Math Colloquium

### Speaker: Þorsteinn Jónsson, University of Guelph

### Title: Variational Techniques for Learning Distributions of Data

Location: V-147 (VR-II)

Time: Monday 2 July at 10:30

### Abstract:

In this talk I will give an overview of a set of techniques that allow us to define latent variable models of data generating distributions.

To this end we introduce neural networks as an efficient and a surprisingly effective way of finding variational parameters that best fit some specified objective.

I will discuss a couple of different approaches that can be taken to define this objective and show you some interesting results.

Math Colloquium

### Speaker: Tony Guttmann, The University of Melbourne

### Title: On the number of Av(1324) permutations

Location: V-147 (VR-II)

Time: Monday 28 May at 10:50

### Abstract:

We give an improved algorithm for counting the number of 1324-avoiding permutations, resulting in 14 further terms of the generating function, which is now known to length 50.

We re-analyse the generating function and find compelling evidence that unlike other classical length-4 pattern-avoiding permutations, the generating function does not have a simple power-law singularity, but rather, the number of 1324-avoiding permutations of length n behaves as \(B\cdot \mu^n \cdot \mu_1^{\sqrt{n}} \cdot n^g\).

We estimate \(\mu = 11.600 \pm 0.003.\) The presence of the stretched exponential term \(\mu_1^{\sqrt{n}}\) is an unexpected feature of the conjectured solution, but we show that such a term is present in a number of other combinatorial problems.

(A.J. Guttmann with A.R. Conway and P. Zinn-Justin)

Math Phys seminar

### Speaker: Kevin John Torres Grosvenor, Niels Bohr Institute

### Title: Nonrelativistic Naturalness and the quest for Emergent Lorentz Symmetry

Location: V-147 (VR-II)

Time: Tuesday 22 May at 10:30

### Abstract:

I will discuss our proposal for a nonrelativistic solution to the Higgs mass hierarchy problem, its dependence on the emergence of various shift symmetries, and the role it might play in the search for a nonrelativistic theory of particle physics in which Lorentz symmetry emerges at “low” energies.

Math Colloquium

### Speaker: Michael Melgaard, University of Sussex

### Title: Rigorous mathematical results on nonlinear PDEs arising in Quantum Chemistry

Location: V-147 (VR-II)

Time: Friday 13 April at 13:30

### Abstract:

An introduction to electronic structure models is given and rigorous results are discussed on the existence of solutions (ground states and excited states) to weakly coupled, semi-linear elliptic PDEs with nonlocal operators arising in Hartree-Fock, Kohn-Sham and multiconfigurative many-particle models in quantum chemistry, in particular for systems with relativistic effects and external magnetic fields.

Math Phys seminar

### Speaker: Matteo Baggioli, Autonomous University of Barcelona

### Title: Holographic quantum phase transitions: time to get dirty!

Location: HB-5 (Háskólabíó)

Time: Monday 9 April at 10:50

### Abstract:

We study the effects of quenched one-dimensional disorder on the holographic Weyl semimetal quantum phase transition (QPT). We observe the smearing of the sharp QPT linked to the appearance of rare regions at the horizon where the local order parameter is non-zero. We discuss the role of the disorder correlation and we compare our results to the weakly coupled expectations from condensed matter theory and simulations. We analyze also the interplay of finite temperature and disorder and we find preliminary indications for the presence of log-oscillatory structures in the order parameter.

Math Colloquium

### Speaker: Anthony Thomas Lyons, Waterford Institute of Technology

### Title: The dressing method for the Camassa-Holm equation

Location: HB-5 (Háskólabíó)

Time: Monday 26 March at 10:50

### Abstract:

The Camassa-Holm equation is a nonlinear shallow water model which has been the focus of a great deal of mathematical research in hydrodynamics for the past two decades. This interest is in part due to the versatility of the system, being relevant as a fluid model possessing solutions which display wave-breaking along with global solutions in the form of soliton, peakon and cuspon solutions.

The inverse scattering transform has been successfully implemented to construct numerous global solutions of this system, and in this talk we present a recently developed variation of this method for the Camassa-Holm equation, known as the dressing method. This efficient implementation allows one to integrate several nonlinear hydrodynamical models, and in particular we shall outline the details of this new dressing method and use it to construct the one and two-soliton solutions of the Camassa-Holm equation.

Math Colloquium

### Speaker: Phillip Wesolek, Binghamton University

### Title: Totally disconnected locally compact groups: from examples to general theory

Location: HB-5 (Háskólabíó)

Time: Monday 5 March at 10:50

### Abstract:

Locally compact groups arise in many areas of mathematics as well as in physics. The study of locally compact groups splits into two cases: the connected groups and the totally disconnected groups. There is a rich and deep theory for the connected groups, which was developed over the last century. On the other hand, the study of totally disconnected locally compact groups groups only seriously began in the last 30 years, and moreover, these groups today appear to admit an equally rich and deep theory. In this talk, we will explore in details a wide variety of examples of totally disconnected locally compact groups. In particular, we discuss Lie groups over over the p-adic numbers, Galois groups, and automorphism groups of locally finite trees. We will then survey some recent results in the theory of totally disconnected locally compact groups.

Math Colloquium

### Speaker: Alexander Wendland, University of Warwick

### Title: Facially restricted graph colouring’s

Location: HB-5 (Háskólabíó)

Time: Monday 19 February at 10:50

### Abstract:

Arguably one of the best known theorems from combinatorics is the four colour theorem, stating that every planar graph can be coloured using at most four colours such that no edge connects two vertices of the same colour. In this talk I will discus variants on this theorem in particular list colouring’s and facial restriction’s on the colouring. In this, I present the method of discharging in Graph Theory, used to finally prove the four colour theorem nearly 140 years after it was first stated, which has been used to prove theorems elsewhere in Mathematics.

Math Colloquium

### Speaker: Hjörtur Björnsson, University of Iceland

### Title: Lyapunov functions for almost sure exponential stability

Location: VRII-158

Time: Monday 27 November at 15:00

### Abstract:

We present a generalization of results obtained by X. Mao in his book “Stochastic Differential Equations and Applications” (2008). When studying what Mao calls “almost sure exponential stability”, essentially a negative upper bound on the almost sure Lyapunov exponents, he works with Lyapunov functions that are twice continuously differentiable in the spatial variable and continuously differentiable in time. Mao gives sufficient conditions in terms of such a Lyapunov function for a solution of a stochastic differential equation to be almost surely exponentially stable. Further, he gives sufficient conditions of a similar kind for the solution to be almost surely exponentially unstable. Unfortunately this class of Lyapunov functions is too restrictive. Indeed, R. Khasminskii showed in his book “Stochastic Stability of Differential Equations” (1979/2012) that even for an autonomous stochastic differential equation with constant coefficients, of which the solution is stochastically stable and such that the deterministic part has an unstable equilibrium, there cannot exists a Lyapunov function that is differentiable at the origin. These restrictions are inherited by Mao’s Lyapunov functions. We therefore consider Lyapunov functions that are not necessarily differentiable at the origin and we show that the sufficiency conditions Mao proves can be generalized to Lyapunov functions of this form.

Math Colloquium

### Speaker: Sigurður Freyr Hafstein, University of Iceland

### Title: Dynamical Systems and Lyapunov functions

Location: VRII-158

Time: Monday 13 November at 15:00

### Abstract:

We discuss dynamical systems and the theory of Lyapunov functions and complete Lyapunov functions. Further, we discuss several different numerical methods for the computation of Lyapunov functions and the corresponding estimation of basins of attraction.