While I was on research leave in autumn 2024, the seminar was organised by
Seminar Talks - Autumn 2024
Thursday, 3rd October 2024 15:10 - 16:00
Shanshan Hua (University of Oxford)
Nonstable K-theory for Z-stable C*-algebras
In Jiang's unpublished paper (1997), it is shown that any $\mathcal{Z}$-stable C*-algebra $A$ is $K_1$-injective and $K_1$-surjective, which means that $K_1(A)$ can be calculated by looking at homotopy equivalence classes of $U(A)$, without matrix amplifications. Moreover, for such $A$, higher homotopy groups of $U(A)$ are isomorphic to $K_0(A)$ or $K_1(A)$, depending on the dimension of the higher homotopy group. In this talk, I will present Jiang's result for $\mathcal{Z}$-stable C*-algebras. Moreover, I will explain briefly our new strategies to reprove his theorems using an alternative picture of the Jiang-Su algebra as an inductive limit of generalized dimension drop algebras.
Nonstable K-theory for Z-stable C*-algebras
In Jiang's unpublished paper (1997), it is shown that any $\mathcal{Z}$-stable C*-algebra $A$ is $K_1$-injective and $K_1$-surjective, which means that $K_1(A)$ can be calculated by looking at homotopy equivalence classes of $U(A)$, without matrix amplifications. Moreover, for such $A$, higher homotopy groups of $U(A)$ are isomorphic to $K_0(A)$ or $K_1(A)$, depending on the dimension of the higher homotopy group. In this talk, I will present Jiang's result for $\mathcal{Z}$-stable C*-algebras. Moreover, I will explain briefly our new strategies to reprove his theorems using an alternative picture of the Jiang-Su algebra as an inductive limit of generalized dimension drop algebras.
Thursday, 10th October 2024 15:10 - 16:00
Prachi Sahjwani (Cardiff University)
Stability of geometric inequalities in various spaces
In this talk, I will discuss the stability of two inequalities: the "Alexandrov-Fenchel inequalities in hyperbolic space" and "Minkowski's inequality in a general warped product space." I will give a brief overview of both inequalities and their respective stability problems. To explain what I mean by stability, I will first discuss it in the context of the isoperimetric inequality, which is a special case of the Alexandrov-Fenchel inequalities. I will also briefly discuss the proofs in both cases. This is joint work with Prof. Dr. Julian Scheuer and is based on the work of Wang/Xia on Alexandrov-Fenchel inequalities and the work of Brendle/Hung/Wang and Scheuer on Minkowski's inequality.
Stability of geometric inequalities in various spaces
In this talk, I will discuss the stability of two inequalities: the "Alexandrov-Fenchel inequalities in hyperbolic space" and "Minkowski's inequality in a general warped product space." I will give a brief overview of both inequalities and their respective stability problems. To explain what I mean by stability, I will first discuss it in the context of the isoperimetric inequality, which is a special case of the Alexandrov-Fenchel inequalities. I will also briefly discuss the proofs in both cases. This is joint work with Prof. Dr. Julian Scheuer and is based on the work of Wang/Xia on Alexandrov-Fenchel inequalities and the work of Brendle/Hung/Wang and Scheuer on Minkowski's inequality.
Thursday, 17th October 2024 15:10 - 16:00
Xavier Crean (Swansea University)
Topological Data Analysis of Abelian Magnetic Monopoles in Gauge Theory
It has been widely argued that non-trivial topological features of the Yang-Mills vacuum are responsible for colour confinement. However, both analytical and numerical progress have been limited by the lack of understanding of the nature of relevant topological excitations in the full quantum description of the model. In this talk, we shall explain how Topological Data Analysis (TDA) may be used to quantitatively analyse Abelian magnetic monopoles across the deconfinement phase transition in lattice gauge theory. The talk will give a background to the physics and introduce the necessary methods from TDA.
Topological Data Analysis of Abelian Magnetic Monopoles in Gauge Theory
It has been widely argued that non-trivial topological features of the Yang-Mills vacuum are responsible for colour confinement. However, both analytical and numerical progress have been limited by the lack of understanding of the nature of relevant topological excitations in the full quantum description of the model. In this talk, we shall explain how Topological Data Analysis (TDA) may be used to quantitatively analyse Abelian magnetic monopoles across the deconfinement phase transition in lattice gauge theory. The talk will give a background to the physics and introduce the necessary methods from TDA.
