07/12/26 3:30 KIN 206: Matthew Jensen
Title: TBA
18/06/26 2pm KIN 625 : Konstanze Rietsch
Title: Totally Positive Toeplitz Matrices: Classical and Modern
Abstract: The subject of upper-triangular totally positive Toeplitz matrices turns out to be a rich one, relating in the infinite case to the infinite symmetric group and its representations (classical theory), and in the finite case to flag varieties and their quantum cohomology rings (modern). We describe the asymptotics of the modern theory and tropicalise both to uncover a third relation, to the parametrisation of Lusztig's canonical basis.
04/11/25 2pm KIN 625: Jean Lagacé
Title: What about a shape can we hear?
Abstract: The abstract of every colloquium about spectral geometry starts with the celebrated question "Can one hear the shape of a drum?", as popularised by Mark Kac in 1966. Invariably, the second (or maybe third) sentence of that abstract will announce that the answer is a resounding "No!", as Carolyn Gordon, David Webb and Scott Wolpert discovered in 1992.
While we cannot hear the shape of a drum, there are still plenty of properties of a shape that can be uncovered from its frequencies and modes of vibration. In this talk, I will lead you through more and more relaxed definitions of what it means to "hear" a shape, and discuss various properties, called spectral invariants, that can be detected via the frequencies, which will be eigenvalues of some differential operators.
Hopefully, we will be able to uncover many spectral invariants, be they geometric, topological or analytic in nature, from those historically found by Lord Rayleigh in the 19th century, up to those I discovered this year, in joint work with Alix Deleporte (Orsay) and Leonid Parnovski (UCL).
12/06/25 2pm K6.29: Rachel Newton
Title: Diophantine equations and when to quit trying to solve them
Abstract: The study of integer or rational solutions to polynomial equations with integer coefficients is one of the oldest areas of mathematics and remains a very active field of research. The most basic question we can ask about such an equation is whether its set of rational solutions is empty or not. This turns out to be a very hard question! I will discuss some modern methods for proving that the set of rational solutions is empty. Along the way, I will describe some joint work with Martin Bright concerning the wild part of the Brauer–Manin obstruction.
08/11/24 3pm S-3.20: Dionysios Anninos
Title: Toward Theories of Accelerated Expansion
Abstract: There are significant indications that at large scales our Universe experiences epochs of accelerated expansion. During such epochs, quantum effects may play a vital role, which in turn lead to foundational challenges. We investigate the problem through the lens of modern developments in theoretical physics. In light of our audience, I will try to emphasise the relevant mathematics as we go along.
20/09/24 3pm K2.31 (Nash Lecture Theatre): Francesca Romana Crucinio
Title: A connection between sampling and optimisation Abstract: This talk explores the connections between tempering (for Sequential Monte Carlo; SMC) and entropic mirror descent to sample from a target probability distribution whose unnormalized density is known. We establish that tempering SMC corresponds to entropic mirror descent applied to the reverse Kullback-Leibler (KL) divergence and obtain convergence rates for the tempering iterates. Our result motivates the tempering iterates from an optimization point of view, showing that tempering can be seen as a descent scheme of the KL divergence with respect to the Fisher-Rao geometry, in contrast to Langevin dynamics that perform descent of the KL with respect to the Wasserstein-2 geometry. We exploit the connection between tempering and mirror descent iterates to justify common practices in SMC and derive adaptive tempering rules that improve over other alternative benchmarks in the literature.
14/06/24 3pm K6.29 (Anatomy Lecture Theatre)
Speaker: Mehdi Yazdi
Title: Unknot Recognition, Three-dimensional Manifolds, and Algorithms
Abstract: One of the oldest problems in low-dimensional topology is the unknot recognition problem, posed by Max Dehn in 1910: Is there an algorithm to decide if a given knot can be untangled? You know that this is a challenging problem if you owned a pair of earphones that are tangled! The unknot recognition problem was highlighted by Alan Turing in his last article in 1954, and the first solution was given by Wolfgang Haken in 1961. However, it remains widely open whether there exists a polynomial time algorithm to detect the unknot. The current state-of-the-art is Lackenby’s announcement for a quasi-polynomial time algorithm, which puts it in similar standing to the graph isomorphism problem. I will discuss what is known about the unknot recognition, how it is related to the theory of foliations on three-dimensional manifolds, as well as recent developments on related algorithmic problems.
17/05/24 3pm K6.29 (Anatomy Lecture Theatre)
Speaker: Benjamin Doyon
Title: The emergence of hydrodynamics in many-body systems
Abstract: One of the most important problems of modern science is that of emergence. How do laws of motion emerge at large scales of space and time, from much different laws at small scales? A foremost example is the theory of hydrodynamics. Take molecules in air, which simply follow Newton’s equations. When there are very many of them, these equations becomes untractable; seeking the knowledge of each molecule’s individual trajectory is completely impractical. Happily it is also unnecessary. At our human scale, new, different equations emerge for aggregate quantities: those of hydrodynamics. And these are apparently all we need to know in order to understand the weather! Despite its conceptual significance, the passage from microscopic dynamics to hydrodynamics remains a notorious open problem of mathematical physics. This goes much beyond molecules in air: similar principles hold very generally, such as in quantum gases and spin lattices, where the resulting equations themselves can be very different. In particular, integrable models, where an extensive mathematical structure allows us to make progress, admit an entirely new universality class of hydrodynamic equations. In this talk, I will discuss in a pedagogical and mathematically precise fashion the general problem and principles of hydrodynamics as an emergent theory, and some recent advances in our understanding, including those obtained in integrable models.