## Analysis, Geometry, and Dynamical Systems

Knotoids and Virtual Knot Theory.

*Louis H. Kauffman*, University of Illinois at Chicago.

## Abstract

Knotoids are open-ended knot diagrams whose endpoints can be in different regions of the diagram. Two knotoids are said to be isotopic if there is a sequence of Reidemeister moves that connects one diagram to the other without moving arcs across endpoints. The definition is due to Turaev. We will discuss three dimensional interpretations of knotoids in terms of projections of open-ended embeddings of intervals into three dimensional space, and we shall discuss a number of invariants of knotoids based on concepts from virtual knot theory. Knotoids are a new branch of classical knot theory and they promise to provide a way to measure the “knottiness” of open interval embeddings in three space. This talk is joint work with Neslihan Gugumcu.

## Geometria em Lisboa

$G_2$-instantons on noncompact $G_2$-manifolds.

*Gonçalo Oliveira*, Duke University.

## Abstract

I will report on joint work with Jason Lotay on some existence and nonexistence results for $G_2$-instantons. I shall compare the behavior of $G_2$-instantons for two distinct $G_2$-holonomy metrics on $\mathbb{R}^4\times S^3$.

## Partial Differential Equations

Minicourse on Microlocal Analysis.

*Jorge Drumond Silva*, Instituto Superior Técnico.

## Abstract

This will be the thirteenth session of a course on Microlocal Analysis.

## Functional Analysis, Linear Structures and Applications

Polytopes, slack ideals and psd-minimality.

*João Gouveia*, Universidade de Coimbra.

## Abstract

The slack ideal is an algebraic object that codifies the geometry of a polytope. This notion was motivated by the study of psd-minimality of polytopes: A d-polytope is said to be psd-minimal if it can be written as a projection of a slice of the cone of d+1 by d+1 positive semidefinite matrices, the smallest possible size for which this may happen. We will show how the slack ideal can be used to extract conditions on psd-minimality, completing the classification of psd-minimal 4- polytopes, settling some open questions and creating new ones. We will proceed to explore the relation of slack ideals and toric ideals of graphs and present some ongoing work and open questions.

## IST Lecture Series in Algebraic Geometry & Physics

A review of Berezin-Toeplitz quantization.

*Martin Schlichenmaier*, University of Luxembourg.

## Abstract

In this lecture course I will introduce the Berezin-Toeplitz (BT) quantization scheme. This scheme is adapted if the phase space manifold is a Kaehler manifold. The BT scheme includes and relates both operator quantization and deformation quantization.

I will define the basic objects and explain the main results. In particular it will turn out that the BT operator quantization has the correct semiclassical limit (at least in the compact Kaehler case).

If time permits I will also discuss coherent states a la Berezin-Rawnsley, covariant and contravariant Berezin symbols and the Berezin transform which is related to the Bergman kernel.

Depending on the wishes of the audience other related topics can be presented.

**See also**: http://dx.doi.org/10.1155/2010/927280

## Analysis, Geometry, and Dynamical Systems

On (special versions of) the Hartshorne Conjecture on Complete Intersections.

*Francesco Russo*, University of Catania.

## Abstract

We shall present some general techniques for studying projective embedded manifolds uniruled by lines, based on the Hilbert scheme of lines passing through a general point of the manifold and contained in it. The main applications will be the proofs of Hartshorne Conjecture for quadratic manifolds, of the classification of quadratic Hartshorne varieties, of the classification of Severi varieties. Our approach will show many connections between these problems, which were overlooked before, and also a uniform way of solving them. If time allows, we shall also discuss some open problems including the Barth-Ionescu Conjecture.

## IST Lecture Series in Algebraic Geometry & Physics

A review of Berezin-Toeplitz quantization.

*Martin Schlichenmaier*, University of Luxembourg.

## Abstract

In this lecture course I will introduce the Berezin-Toeplitz (BT) quantization scheme. This scheme is adapted if the phase space manifold is a Kaehler manifold. The BT scheme includes and relates both operator quantization and deformation quantization.

I will define the basic objects and explain the main results. In particular it will turn out that the BT operator quantization has the correct semiclassical limit (at least in the compact Kaehler case).

If time permits I will also discuss coherent states a la Berezin-Rawnsley, covariant and contravariant Berezin symbols and the Berezin transform which is related to the Bergman kernel.

Depending on the wishes of the audience other related topics can be presented.

**See also**: http://dx.doi.org/10.1155/2010/927280

## IST Lecture Series in Algebraic Geometry & Physics

A review of Berezin-Toeplitz quantization.

*Martin Schlichenmaier*, University of Luxembourg.

## Abstract

In this lecture course I will introduce the Berezin-Toeplitz (BT) quantization scheme. This scheme is adapted if the phase space manifold is a Kaehler manifold. The BT scheme includes and relates both operator quantization and deformation quantization.

I will define the basic objects and explain the main results. In particular it will turn out that the BT operator quantization has the correct semiclassical limit (at least in the compact Kaehler case).

If time permits I will also discuss coherent states a la Berezin-Rawnsley, covariant and contravariant Berezin symbols and the Berezin transform which is related to the Bergman kernel.

Depending on the wishes of the audience other related topics can be presented.

**See also**: http://dx.doi.org/10.1155/2010/927280

## Analysis, Geometry, and Dynamical Systems

Some loci of rational cubic fourfolds.

*Francesco Russo*, University of Catania.

## Abstract

We shall report on joint work with Michele Bolognesi and Giovanni Staglianò on the irreducible divisor $\mathcal C_{14}$ inside the moduli space of smooth cubic hypersurfaces in $\mathbb P^5$. A general point of $\mathcal C_{14}$ is, by definition, a smooth cubic fourfold containing a smooth quartic rational normal scroll (or, equivalently, a smooth quintic del Pezzo surfaces) so that it is rational. We shall prove that **every** cubic fourfold contained in $\mathcal C_{14}$ is rational.

In passing we shall review and put in modern terms some ideas of Fano, yielding a geometric insight to some known results on cubic fourfolds, e.g. the Beauville-Donagi isomorphism, and discuss also the connections of our results with the recent examples about the bad behavior of rationality in smooth families of fourfolds.

## String Theory

Multi matrix systems.

*João Rodrigues*, University of the Witwatersrand.

## Abstract

We consider the quantum mechanics of an even number of space indexed hermitian matrices. Upon complexification, we show that a closed subsector naturally parametrized by a matrix valued radial coordinate has a description in terms of non interacting

## Analysis, Geometry, and Dynamical Systems

To be announced.

*David Krejciric*, Czech Technical University.