Persistent homology is an algebraic topology tool that has gained a lot of relevance in topological data analysis during the past few years. During this lecture we will introduce this concept from a categorical point of view and use category theory techniques in order to prove what is known as stability theorem.

Persistent homology is an algebraic topology tool that has gained a lot of relevance in topological data analysis during the past few years. During this lecture we will introduce this concept from a categorical point of view and use category theory techniques in order to prove what is known as stability theorem.

The set of all extreme points of the closed unit ball of a unital \(C^*\)-algebra \(A\) was identied by R.V. Kadison as the maximal partial isometries in \(A\). Let \(u\) be an element in \(A\), it is said that \(u\) is unitary if \(uu^* = u^*u = 1_A\), that is, if \(u\) is invertible in \(A\) with \(u^{-1}= u\). It is clear that any unitary is an extreme point in \(A\), however the reciprocal statement is not in general true. In 1991 G.K. Pedersen provided a characterisation of the unitary elements of a unital \(C^*\)-algebra among the extreme points of its closed unit ball in terms of the distance to the group of invertible elements. We present an alternative proof of this result based on the dierent natures coexisting in a unital \(C^*\)- algebra. It is a nice example of how to combine all the knowledge existing in the setting of associative and Jordan algebras with the more general triple theory.

The set of all extreme points of the closed unit ball of a unital \(C^*\)-algebra \(A\) was identied by R.V. Kadison as the maximal partial isometries in \(A\). Let \(u\) be an element in \(A\), it is said that \(u\) is unitary if \(uu^* = u^*u = 1_A\), that is, if \(u\) is invertible in \(A\) with \(u^{-1}= u\). It is clear that any unitary is an extreme point in \(A\), however the reciprocal statement is not in general true. In 1991 G.K. Pedersen provided a characterisation of the unitary elements of a unital \(C^*\)-algebra among the extreme points of its closed unit ball in terms of the distance to the group of invertible elements. We present an alternative proof of this result based on the dierent natures coexisting in a unital \(C^*\)- algebra. It is a nice example of how to combine all the knowledge existing in the setting of associative and Jordan algebras with the more general triple theory.

Calabi-Yau (CY) manifolds have received increasing interest with the advent of string theory, where they serve as a promising candidate for the extra six dimensions of the ten-dimensional theory. A particular interesting phenomenon is that of mirror symmetry. This is a duality between the complex structure of a CY manifold and the symplectic structure of its mirror CY manifold. While mirror symmetry was discovered in physics, it has also become a research area onto itself in geometry which have led to a fruitful collaboration between physicists and mathematicians.

We will begin by introducing some of the basics of symplectic and complex geometry. This will allow us to define CY manifolds and discuss some of their properties. We will end by speaking about mirror symmetry and some of the conjectures and approaches towards it.

Calabi-Yau (CY) manifolds have received increasing interest with the advent of string theory, where they serve as a promising candidate for the extra six dimensions of the ten-dimensional theory. A particular interesting phenomenon is that of mirror symmetry. This is a duality between the complex structure of a CY manifold and the symplectic structure of its mirror CY manifold. While mirror symmetry was discovered in physics, it has also become a research area onto itself in geometry which have led to a fruitful collaboration between physicists and mathematicians.

We will begin by introducing some of the basics of symplectic and complex geometry. This will allow us to define CY manifolds and discuss some of their properties. We will end by speaking about mirror symmetry and some of the conjectures and approaches towards it.

We will give a brief overview of some elementary Fourier analytical techniques in arithmetic combinatorics, and apply them to the case of large Sidon sets. More concretely, we will sketch why extremal Sidon sets are pseudorandom and what this means, and explain how one can use a transference principle to prove the existence of solutions to equations in large Sidon sets.

In 2012, Voronov proved a Poincaré lemma for differential forms on a contractible manifold taking values in a differential graded Lie algebra (dgla) and suggested that an analogous result may hold when the dgla is replaced by an L-infinity algebra. This talk aims to explain this result and how some model category ideas can help to prove it. If time permits, we will also see how Voronov's result can be generalized to L-infinity algebras. No knowledge of L-infinity algebras or model categories will be assumed.

In 2012, Voronov proved a Poincaré lemma for differential forms on a contractible manifold taking values in a differential graded Lie algebra (dgla) and suggested that an analogous result may hold when the dgla is replaced by an L-infinity algebra. This talk aims to explain this result and how some model category ideas can help to prove it. If time permits, we will also see how Voronov's result can be generalized to L-infinity algebras. No knowledge of L-infinity algebras or model categories will be assumed.

The initial motivation for the study of quantum groups was a certain problem in statistical mechanics – the quantum Yang-Baxter equation – which concerns the scattering of particles on a line. Looking at the same problem but with an added boundary, the analogue of the QYBE is the (quantum) reflection equation. A family of universal solutions to the reflection equation can be constructed for every representation of a finite dimensional Lie algebra, depending on certain combinatorial data (X,). The aim of this talk is to give an overview of the above, focussing primarily on the simplest case: U_q(sl_2).

The initial motivation for the study of quantum groups was a certain problem in statistical mechanics – the quantum Yang-Baxter equation – which concerns the scattering of particles on a line. Looking at the same problem but with an added boundary, the analogue of the QYBE is the (quantum) reflection equation. A family of universal solutions to the reflection equation can be constructed for every representation of a finite dimensional Lie algebra, depending on certain combinatorial data (X,). The aim of this talk is to give an overview of the above, focussing primarily on the simplest case: U_q(sl_2).

This talk will be split in two parts. The first one will be dedicated to introducing the definitions of compact quantum groups in the sense of Woronowicz, and some objects of noncommutative probability. In the second one it will be extended the purely algebraic setting of noncommutative Lévy processes to the analytic case by simply exploiting the fact that in the classical version the whole information is encoded in the corresponding convolution semigroup. In fact, we prove that Lévy processes on compact quantum groups actually exist via the generalization of stochastic Itô integrals, set by Hudson and Parthasarathy. Further, an analogue to the characterization of Lévy processes on involutive bialgebras, made by means of Schürmann’s theory, will be presented for the context of compact quantum groups. Finally, it is shown that there exists a one-to-one correspondence between norm continuous quantum Lévy processes and the Fock-space ones.

In this talk we will derive the Frenet-Serret formulas of differential geometry. We will then describe an interesting application of them as a tool to produce a lax pair for a nonlinear integrable system (nonlinear Schrodinger). Finally, we will try to give some geometric motivation for lax pairs of this form - they can define a connection which is in some sense curvature free.

Theorems from pure differential geometry are the foundation on which a field as applied as computer graphics is built. We will look at an example of this from the lens of discrete differential geometry: a concept (curvature or developability of surfaces) will be introduced theoretically, motivated by its practical applications and then discretized for its manipulation in the form of computer code. We will see how, surprisingly, theoretically equivalent formulations lead to very different discrete behaviours, and analyze the benefits and drawbacks of each.

In the 1920s, Élie Cartan asked: which are the Riemannian manifolds whose curvature tensor is invariant under the action of its holonomy group? The answer to this question brought about the notion of symmetric space which arises in a broad diversity of situations in both mathematics and physics. Moreover, totally geodesic submanifolds in symmetric spaces are those submanifolds with the simplest geometry and admit a nice algebraic characterization in terms of Lie algebras. In this talk, I will present a classification of totally geodesic submanifolds in products of symmetric spaces of rank one.