Algebraic geometry

Thursday, November 20, 2008 - 10:30am - 11:15am
Shmuel Onn (Technion-Israel Institute of Technology)
We develop an algorithmic theory of nonlinear optimization over sets
of integer points presented by inequalities or by oracles. Using a
combination of geometric and algebraic methods, involving zonotopes,
Graver bases, multivariate polynomials and Frobenius numbers, we provide
polynomial-time algorithms for broad classes of nonlinear combinatorial
optimization problems and integer programs in variable dimension.
I will overview this work, joint with many colleagues over the last few
Friday, June 1, 2007 - 9:00am - 9:50am
Manfred Husty (Leopold-Franzens Universität Innsbruck)
Algebraic methods in connection with classical multidimensional geometry have
proven to be very efficient in the computation of direct and inverse kinematics
of mechanisms as well as the explanation of strange, pathological behaviour of
mechanical systems. Generally one can say that every planar, spherical or
spatial mechanism having revolute or prismatic joints can be described by
systems of algebraic equations. In this talk we give an overview of the results
achieved within the last years using algebraic geometric methods, geometric
Thursday, March 8, 2007 - 10:30am - 11:20am
Marta Casanellas (Polytechnical University of Cataluña (Barcelona))
Many statistical models of evolution can be viewed as
algebraic varieties. The generators of the ideal associated to a model
and a phylogenetic tree are called invariants. The invariants of an
statistical model of evolution should allow to determine what is the
tree formed by a set of living species.

We will present a method of phylogenetic inference based on invariants
and we will discuss why algebraic geometry should be considered as a
powerful tool for phylogenetic reconstruction. The performance of the
Wednesday, March 7, 2007 - 10:30am - 11:20am
Niko Beerenwinkel (Harvard University)
The relationship between the shape of a fitness landscape and the underlying gene interactions, or epistasis, has been extensively studied in the two-locus case. Epistasis has been linked to biological important properties such as the advantage of sex. Gene interactions among multiple loci are usually reduced to two-way interactions. Here, we present a geometric theory of shapes of fitness landscapes for multiple loci. We investigate the dynamics of evolving populations on fitness landscapes and the predictive power of the geometric shape for the speed of adaptation.
Monday, January 30, 2012 - 3:40pm - 4:40pm
Peter Kronheimer (Harvard University)
In his 1968 book on singularities of complex hypersurfaces, Milnor asked a question about the unknotting number of knots that arise as the links of singular points of complex plane curves. The question was eventually answered in the affirmative, using gauge theory, by Kronheimer and Mrowka in 1992. A proof requiring only combinatorial techniques was found much later, by Rasmussen, using Khovanov homology. In this talk we will explore a surprising relationship between these two proofs: an interplay between gauge theory and Khovanov homology.
Tuesday, January 16, 2007 - 9:30am - 10:20am
Marie-Francoise Roy (Université de Rennes I)
Global optimization of polynomial functions under polynomial constraints will be related to general algorithmic problems in real algebraic geometry and the current existing complexity results discussed.
The results in the special case of quadratic polynomials will be described.

Main reference for the talk: S. Basu, R. Pollack, M.-F. Roy: Algorithms in real algebraic geometry, Springer, second edition (2006)
Friday, October 27, 2006 - 3:00pm - 3:50pm
Jürgen Gerhard (Maplesoft)
A preview to some of the new features
of the next version of Maple will be given, in particular,
to the improvements in the area of algebraic geometry and
polynomial system solving resulting from integrating
Jean-Charles Faugere's package FGb and Fabrice Rouillier's package RS.
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