Geometry and Topology

Speaker: Tomoo Matsumura (Cornell)

"Hamiltonian Torus Actions on Orbifolds"

Time: 15:30

Room: MC 108

When a symplectic manifold M carries a Hamiltonian torus R action, the injectivity theorem states that the R-equivariant cohomology of M is a subring of the one of the fixed points and the GKM theorem allows us to compute this subring by only using the data of 1-dimensional orbits. The results in the first part of this talk are a generalization of this technique to Hamiltonian R actions on orbifolds and an application to the computation of the equivariant cohomology of compact toric orbifolds. In the second part, we will introduce the equivariant Chen-Ruan cohomology ring which is a symplectic invariant of the action on the orbifold and explain the injectivity/GKM theorem for this ring.

Analysis Seminar

Speaker: Ekaterina Shemyakova (Western)

"Differential Transformations for Integrable PDEs"

Time: 15:30

Room: MC 107

Transformational Methods are known to be one of the most efficient methods for finding exact solutions of Partial Differential Equations. In this talk we shall be concentrated on the differential transformations introduced by Darboux (DT). DT can be defined by so-called (m,n)-transformations which are Linear Partial Differential Operators without mixed derivatives. The (m,n)-transformations have interesting algebraic structure. The (m,n)-transformations can help us to solve the problem of the generality of the Darboux Wronskian formulas. Namely, Darboux stated and different authors proved for different cases that given some number of partial solutions, a DT can be defined via some Wronskians. Darboux believed that the reverse statement will be true "generally speaking". In this talk we show several results on our way to prove this reverse statement and to decide what is "the general case" in this context. The second part of the talk will be devoted to an invariant description of the DT. We start with an idea that in view of the said above it would be more efficient to defined DT in terms of invariants of the pair (L,z), where L is a Linear Partial Differential Operator, and z is an element of its kernel. We show that such invariants is in correspondence with solutions of certain PDE, and that instead of a chain of DT we can consider mappings of invariants.

Colloquium

Speaker: Gene Freudenberg (University of Western Michigan)

"Locally Nilpotent Derivations of Rings with Roots Adjoined"

Time: 15:30

Room: MC 107

Working over a ground field k of characteristic zero, this talk will discuss locally nilpotent derivations of rings of the form $B = R[z ]$, where $R$ is a commutative $k$-domain, and $z^n\in R$ for some positive integer $n$. Such a ring has a natural grading by $Z_n$ . We give basic properties of locally nilpotent derivations $D$ of $B$ which are homogeneous relative to this grading. In particular, $D$ is always a quasi-extension of a locally nilpotent derivation $\delta$ of $R$, and $D^2 z = 0$. This approach yields strong sufficient conditions for a ring of this type to be rigid, using in particular the absolute degree $|f |_R$ of elements of $R$. For example, we show that if $R$ is $Z$-graded, $f ∈ R$ is $Z$-homogeneous of degree coprime to $n$, and $|f |_R\ge 2$, then the ring $B = R[f^{1/n}]$ is rigid. The main idea is to study the locally nilpotent derivations of $B$ by looking at those of $R$.

Several applications of our results will be discussed. For example, we study $G_a$ -actions of Pham- Brieskorn surfaces and threefolds, with particular interest in questions of rigidity and stable rigidity. Our methods permit us to show rigidity for many cases which were previously open. This talk represents the speaker’s joint work with L. Moser-Jauslin.

Algebra Seminar

Speaker:

"Algebra Seminar cancelled "

Time: 14:30

Room: MC 107