Autumn 2006 schedule

# MathematicsDepartmentColloquium : Spring 2007 schedule

 Monday, January 22, 1530-1630

SPEAKER : Douglas Rogers.

TITLE : Golomb rulers, radiotelescopes and other uses for integer difference triangles.

ABSTRACT : A Golomb ruler is a measuring rod on which the marks are set so that the distance between any pair is an integral multiple of some unit distance and no two pairs of marks give the same measurement. While this idea is comparatively simple, it is useful in a wide variety of practical applications: radio telescopes, telecommunications and coding theory, missile guidance, crystallography, graph labeling and block designs. Usually some optimizing constraint is involved.

In this expository talk, we look especially at the case of a collection of Golomb rulers that yields every integral multiple of the unit distance from some run of consecutive integers once and only once is of special interest. The classic problem of this type is to partition the run of consectutive integers

c, c+1, ... , c+3m-1

into m sets of the form {x, y, x+y}, which is related to some work of the Norwegian mathematician, Thoralf Albert Skolem (1887--1963) (for whom see here).

The talk draws only on elementary number theory and graph theory. But the topic remains active, as can be seen from the recent discussion of linear programming bounds here, while a table of lengths of shortest known Golomb rulers appears here. Brian Hayes devoted his regular column in American Scientist to Golomb Rulers in the issue for March/April 1998; the distributed computer search he mentioned is discussed here. Information on Solomon Wolf Golomb (1932--) can be found here or here.

 Monday, February 19, 1530-1630

VIDEO PRESENTATION : "N is a Number" (the life and work of Paul Erdös).

 Monday, Febuary 26, 1530-1630

TITLE : An easy proof of the zeta(2) limit in the random assignment problem.

ABSTRACT : Let the entries $a_{i,j}$ of an $n$ by $n$ matrix be independent random numbers taken from uniform distribution on the interval $[0,1]$. The assignment problem asks for the minimum of $\sum_i a_{i, \sigma(i)}$ taken over all permutations $\sigma$. In 2000 David Aldous proved the 15 year old conjecture that this minimum converges in probability to $\pi^2/6$ as $n\to\infty$. If the matrix entries are instead taken with exponential distribution (this gives similar behavior in the large $n$ limit), there is an exact formula for the expected value of the minimum assignment: $1+1/4+1/9+\dots+1/n^2$. There are now several proofs and generalizations of this formula. I will give a particuarly simple proof that must be quite close to "the book".

 Monday, March 19, 1530-1630

SPEAKER : Måns Henningson, Fysik Chalmers.

TITLE : Four-dimensional quantum field theory : some surprises of the simplest example.

ABSTRACT : Quantum field theory has been immensely successful when applied to physics: Predictions from quantum electrodynamics agree with experiments to a precision of up to ten significant digits. The standard model of elementary particle physics has correctly predicted and described a wide variety of phenomena. Further exciting developments are expected when the Large Hadron Collider at CERN starts operating next year.

Mathematically the situation is more troublesome: Despite many efforts, no non-trivial example of a quantum field theory in four space-time dimensions has yet been rigorously constructed. To make progress on this issue, it appears fruitful to consider relatives of the theories that appear in nature, but with larger amounts of (super)symmetry. Yang-Mills theories with maximally extended supersymmetry are then a natural place to start.

In this talk, we will review some aspects of these beautiful and fascinating theories. In particular, we will describe the by now well established, but still rather mysterious, S-duality property, which indicates that maybe these quantum theories are neither four-dimensional nor field theories after all...

 Monday, April 16, 1530-1630

SPEAKER : Peter Bro Miltersen, University of Århus, Denmark.

TITLE : Strategic game playing and equilibrium refinements.

ABSTRACT : Koller, Megiddo and von Stengel showed in 1994 how to efficiently find minimax behavior strategies of two-player imperfect information zero-sum extensive games using linear programming. Their algorithm has been widely used by the AI-community to solve very large games, in particular variants of poker. However, it is a well known fact of game theory that the Nash equilibrium concept has serious deficiencies as a prescriptive solution concept, even for the case of zero-sum games where Nash equilibria are simply pairs of minimax strategies. That these deficiencies show up in practice in the AI-applications was documented by Koller and Pfeffer. In this talk, we argue that the theory of equilibrium refinements of game theory provides a satisfactory framework for repairing the deficiencies, also for the AI-applications. We describe a variant of the Koller, Megiddo and von Stengel algorithm that computes a *quasi-perfect* equilibrium and another variant that computes all *normal-form proper* equilibria. Also, we present a simple yet non-trivial characterization of the normal form proper equilibria of a two-player zero-sum game with perfect information.

The talk is based on joint work with Troels Bjerre Sørensen.

 Monday, April 30, 1530-1630

SPEAKER : Sverker Lundin.

TITLE : Svensk skolmatematik i historisk belysning.

ABSTRACT :

Obs! Föredraget är på svenska.

