Tuesdays at 4:10 in
Monday, January 26 in 298 Carver PDE candidate Alden Waters(Ecole Normale Superieure) Gaussian beams and reconstruction methods in inverse problems
One of the central questions in inverse problems is the reconstruction of waves emitted from the boundary of an obstacle. The Gaussian beam Ansatz which accurately approximates waves in the high frequency regime has been useful to numerical analysts and geophysicists since the 1960s. We show that using these suitably localized approximations to solutions that new questions of reconstruction of materials can be addressed.
The problem of describing the set of hypersurfaces passing through a finite set of points with given multiplicity leads to challenging mathematical questions. For example, one can ask what the minimum degree of such a hypersurface is or how many independent hypersurfaces there are of any given degree. The most general forms of these questions are still open and have given rise to longstanding conjectures in algebraic geometry.
Searching for structural reasons to explain some of the these conjectures, Harbourne and Huneke proposed an approach based on comparisons between the set of all polynomials vanishing at the points to a prescribed order, which is called a symbolic power ideal, and algebraically better understood counterparts, namely the ordinary powers of the ideal of base points. Two questions will be shown to be related: How tight can this comparison be made? Which arrangements of lines in the plane have no points where only two lines meet? I will answer these and many more questions while considering some special arrangements of lines with unexpected combinatorial and algebraic properties.
Quantized enveloping algebras originally arose in the context of quantum integrable systems in the early 1980's. Since then, it has gradually been understood that these algebras are intimately connected to many areas of mathematics and physics. In this talk I will discuss some recent results about the structure and representation theory for quantized enveloping algebras, while highlighting the role played by various generalized Weyl algebras.
The classical Nirenberg problem asks for which functions on the sphere arise as the scalar curvature of a metric that is conformal to the standard metric. In this talk, we will discuss similar questions for fractional Q-curvatures. This is equivalent to solving a family of nonlocal nonlinear equations of order less than n, where n is the dimension of the sphere. We will give a unified approach to establish existence and compactness of solutions. The main ingredient is the blow up analysis for nonlinear integral equations with critical Sobolev exponents. We will also discuss related topics including solutions with isolated singularities. This talk is based on joint works with L. Caffarelli, Y.Y. Li, Y. Sire and J. Xiong.
Many topological spaces come equipped with an action by a group of symmetries and we naturally want to understand phenomena preserved by these actions. In fact, analyzing such group actions can be fruitful even in contexts that initially appear nonequivariant. Algebraic topology uses tools such as cohomology and one of the challenges of equivariant homotopy theory is to develop cohomology theories adapted to the setting where groups act. In this talk, I will describe new work that allows us to construct equivariant cohomology theories. These methods take as their input algebraic data familiar to representation theorists, and provide better access to the underlying algebra of the cohomology theory than previous constructions.
Tuesday, January 20 in 204 Carver- ALG candidate Alexandra Seceleanu (UNL) Symbolic versus ordinary powers for ideals of points
Wednesday, January 21 in 298 Carver- ALG candidate Jonas Hartwig (UC Riverside) Quantized enveloping algebras and generalized Weyl algebras
Thursday, January 22 in 232 Carver - PDE candidate Tianling Jin (University of Chicago) The Nirenberg problem and its generalizations: A unified approach