Thursday, 24th October 2024 15:10 - 16:00
Lewis Topley (University of Bath)
Lusztig-Spaltenstein induction of nilpotent orbits
Parabolic induction is the most ubiquitous construction in Lie theoretic representation theory, allowing us to construct representations from a nice class subobjects, whenever we work with Lie algebras, algebraic groups, quantum groups, Weyl groups or finite groups of Lie type. In fact, this may be the only technique for generating interesting representations (discuss). At the same time representations are often associated to nilpotent orbits in Lie algebras. Examples of the latter relationship include Springer theory, Whittaker modules and finite W-algebras. The analogue of parabolic induction for nilpotent orbits is called Lusztig-Spaltenstein induction, and I like it very much. In this talk I will introduce this construction, and explain a theorem of mine (joint with Neil Saunders) which describes the combinatorial behaviour with respect to Springer fibres in type A.
Lusztig-Spaltenstein induction of nilpotent orbits
Parabolic induction is the most ubiquitous construction in Lie theoretic representation theory, allowing us to construct representations from a nice class subobjects, whenever we work with Lie algebras, algebraic groups, quantum groups, Weyl groups or finite groups of Lie type. In fact, this may be the only technique for generating interesting representations (discuss). At the same time representations are often associated to nilpotent orbits in Lie algebras. Examples of the latter relationship include Springer theory, Whittaker modules and finite W-algebras. The analogue of parabolic induction for nilpotent orbits is called Lusztig-Spaltenstein induction, and I like it very much. In this talk I will introduce this construction, and explain a theorem of mine (joint with Neil Saunders) which describes the combinatorial behaviour with respect to Springer fibres in type A.
Thursday, 31st October 2024 15:10 - 16:00
Mykola Matviichuk (Imperial)
New quantum projective spaces from deformations of q-polynomial algebras
I will discuss how to construct a large collection of “quantum projective spaces”, in the form of Koszul, Artin-Schelter regular quadratic algebras with the Hilbert series of a polynomial ring. I will do so by starting with the toric ones (the q-polynomial algebras), and then deforming their relations using a diagrammatic calculus, proving unobstructedness of such deformations under suitable nondegeneracy conditions. Time permitting, I will show that these algebras coincide with the canonical quantizations of corresponding families of quadratic Poisson structures. This provides new evidence to Kontsevich's conjecture about convergence of his deformation quantization formula. This is joint work with Brent Pym and Travis Schedler.
New quantum projective spaces from deformations of q-polynomial algebras
I will discuss how to construct a large collection of “quantum projective spaces”, in the form of Koszul, Artin-Schelter regular quadratic algebras with the Hilbert series of a polynomial ring. I will do so by starting with the toric ones (the q-polynomial algebras), and then deforming their relations using a diagrammatic calculus, proving unobstructedness of such deformations under suitable nondegeneracy conditions. Time permitting, I will show that these algebras coincide with the canonical quantizations of corresponding families of quadratic Poisson structures. This provides new evidence to Kontsevich's conjecture about convergence of his deformation quantization formula. This is joint work with Brent Pym and Travis Schedler.
Thursday, 7th November 2024 15:10 - 16:00
Angela Capel Cuevas (Cambridge)
Quantum Markov Semigroups and Modified Logarithmic Sobolev Inequalities
A dissipative evolution of an open quantum many-body system weakly coupled to an environment can be modelled by a quantum Markov semigroup, and its mixing time can be bounded using optimal constants of certain quantum functional inequalities, such as the modified logarithmic Sobolev constant. In this talk, we will review the mathematical formalism of dissipative evolutions governed by Lindbladians, and we will summarize the current state of the art on mixing times when the system has an associated commuting Hamiltonian.
Quantum Markov Semigroups and Modified Logarithmic Sobolev Inequalities
A dissipative evolution of an open quantum many-body system weakly coupled to an environment can be modelled by a quantum Markov semigroup, and its mixing time can be bounded using optimal constants of certain quantum functional inequalities, such as the modified logarithmic Sobolev constant. In this talk, we will review the mathematical formalism of dissipative evolutions governed by Lindbladians, and we will summarize the current state of the art on mixing times when the system has an associated commuting Hamiltonian.
