# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

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### Partial symmetry of solutions to some variational problems

We study symmetry properties of several radially symmetric minimization problems. The minimizers which we obtain are sign changing solutions of superlinear elliptic problems or eigenfunctions of weighted asymmetric eigenvalue problems. In both instances we prove that the minimizers have a foliated Schwarz symmetry, but in general they are not radially symmetric. The basic tool which we use is polarization, a two point rearrangement which is useful especially for the study of symmetry properties of sign changing functions. This is joint work with T. Bartsch and M. Willem.

### Sparse changes of coordinates for computing the Castelnuovo-Mumford regularity

Let $I$ be a homogeneous ideal of $R$, the polynomial ring in $n+1$ variables over an arbitrary field $K$. The Castelnuovo-Mumford regularity of $I$ is a numerical invariant related, on one hand, to the minimal graded free resolution of $I$, and on the other to the graded cohomology modules of $R/I$. In this work, avoiding the construction of a minimal graded free resolution of $I$, we provide effective methods for computing the Castelnuovo-Mumford regularity of $I$ that also compute other cohomological invariants of $R/I$. We do this following the philosophy of Bayer and Stillman in their celebrated paper (Inventiones, 1987) making changes of coordinates. The problem with generic projective changes of coordinates is that they usually destroy the sparsity of the ideal and hence are not useful from the computational point of view. In this work, the changes of coordinates we make depend on the ideal $I$ and take advantage of some properties of the ideal. In the worst case, the changes of coordinates are sparse enough to be used for computing the regularity. We will give several examples where the regularity is obtained in a few seconds using our methods while a minimal graded free resolution could not be obtained. When the field K is infinite, we also obtain for free a new algorithm for computing a Noether normalization of $R/I$ that provides a significant improvement of the methods known until now.
This is joint work with Isabel Bermejo (University of La Laguna, Spain). The results presented in this talk have been implemented in the package SINGULAR 2.0.5 (http://www.singular.uni-kl.de/) in a specific distributed library co-written with Gert-Martin Greuel (University of Kaiserslautern, Germany).

### Well-Posedness of the Water-Waves Equations

The water-waves problem consists in finding the motion of the free surface of a perfect, incompressible and irrotational fluid under the influence of gravity. Such a motion is described by the Euler Equations with free surface. I will propose a proof of the well-posedness of these equations, which is quite elementary. I will comment on various of the tools involved in the proof: Dirichlet-to-Neuman operators, regularizing diffeomorphisms, shape optimization, Nash-Moser iterative scheme, etc.

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