Colloquium

Speaker: Matthew Satriano (Waterloo)

"Galois closures and small components of the Hilbert schemes of points"

Time: 15:30 - 16:30

Room: MC 108

Bhargava and the speaker introduced a functorial Galois closure operation for finite-rank ring extensions, generalizing constructions of Grothendieck and Katz-Mazur. In this talk, we use Galois closures to construct new components of Hilbert schemes of points, which are fundamental objects in algebraic geometry whose component structure is largely mysterious. We answer a 35 year old open problem posed by Iarrobino by constructing an infinite family of low dimensional components. This talk is based on joint work with Andrew Staal.

Ph.D. Presentation

Speaker: Oussama Hamza (Western)

"Special quotients of Absolute Galois Groups with applications in Number Theory and Pythagorean fields."

Time: 08:00 - 09:00

Room: Zoom

This talk aims to present the results obtained by Oussama Hamza, during his PhD studies, and his collaborators: Christian Maire, Jan Minac and Nguyen Duy Tân.

Their work precisely focuses on realisation of pro-p Galois groups over some fields with specific properties for a fixed prime p: especially filtrations and cohomology. Hamza was particularly interested on Number and Pythagorean fields.

This talk will mostly deal with the last results obtained by Hamza and his collaborators on Formally real Pythagorean fields of finite type (RPF). For this purpose, they introduced a class of pro-2 groups, which is called $\Delta$-RAAGs, and studied some of their filtrations. Using previous work of Minac and Spira, Hamza and his collaborators showed that every pro-2 Absolute Galois group of a RPF is $\Delta$-RAAG. Conversely if a group is $\Delta$-RAAG and a pro-2 Absolute Galois group, then the underlying field is necessarily RPF. This gives us a new criterion to detect Absolute Galois groups.

Finally, we also show that the pro-2 Absolute Galois group of a RPF satisfy the Kernel unipotent conjecture jointly introduced, by Minac and Tân, with the Massey vanishing conjecture, which attracted a lot of interest.

Colloquium

Speaker: Rui Dong (Leiden Linguistics Center)

"Linguistics from a topological viewpoint"

Time: 15:30 - 16:30

Room: MC 107

Typological databases in linguistics are usually categorical-valued. As a result, it is difficult to have a clear visualization of the data. In this talk, I will describe a workflow to analyse the topological shapes of South American languages by applying multiple correspondence analysis technique and topological data analysis methods

Please note: This is a Zoom talk!

Ph.D. Presentation

Speaker: Manimugdha Saikia (Western)

"Analytic properties of quantum states on manifolds"

Time: 09:00 - 10:00

Room: MC 204

The aim of this talk is to outline the results obtained by me (during my PhD studies) and my collaborators.

In quantum information theory, there is a rich collection of analytic tools to study tensor product of Hilbert spaces. Geometric quantization attaches Hilbert spaces to symplectic manifolds. The principal objective of the first set of study, supervised by T Barron, is to investigate how the geometry of the manifold influences the quantum information theoretic aspects of the Hilbert space and vice versa. For instance, in one of the works, we presented an asymptotic result for the average entropy over all the pure states on the Hilbert space , where is a Hermitian ample line bundle over a compact complex manifold . In another work, we associated states with subsets of a product of two integral Kahler manifolds using a quantum line bundle in a particular manner. We proved that the state associated this way is separable when the subset is a finite union of products.

In the second part of the talk, we present a work on quantum circuit synthesis, joint with A.R. Kalra, D. Valluri, S. Winnick and J. Yard. In classical computing, we choose a small set of special gates (known as a universal gate set) and make circuits using these gates to generate any classical gates (Boolean functions). However, the quantum version of circuit synthesis is a bit more complicated. In this talk, we will introduce what it means to be a universal gate set in quantum computing and see some examples. Finally, we shall present our construction of an exact circuit synthesis algorithm of unitaries in the groups and over the multi-qutrit Clifford+T universal gate set with the of help of ancilla.

Ph.D. Presentation

Speaker: Alejandro Santacruz Hidalgo (Western)

"Monotone functions on general measure spaces"

Time: 11:00 - 12:00

Room: MC 108

Monotone functions over the real numbers are very well-behaved compared to general measurable functions. Consequently, a wide variety of techniques and applications are in place for working with them. In this talk, we explore the notion of an ordered core, which allows us to define core decreasing functions and generalize monotone functions to general measure spaces without reliance on a strict ordering among elements.

We will begin by introducing a definition of monotone functions compatible with the ordered core. This allows us to extend the down space construction, a variant of the Köthe dual restricted to positive decreasing functions, to all measure spaces. We will look at their associate spaces and their relationship with a suitable version of the least decreasing majorant construction in this more general setting. We will discuss the interpolation structure of these spaces and find strong similarities to the real line case; the down spaces corresponding to L1 and L∞ form an exact Calderón couple and as a consequence, we can describe all their exact interpolation spaces in terms of the K−functional.

We illustrate the versatility and adaptability of this generalized perspective on decreasing functions by proving a new characterization for the boundedness of an abstract Hardy operator between L^1 to L^q with general measures.