Thu, 22 May 2025
16:00
L5

Liquidity Competition Between Brokers and an Informed Trader

Ryan Donnelly
(King's College London)
Abstract

We study a multi-agent setting in which brokers transact with an informed trader. Through a sequential Stackelberg-type game, brokers manage trading costs and adverse selection with an informed trader. In particular, supplying liquidity to the informed traders allows the brokers to speculate based on the flow information. They simultaneously attempt to minimize inventory risk and trading costs with the lit market based on the informed order flow, also known as the internalization-externalization strategy. We solve in closed form for the trading strategy that the informed trader uses with each broker and propose a system of equations which classify the equilibrium strategies of the brokers. By solving these equations numerically we may study the resulting strategies in equilibrium. Finally, we formulate a competitive game between brokers in order to determine the liquidity prices subject to precommitment supplied to the informed trader and provide a numerical example in which the resulting equilibrium is not Pareto efficient.

Tue, 06 May 2025

14:00 - 15:00
L4

Optimally packing Hamilton cycles in random directed digraphs

Adva Mond
(King's College London)
Abstract

At most how many edge-disjoint Hamilton cycles does a given directed graph contain? It is easy to see that one cannot pack more than the minimum in-degree or the minimum out-degree of the digraph. We show that in the random directed graph $D(n,p)$ one can pack precisely this many edge-disjoint Hamilton cycles, with high probability, given that $p$ is at least the Hamiltonicity threshold, up to a polylog factor.

Based on a joint work with Asaf Ferber.

Wed, 19 Feb 2025
16:00
L6

Graph manifolds and their Thurston norm

Alessandro Cigna
(King's College London)
Abstract

A classical approach to studying the topology of a manifold is through the analysis of its submanifolds. The realm of 3-manifolds is particularly rich and diverse, and we aim to explore the complexity of surfaces within a given 3-manifold. After reviewing the fundamental definitions of the Thurston norm, we will present a constructive method for computing it on Seifert fibered manifolds and extend this approach to graph manifolds. Finally, we will outline which norms can be realized as the Thurston norm of some graph manifold and examine their key properties.

Thu, 30 Jan 2025
16:00
Lecture Room 4

3-descent on genus 2 Jacobians using visibility

Lazar Radicevic
(King's College London)
Abstract

We show how to explicitly compute equations for everywhere locally soluble 3-coverings of Jacobians of genus 2 curves with a rational Weierstrass point, using the notion of visibility introduced by Cremona and Mazur.  These 3-coverings are abelian surface torsors, embedded in the projective space $\mathbb{P}^8$ as degree 18 surfaces. They have points over every $p$-adic completion of $\mathbb{Q}$, but no rational points, and so are counterexamples to the Hasse principle and represent non-trivial elements of the Tate-Shafarevich group.  Joint work in progress with Tom Fisher.

Tue, 11 Mar 2025
14:00
L6

Gelfand--Kirillov dimension and mod p cohomology for quaternion algebras

Andrea Dotto
(King's College London)
Abstract

The Gelfand--Kirillov dimension is a classical invariant that measures the size of smooth representations of p-adic groups. It acquired particular relevance in the mod p Langlands program because of the work of  Breuil--Herzig--Hu--Morra--Schraen, who computed it for the mod p cohomology of GL_2 over totally real fields, and used it to prove several structural properties of the cohomology. In this talk, we will present a simplified proof of this result, which has the added benefit of working unchanged for nonsplit inner forms of GL_2. This is joint work with Bao V. Le Hung.

Thu, 07 Nov 2024

17:00 - 18:00
L3

Ramification Theory for Henselian Valued Fields

Vaidehee Thatte
(King's College London)
Abstract

Ramification theory serves the dual purpose of a diagnostic tool and treatment by helping us locate, measure, and treat the anomalous behavior of mathematical objects. In the classical setup, the degree of a finite Galois extension of "nice" fields splits up neatly into the product of two well-understood numbers (ramification index and inertia degree) that encode how the base field changes. In the general case, however, a third factor called the defect (or ramification deficiency) can pop up. The defect is a mysterious phenomenon and the main obstruction to several long-standing open problems, such as obtaining resolution of singularities. The primary reason is, roughly speaking, that the classical strategy of "objects become nicer after finitely many adjustments" fails when the defect is non-trivial. I will discuss my previous and ongoing work in ramification theory that allows us to understand and treat the defect.

Mon, 25 Nov 2024
16:00
C3

Gap distributions and the Metric Poissonian Property 

Sophie Maclean
(King's College London)
Abstract
When studying dilated arithmetic sequences, it is natural to wonder about their distribution. Whilst it is relatively achievable to ascertain whether the resulting sequence is equidistributed, is it much more difficult to say much about gap size between consecutive elements of the new set? In this talk I will explore the gap distributions in dilated arithmetic sequences modulo 1, including what it means for a sequence to have the metric poissonian property. I will also give an overview of the current progress and what I am aiming to discover in my own work.
 
 
Tue, 29 Oct 2024

14:00 - 15:00
L4

Lower tails for triangle counts in the critical window

Matthew Jenssen
(King's College London)
Abstract

The classical lower-tail problem for triangles in random graphs asks the following: given $\eta\in[0,1)$, what is the probability that $G(n,p)$ contains at most $\eta$ times the expected number of triangles?  When $p=o(n^{-1/2})$ or $p = \omega(n^{-1/2})$ the asymptotics of the logarithm of this probability are known via Janson's inequality in the former case and regularity or container methods in the latter case.

We prove for the first time asymptotic formulas for the logarithm of the lower tail probability when $p=c n^{-1/2}$ for $c$ constant.  Our results apply for all $c$ when $\eta \ge 1/2$ and for $c$  small enough when $\eta < 1/2$.  For the special case $\eta=0$ of triangle-freeness, our results prove that a phase transition occurs as $c$ varies (in the sense of a non-analyticity of the rate function), while for $\eta \ge 1/2$ we prove that no phase transition occurs.

Our method involves ingredients from algorithms and statistical physics including rapid mixing of Markov chains and the cluster expansion.  We complement our asymptotic formulas with efficient algorithms to approximately sample from $G(n,p)$ conditioned on the lower tail event.

Joint work with Will Perkins, Aditya Potukuchi and Michael Simkin.

Thu, 07 Nov 2024
16:00
L3

E-functions and their roots

Peter Jossen
(King's College London)
Abstract
E-functions are a special class of entire function given by power series with algebraic coefficients, particular examples of which are the exponential function or Bessel functions. They were introduced by Siegel in the 1930's.
 
While special values of E-functions are relatively well understood, their roots remain mysterious in many ways. I will explain how roots of E-functions are distributed in the complex plane (essentially a Theorem of Pólya), and discuss a couple of related questions and conjectures. From the roots of an E-function one may also fabricate a "spectral" zeta function, which turns out to have some interesting properties.
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