into levels and the univalence axiom developed by Voevodsky, provides both a new logical foundation for mathematics (Univalent Foundations) and a formal language usable with computers for checking the proofs mathematicians make daily.  As a foundation, it replaces set theory with a framework where propositions and sets are defined in terms of a more primitive notion called "type" -- in this framework the notion of symmetry arises at the most basic level: from the logic.  As a formal language, it encodes the axioms of mathematics and the rules of logic simultaneously, and promises to make the extraction of algorithms and values from constructive proofs easy.  As a mathematical topic, it offers an intriguing range of open problems at all levels of accessibility.

I will give an intuitive introduction to these recent developments.

NB! Coffee/Tea/Biscuits from 14.00.

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