# 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

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

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

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)

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.

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.