Skolmatematik definieras här som ett system av praktiker, varav vissa är undervisningspraktiker. För att utgöra skolmatematik måste dessa praktiker åtföljas av en skolmatematisk diskurs, i vilken undervisningen framställs som en väg mot mål som bara indirekt har med matematik att göra, till exempel demokrati, självförtroende, och förmåga att tänka redigt och klart. Fenomenet skolmatematik blev allmänt utbrett i Sverige kring mitten av 1800-talet. I texten presenteras två teser. Den första är att undervisningspraktiken \u2013 det som händer i klassrummen \u2013 varit relativt konstant sedan 1880-talet: vad eleverna gör är huvudsakligen att lösa långa rader av övningsuppgifter. Den andra tesen säger att denna undervisningspraktik varken idag eller någonsin bidragit till att i högre utsträckning realisera de mål som presenteras i den skolmatematiska diskursen. Skolmatematiken är tvärt om \u2013 vad gäller den kompetens den befordrar \u2013 ett relativt slutet system. Nytta av det man lär sig på matematiklektionerna, har man nästan uteslutande på de åtföljande matteproven. Relationen mellan den skolmatematiska undervisningspraktiken och det omgivande samhället förmedlas istället genom betyg och examina, vilka utgör en form av symboliskt kapital. Den skolmatematiska diskursen tjänar huvudsakligen till att legitimera denna kapitalform, samt till att reproducera skolmatematiken som socialt fenomen. Texten utgörs huvudsakligen av två redogörelser: dels för undervisningspraktikens-, dels för den skolmatematiska diskursens utveckling från 1830 till idag, genom vilka teserna knyts till skolmatematikens historia. I texten väcks frågan om den akademiska matematikens relation till skolmatematiken. Tentativt sägs akademisk matematik utgöra en specialitet, som varken har eller bör ha särskilt mycket med skolmatematik att göra.

 Monday, May 7, 1345-1445 (OBS! unusual time).

SPEAKER : Ian Sloan, University of New South Wales, Australia (President ICIAM).

TITLE : Lifting the curse of dimensionality : Numerical integration in very high dimensions.

ABSTRACT : Richard Bellmann coined the phrase "the curse of dimensionality" to describe the extraordinarily rapid increase in the difficulty of most problems as the number of variables increases. A typical problem is numerical multiple integration, where the cost of any integration formula of product type obviously rises exponentially with the number of variables. Nevertheless, problems with hundreds or even thousands of variables do arise, and are now being tackled successfully. In this talk I will the story of recent developments, in which in less than a decade the focus has turned from existence theorems to concrete constructions that achieve the theoretically predicted results even for integrals in thousands of dimensions with many thousands of points. Suitable integration rules of this kind are now being applied to applications from mathematical finance.

 Monday, May 14, 1530-1630.

SPEAKER : Kyril Tintarev, Uppsala Universitet.

TITLE : Mountain pass theorem and a windy road to a saddle point.

ABSTRACT : Most mathematicians today never rode a horse or held a saddle in their hands. This historic change deprives the term "saddle point" of its transparency. But suppose you live on a coast and want to get to an inland valley by taking the lowest elevation on your way. You will not go through the mountain tops but will chose a mountain pass, where the point of highest elevation will be indeed a saddle point (which for lack of saddles in our lives we would rather call it a mountain pass point). The height function at the mountain pass point will be then given by

c=\inf_{paths t\to x_t}\sup_t h(x_t).

A prerequisite for finding a mountain pass point is that there is a mountain ridge to cross, that is that $\inf h$ over a certain set (that every path enters) is larger than the values of $h$ at the endpoints of the trip. One says then that the function has a "mountain pass geometry". The mountain pass theorem shows that if a function $h$ has a mountain pass geometry with a minimax value $c$, then there exists a sequence $x_k$ such that h(x_k) -> c and h'(x_k) -> 0. This sequence may or may not converge: a sequence $(0,\pm k)$ is critical for $h(x,y)=(1+y^2)^{-1}-x^2$.

The mountain pass theorem is often the only method to find solutions to semilinear elliptic equations that are mountain pass points of the functional

G(u)=\int_{\R^N}(\frac12|\nabla u|^2-F(x,u)).

It is easy to trace, under general conditions on $F$, the mountain pass geometry for such functionals. Then the really difficult work begins -- to verify convergence of the critical sequence. We will follow some steps on the windy road towards the existence of a mountain pass point -- and therefore a solution to the equation

-\Delta u=F_u(x,u).

 Monday, May 21, 1530-1630.

SPEAKER : Björn Walther, Högskolan i Kalmar.

TITLE : Space-time estimates and operator monotone functions.

ABSTRACT : Joint work with Yacin Ameur on decay and regularity estimates for the time-dependent Schrödinger equation will be described. The estimates in an example for a Hamiltonian with a singular potential (related to the Aharonov-Böhm Hamiltonian) in most cases coincide with previously known estimates for the free Hamiltonian. To prove our results we combine techniques for integral estimates of Bessel functions with spectral theory and theory of operator monotone functions.