Department of

Mathematics


Seminar Calendar
for events the day of Monday, November 4, 2013.

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Monday, November 4, 2013

10:00 am in 145 Altgeld Hall,Monday, November 4, 2013

Semi-toric systems as Hamiltonian S^1-spaces

Daniele Sepe (Utrecht University Math)

Abstract: The classification of completely integrable Hamiltonian systems on symplectic manifolds is a driving question in the study of Hamiltonian mechanics and symplectic geometry. From a symplectic perspective, such systems correspond to Hamiltonian R^n-actions which are locally toric. The class of integrable Hamiltonian systems on 4-dimensional symplectic manifolds corresponding to Hamiltonian S^1 x R actions (with some extra assumptions on the singularities) is known as semi-toric: it was introduced by Vu Ngoc, and Pelayo and Vu Ngoc obtained a classification for `generic' semi-toric systems. From such a system one obtains a 4-dimensional manifold with a Hamiltonian S^1-action by restricting the action: when the underlying symplectic manifold is closed, Karshon classified these spaces in terms of a labelled graph. This talk aims at explaining how, starting from a semi-toric system on a closed 4-dimensional symplectic manifold, Karshon's invariants of the underlying Hamiltonian S^1-space can be recovered using the notion of `polygons with monodromy' introduced by Vu Ngoc. This should be thought of as analogous to the procedure to obtain Karshon's invariants from Delzant polygons in the case of symplectic toric manifolds. This is joint work with Sonja Hohloch (EPFL) and Silvia Sabatini (IST Lisbon), and part of a longer term project to study Hamiltonian S^1 x R actions on closed 4-dimensional manifolds.

1:30 pm in 345 Altgeld Hall,Monday, November 4, 2013

Generalized Barycentric Coordinates

Hal Schenck   [email] (UIUC Math)

Abstract: In approximation theory, it is often useful to have a representation of a point in terms of some specificed vertices. When the vertices are those of a triangle, these are the familiar barycentric coordinates introduced by Moebius in 1827. About 30 years ago, Wachspress introduced a version for an arbitrary polygon; this was generalized by Warren to higher dimensions. The Wachspress coordinates of a point are rational functions determined by the vertex positions; it is useful to determine the algebraic relations among them. Interestingly, this leads to combinatorial problems involving the Stanley-Reisner ring, as well as some fancy algebraic geometry (which I will mention but not dwell on). Joint work with Corey Irving, Santa Clara University.

4:00 pm in 241 Altgeld Hall,Monday, November 4, 2013

The Ricci Flow for Homogeneous Spaces

Tracy Payne (Idaho State)

Abstract: The Ricci flow evolves a metric g_0 on a manifold M according to the PDE g_t = -2 ric(g_t). If M is a simply connected homogeneous space, the Ricci flow preserves the homogeneity of M. Let X be a class of simply connected homogeneous spaces that remains invariant under the Ricci flow. It is possible to encode the Ricci flow on X as a system of ODEs. We will describe the general set-up of the system of ODEs and describe methods for qualitative analysis of such systems, giving examples of the analysis for various specific subclasses.

5:00 pm in 241 Altgeld,Monday, November 4, 2013

Explicit Kazhdan constants for some property (T) group inclusions (following Shalom) Part II

Stephen Longfield (UIUC Math)