Monday, February 11, 1530 - 1630 (OBS! This is a promotion lecture).
TITLE: From transportation theory to Kähler-Einstein metrics and negative temperature states.
ABSTRACT: Transportation theory, as developed in the classical works of Monge and Kantorovich, concerns the optimal transportation and relocation of resources and it has been widely applied in such diverse areas as image processing, economics and even metereology. In mathematical terms the problem is, given two measures in Euclidean space, to find the optimal map (the so called transport plan) which pushes forward the first "initial measure" to the second "target measure". Here optimality is defined with respect to a given cost function, typically the squared distance function.
In this talk I will present a new solution to this problem which involves a mathematical "in vitro experiment"; in physical or statistical mechanical terms this means that the optimal solution is recovered from the equililbrium state of an "imaginary" gas of particles in the limit of many particle and large temperature. More precisely, in quantum mechanical terms, the particles in question are free bosons, confined to the support of the first measure and their momenta is supposed to be distributed according to the second measure, which will be taken to be the uniform measure on a given convex polytope P.
In fact, the motivation for this setup comes from a completely different direction, namely a probabilistic approch to complex geometry and, more precisely, the existence problem for Kähler-Einstein metrics on complex algebraic varieties. Indeed, as will be explained in the lecture, lowering the temperature of the gas below absolute zero, all the way down to a certain critical negative temperature, the corresponding new equilibrium state of the gas yields a solution to the Kähler-Einstien equation on the toric algebraic variety X corresponding to the given polytope P. Or, to be more precise: this happens when a Kähler-Einstein metric exists and otherwise the gas evaporates into thin air.
The talk will, hopefully, be accessible to a wide audience.
Monday, February 18, 1530 - 1630.
TITLE: What is the abc-conjecture (Vad är abc-förmodan ?).
ABSTRACT (english): If a,b,c are positive integers without a common factor, then it seems that the equality a+b=c puts strong restrictions on the simultaneous appearance of several identical prime factors in the numbers a,b,c. This observation is the basis for the abc-conjecture, which was formulated by Oesterlé and Masser in the mid-1980s. Their conjecture has far-reaching consequences in number theory (e.g. in a suitable version, it implies Fermat's Last Theorem). A few months ago, the Japanese mathematician Shinichi Mochizuki announced a proof of the abc-conjecture, which attracted much attention both among mathematicians and in the media all over the world. The purpose of the lecture is to discuss the abc-conjecture, in particular, in connection with the wide interest in it caused by Mochizuki's claim.
ABSTRAKT (svenska): Om a,b,c är positiva heltal utan gemensamma delare så verkar likheten a+b=c sätta starka begränsningar på samtidig förekomst av flera identiska primtalsdelare i alla tre talen a,b,c. Denna observation är grunden för abc-förmodan som formulerades av Oesterlè och Masser i mitten av 1980-talet. Deras förmodan har mycket långtgående konsekvenser i talteorin (t.ex. implicerar dess lämplig version Fermats stora sats). För några månader sedan annonserade den japanske matematikern Shinichi Mochizuki ett bevis av abc-förmodan, vilket uppmärksammades flitigt både i matematiska kretsar och i flera tidningar runt om i världen. Syftet med föredraget är att berätta om abc-förmodan, speciellt med tanke på det bredda intresse som Mochizukis annonsering har föranlett.
Monday, April 22, 1530 - 1630.
TITLE: Counting rational points on varieties using dynamics
ABSTRACT: The talk will begin with a short survey of the "dynamical/probabilistic" approach to counting integer and rational points on homogeneous varieties. Some examples how the techniques in this approach are applied to "classical settings" such as the Gauss' circle problem and its Fuchsian analogues will then be described. Finally, and if time permits, we shall discuss recent work (joint with M. Einsiedler and A. Gorodnik) on counting rational points on "mod-diagonal varieties".
The talk will be accesible to advanced undergraduates.
Monday, April 29, 1530 - 1630 (OBS! This is a promotion lecture).
TITLE: Automated Solution of Differential Equations
ABSTRACT: For more than two centuries, partial differential equations have been
the most important tool available to scientists and engineers to model
and understand a wide range of physical phenomena. The solution of a
partial differential equation is often a very challenging and
time-consuming task, involving mathematical analysis, numerical
analysis and the development of sophisticated computer programs.
In this lecture, I demonstrate how the solution process can be
completely automated, to the point where complicated systems of
nonlinear partial differential equations can be solved with ease in a
Monday, May 20, 1530 - 1630.
TITLE: Global Geometry and Analysis on Locally pseudo-Riemannian Homogeneous Spaces
ABSTRACT: The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as
Lorentz geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry, surprisingly little is known about global properties of the geometry even if we impose a
locally homogeneous structure. Further, almost nothing is known about global analysis on such spaces. Taking anti-de Sitter manifolds, which are locally modelled on AdS^n as an example, I plan to explain two
programs: 1. (global shape) Is a locally homogeneous space closed? 2. (spectral analysis) Does the spectrum of the Laplacian vary when we deform the geometric structure?