Thursday, 14th November 2024 15:10 - 16:00
Mohammad Al Attar (Durham University)
Deformations of Lipschitz Homeomorphisms
The theory of deforming homeomorphisms dates back to the 60’s with Černavskii showing the local contractibility of homeomorphism groups of certain classes of manifolds. In this talk I will discuss the history of the theory of deforming homeomorphisms and then discuss my research on deforming Lipschitz homeomorphisms, which extends the work of Siebenmann to the Lipschitz category.
Deformations of Lipschitz Homeomorphisms
The theory of deforming homeomorphisms dates back to the 60’s with Černavskii showing the local contractibility of homeomorphism groups of certain classes of manifolds. In this talk I will discuss the history of the theory of deforming homeomorphisms and then discuss my research on deforming Lipschitz homeomorphisms, which extends the work of Siebenmann to the Lipschitz category.
Thursday, 21st November 2024 15:10 - 16:00
Gillian Grindstaff (Oxford)
Applications of topology in tumours, glaciers, and Chicago
Topological data analysis (TDA) has been widely applied to perform generalised hole detection and shape profiling across a range of disciplines. The flagship technique in TDA is persistent homology of a Vietoris-Rips complex, where the shape of data is completely unknown, but filtrations over a known domain or incorporating additional structure can also be incredibly powerful in isolating topological signal from noise. In this talk we will tour three spatial applications: agent-based tumour-immune modeling, arctic melt-pond evolution, and resource deserts in urban areas. For each, we craft filtrations tailored to the available information, assumptions, and hypotheses, and interpret the results. We highlight computational limits and scientific questions such as the interpretation of persistence diagrams, comparison with benchmark techniques, model fitting, and feature extraction.
Applications of topology in tumours, glaciers, and Chicago
Topological data analysis (TDA) has been widely applied to perform generalised hole detection and shape profiling across a range of disciplines. The flagship technique in TDA is persistent homology of a Vietoris-Rips complex, where the shape of data is completely unknown, but filtrations over a known domain or incorporating additional structure can also be incredibly powerful in isolating topological signal from noise. In this talk we will tour three spatial applications: agent-based tumour-immune modeling, arctic melt-pond evolution, and resource deserts in urban areas. For each, we craft filtrations tailored to the available information, assumptions, and hypotheses, and interpret the results. We highlight computational limits and scientific questions such as the interpretation of persistence diagrams, comparison with benchmark techniques, model fitting, and feature extraction.
Thursday, 28th November 2024 15:10 - 16:00
Matthew Westaway (Bath)
Parabolic Induction in Lie Theory
Parabolic induction is an important tool in Lie theory arising in different contexts including nilpotent orbits, ideals in universal enveloping algebras, representations of reduced enveloping algebras, and representations of finite W-algebras. This talk will provide a general introduction to parabolic induction, requiring no prior background. Then in some of these contexts, we will look at more recent results regarding when this procedure is injective or surjective. This talk will be based on joint work with S. Goodwin and L. Topley.
Parabolic Induction in Lie Theory
Parabolic induction is an important tool in Lie theory arising in different contexts including nilpotent orbits, ideals in universal enveloping algebras, representations of reduced enveloping algebras, and representations of finite W-algebras. This talk will provide a general introduction to parabolic induction, requiring no prior background. Then in some of these contexts, we will look at more recent results regarding when this procedure is injective or surjective. This talk will be based on joint work with S. Goodwin and L. Topley.
Thursday, 5th December 2024 15:10 - 16:00
Jennifer Pi (University of Oxford)
From Classical to Free Entropy
There are two main notions of entropy from probability theory, which coincide in the classical setting. These two constructions led Voiculescu to define different notions of entropy in the setting of free probability, called microstates and non-microstates free entropy. In this talk, I'll discuss the classical setting as motivation, then move to the free setting. In joint work with David Jekel, we relate each notion of free entropy back to an appropriate limit of classical entropic quantities, which proves an inequality between the two notions of free entropy.
From Classical to Free Entropy
There are two main notions of entropy from probability theory, which coincide in the classical setting. These two constructions led Voiculescu to define different notions of entropy in the setting of free probability, called microstates and non-microstates free entropy. In this talk, I'll discuss the classical setting as motivation, then move to the free setting. In joint work with David Jekel, we relate each notion of free entropy back to an appropriate limit of classical entropic quantities, which proves an inequality between the two notions of free entropy.
