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for events the day of Friday, September 18, 2015.

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Friday, September 18, 2015

2:00 pm in 447 Altgeld Hall,Friday, September 18, 2015

Various definitions of Sobolev spaces in metric-measure spaces

Derek Jung (UIUC Math)

Abstract: The Sobolev space $W^{1,p}(\mathbb{R}^n)$ is defined as the space of $L^p$ functions on $\mathbb{R}^n$ with weak derivatives in $L^p$. An equivalent way to define this space is as the completion of the space of smooth functions in $L^p$ with $L^p$-integrable gradient, using a suitable norm. Now suppose we replace $\mathbb{R}^n$ and Lebesgue measure with a more general metric space and measure. How can one define Sobolev spaces without the notion of partial derivatives? Based on a paper by Piotr Hajlasz, my talk will describe a few ways to define Sobolev spaces on metric-measure spaces and under what conditions on the space these definitions coincide.

2:00 pm in 143 Altgeld Hall,Friday, September 18, 2015

Symplectic embeddings of products

Richard Hind (Notre Dame)

Abstract: McDuff and Schlenk have determined precisely when a 4-dimensional ellipsoid can be symplectically embedded in a ball. In joint work with Dan Cristofaro-Gardiner we consider the higher dimensional embedding problem given by taking products of the ellipsoid and balls with a Euclidean space. It turns out that an infinite staircase determined by ratios of odd index Fibonacci numbers, which appears in the solution to the 4-dimensional problem, is also present in higher dimensions. Beyond this however we have significant additional flexibility.

4:00 pm in 243 Altgeld Hall,Friday, September 18, 2015

Pants, Primes and Projective Planes

Peter Nelson (UIUC Math)

Abstract: I'll talk about a basic but coarse notion of "equivalence" for manifolds, and then tease a connection between algebraic topology and number theory.

4:00 pm in 345 Altgeld Hall,Friday, September 18, 2015

On "Surreal numbers, derivations and transseries" by A. Berarducci and V. Mantova

Nigel Pynn-Coates (UIUC Math)

Abstract: This is the second in the upcoming series of talks on the paper mentioned in the title. The goal of this talk will be to show how the class of surreal numbers can be construed as a Hahn field over $\mathbb R$ using the $\omega$ map and Conway normal form.

4:00 pm in 241 Altgeld Hall,Friday, September 18, 2015

Formulas for quiver loci and Grothendieck polynomials as iterated residues

Justin Allman   [email] (Duke University)

Abstract: To any quiver one associates a dimension vector and the corresponding space of quiver representations. The representation space carries the natural action of a so-called base-change group. In the case that the underlying non-oriented graph of the quiver is a simply laced Dynkin diagram the number of orbits is finite. The closure of such an orbit (a so-called quiver locus) defines a class in the equivariant K-Theory (respectively the equivariant cohomology) of the representation space. This class can be expressed in terms of double stable Grothendieck polynomials (resp. Schur functions) where it displays conjectural positivity properties. We provide a new description of quiver loci in terms of iterated residues, a technique which has recently been useful in proving positivity and stability results in analogous problems.