Speakers

Do we really need strictness?

Around 2006, Vladimir Voevodsky imagined a new foundation for mathematics where objects have natively an higher homotopical structure. This idea is embodied in the univalence axiom stating roughly that isomorphic structures are equal. This was the starting point of a new foundation for mathematics: the univalent foundation. 

Although a huge amount of mathematics was successfully established in univalent mathematics, not having a way to talk about the strict structure of the universe can also be a handicap. Some important constructions are still not available; they usually take the form of an infinite amount of data that we don't know how to fully encode internally to the theory. It seems that all these problems are different formulations of one and unique problem, famously stated in this way: are semi-simplicial types definable in univalent foundations?

In this talk I will first give a brief introduction to the univalent foundations and then explain the fundamental role of the semi-simplicial type problem and why it is so hard.

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Reproducing kernel Hilbert spaces

The aim of this talk is to provide a introduction to the theory of reproducing kernel spaces. Specifically, we will review their definition and basic properties and characterize the occurring kernel functions. Furthermore, we will discuss an example from the extension theory of

symmetric operators, which highlights the widespread use of reproducing kernels in contemporary mathematics.

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Quantum Contextuality for Beginners

Towards a local theory of probability

In quantum mechanics, not everything that can be measured can be measured at once. The predictions of quantum mechanics instead form a patchwork of probability distributions over different "contexts'' of simultaneously measurable variables ("observables''). Contextuality is the phenomenon that these ``local'' probability distributions may not always be realised as the marginals of some underlying global distribution: Not only can we not know the values of all observables simultaneously -- they cannot all even have values simultaneously.

To us mathematicians, contextuality is a classic local-to-global gluing problem, familiar from algebra, geometry, and topology -- but now for probability distributions. In this talk, I will give an introduction to contextuality (with examples!) and show how it naturally leads us to consider generalisations of the traditional foundations of probability theory, using the language of categories and sheaves. I promise it will not be scary.

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A generalized Liouville equation and magnetic stability

In this talk, we are going to see a connection between two different topics in mathematics and physics. The first one is Liouville equation which is motivated by geometry and seeks to classify conformally flat 2 dimensional manifolds. The second topic concerns the stability of a special type of quasi-particles, called anyons, which appear only in a 2 dimensional plane. We discuss our joint work with Douglas Lundholm and Dinh-Thi Nguyen, which develops a rigid analysis to resolve the stability problem. Then, we present some further developments and open problems in this area.

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Machine Learning, PDEs, and Quasistationary Distributions

I will give some general background on Machine Learning (ML) methods for solving Partial Differential Equations, with emphasis on Physics-Informed Neural Networks (PINNs). We will see how this approach can be used to approximate so-called Quasistationary Distributions of Ito diffusions with boundary extinction. I will highlight some difficulties that can arise, both with the ML method, as well as more generally with penalty-based approximation schemes. The talk is intended for a general mathematical audience and I will introduce the main players as they come up.

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Recent Quantum Algorithm Developments

In 1994 Peter Shor developed an efficient quantum algorithm for integer factorisation and computing discrete logarithms. These problems are believed to be hard to solve with a classical computer, and the security of much of the cryptography that is used today relies on this assumed hardness. Luckily, although quantum computers are under active development, they are still far too weak to run Shor's algorithm to solve any cryptographically interesting problem instances. It was therefore of great interest when Oded Regev last year introduced a quantum factoring algorithm with an asymptotic advantage over Shor's algorithm. In this talk, I will present the idea behind Shor's algorithm and this new, asymptotically more efficient, algorithm. I will also mention my joint work with Martin Ekerå, where we provide an extension of Regev's new algorithm to computing discrete logarithms.

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An introduction to topological data analysis

Introduced at the turn of the century, topological data analysis (TDA) is a relatively new field at the intersection of homological algebra and statistics. Its guiding principle is that the shape of data is something worth studying. In this talk, I will introduce the tools of homological algebra that topologists have developed to capture the idea of shape, show examples of data with interesting shapes, and explain the pipeline of persistent homology, one of the most popular methods in TDA. Only basic knowledge of linear algebra (mostly vector spaces over the field of two elements) is needed to follow this talk.

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