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Monday, February 19, 1530-1630 |
VIDEO PRESENTATION :
"N is a Number" (the life and work of Paul Erdös).
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Monday, Febuary 26, 1530-1630 |
SPEAKER :
Johan Wästlund, Linköping.
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".
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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...
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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.
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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.
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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.
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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).
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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.