Date: July 26-30, 2021
Speaker: Mahir Bilen Can (Tulane University)
Title: Lie Groups and Algebraic Groups in Action
Abstract: The purpose of our lectures is to give a short but self-contained overview of some well-known results about the geometry of algebraic group actions. We will focus mainly on the actions of connected reductive groups. Our main goals are 1) introducing some interesting examples of equivariant completions of homogeneous spaces, 2) explaining several combinatorial gadgets such as valuation cones, weight monoids, colors, etc. that are not only useful for classifying algebraic actions of low complexity but also essential for understanding these equivariant completions. Along the way, we will review some representation theory. In addition, we will analyze some concrete examples of combinatorial varieties such as toric and Schubert varieties.
- Lecture 1 Monday, July 26 at 6 AM (PT), 9 AM (EDT), 4 PM (TRT)
- Lecture 2 Tuesday, July 27 at 6 AM (PT), 9 AM (EDT), 4 PM (TRT)
- Lecture 3 Wednesday, July 28 at 6 AM (PT), 9 AM (EDT), 4 PM (TRT)
- Lecture 4 Thursday, July 29 at 6 AM (PT), 9 AM (EDT), 4 PM (TRT)
- Lecture 5 Friday, July 30 at 6 AM (PT), 9 AM (EDT), 4 PM (TRT)
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Date: August 23-26, 2021
Speakers: Joé Brendel (University of Neuchâtel) and Felix Schlenk (University of Neuchâtel)
Title: Toric Symplectic Manifolds
Abstract: Toric symplectic manifolds are symplectic manifolds with an effective Hamiltonian torus action of maximal dimension. Toric manifolds are distinguished by the property that they can be reconstructed from a combinatorial object called the moment polytope. Thus they are a great playground for symplectic topology and the study of Lagrangian submanifolds, since complicated invariants may be reduced to combinatorial properties of the corresponding moment polytope. In recent years, there has been much interest in a generalization called “almost toric” structures.
In these four lectures, we will introduce these two classes of symplectic manifolds, and use their special structure to study Lagrangian tori and symplectic embedding problems.
- Lecture 1. Monday, August 23 at 6 AM (PT), 9 AM (EDT), 4 PM (TRT)
The Delzant construction by Joé Brendel
- Lecture 2. Tuesday, August 24 at 6 AM (PT), 9 AM (EDT), 4 PM (TRT)
Versal deformations and the Chekanov torus by Joé Brendel
- Lecture 3. Wednesday, August 25 at 6 AM (PT), 9 AM (EDT), 4 PM (TRT)
Almost toric symplectic fibrations by Felix Schlenk
- Lecture 4. Thursday, August 26 at 6 AM (PT), 9 AM (EDT), 4 PM (TRT)
Three applications (maximal embeddings of ellipsoids, exotic Lagrangian tori, and non-isotopic cube embeddings) by Felix Schlenk
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Date: September 27-28, 2021
Speakers: Selman Akbulut (GGTI) and Eylem Zeliha Yildiz (Duke University)
Title: Shaking Knots
Abstract:
"Knot Shaking" is a technique introduced $44$ years ago as a tool to study exotic smoothings of $4$-manifolds with boundary. Let be $K$ be a knot, and $K^{r}$ be the $4$-manifold obtained by attaching a $2$-handle to $B^{4}$ along $K$ with framing $r$. We say that $K$ is $r$-shake slice if a generator of $H_{2}(K^{r})=Z$ is represented by a smoothly imbedded $2$-sphere; this is equivalent to saying that the link consisting of $K$ and an even number of oppositely oriented parallel copies of $K$ (parallel with respect to $r$-framing) to bound disk with holes in $B^4$. Clearly slice knots are $r$-shake slice. It is known that when $r\neq 0$ not all $r$-shake slice knots are slice, and there are knots that are not $r$-shake slice. We address the important remaining case of $r=0$, and prove that $0$-shake slice knots are slice. Along the way, we discuss how shaking is related to the exotic smooth structures and corks.
- Lecture 1. Monday, September 27 at 6 AM (PT), 9 AM (EDT), 4 PM (TRT) by Eylem Zeliha Yildiz
- Lecture 2. Tuesday, September 28 at 6 AM (PT), 9 AM (EDT), 4 PM (TRT) by Selman Akbulut
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