Math Colloquium

### Speaker: Hermann Þórisson

Title: On the Skorohod Representation

Location: Naustið, Tæknigarði

Time: Tuesday, Ocober 11, 12:30-14:00

### Abstract:

According to the Skorohod representation theorem, convergence in distribution to a limit in a separable set is equivalent to the existence of a coupling with elements converging a.s. in the metric. A density analogue of this theorem says that a sequence of probability densities on a general measurable space has a probability density as a pointwise lower limit if and only if there exists a coupling with elements converging a.s. in the discrete metric. In this talk the discrete-metric theorem is extended to stochastic processes considered in a widening time window. The extension is then used to prove the Skorohod representation theorem.

Math Colloquium

### Speaker: Séverine Biard, University of Iceland

### Title: On pseudoconvex domains and some of their applications

Location: V-158, VRII

Time: Monday 10 October at 15:00

### Abstract:

One of the most commonly studied object in several complex variables is pseudoconvex domains, which I will introduce. Domains of holomorphy in \(\mathbb{C}^n\), they are studied in more general complex manifolds thanks to a powerful notion: plurisubharmonicity. Those domains are the support of the famous inhomogeneous Cauchy-Riemann equation or also called \(\bar\partial\)-equation. Combined with the existence of a bounded plurisubharmonic function, I will also talk about some applications to complex geometry and complex dynamics.

Math Colloquium

### Speaker: Erik Broman, Chalmers University of Technology and Gothenburg University

### Title: Infinite range continuum percolation models

Location: V-158, VRII

Time: Monday 3 October at 15:00

### Abstract:

In the classical Boolean percolation model, one distributes balls in \(R^d\) in a random, homogeneous way. The density of these balls is controlled by a parameter \(\lambda.\) Depending on this density, the collection of balls then either form an infinite cluster (\(\lambda\) large) or consists of only small components (\(\lambda\) small).

In the talk I will discuss two variants of this model, both which are infinite range. In the first case, the balls are replaced by bi-infinite cylinders with radius 1. We then investigate what the connectivity structure of the resulting set is, and how this depends on \(\lambda\) as well as on the underlying geometry (Euclidean vs hyperbolic).

In the second case, we replace the balls with attenuation functions. That is, we let \(l:(0,\infty) \to (0,\infty)\) be some non-increasing function and for every \(y\in R^d\) we define \(\Psi(y):=\sum_{x\in \eta}l(|x-y|)\). We study the level sets \(\Psi_{\geq h}\), which is simply the set of points where the random field \(\Psi\) is larger than or equal to \(h.\) We determine for which functions \(l\) this model has a non-trivial phase transition in \(h.\) In addition, we will discuss some classical results and whether these can be transferred to this setting.

The aim is that the talk should be accessible to anyone with a mathematical, but not necessarily probabilistic, background.

Math Colloquium

### Speaker: Rögnvaldur Möller, University of Iceland

### Title: Highly-arc-transitie digraphs of prime out-valency

Location: V-158, VRII

Time: Monday 19 September at 15:00

### Abstract:

Joint work with Primoz Potocnik, Ljubljana, and Norbert Seifter, Leoben.

The concept of a highly-arc-transitive digraph was defined by Cameron, Praeger and Wormald in a paper that appeared in 1993. Examples constructed by various people have shown that suggestions put forward in that paper are wrong. But if it assumed that the highly-arc-transitive digraph has prime out-valency then some of the suggestions of Cameron, Praeger and Wormald are correct. The second part of the talk is about a general method to construct k-arc-transitive digraphs that are not (k+1)-arc-transitive. This construction gives examples that limit the possibilities of extending the results in the first part and also give examples of digraphs with polynomial growth that are k-arc-transitive but not (k+1)-arc-transitive

Math Colloquium

### Speaker: Daniel Friedan, Rutgers University and University of Iceland

Title: Quasi Riemann Surfaces

Location: TG-227 (Tæknigarður, 2nd floor)

Time: Friday, August 26 at 13:20.

### Abstract:

This will be a talk about some speculative mathematics (analysis) with

possible applications in quantum field theory. I will leave any mention

of quantum field theory to the end. I will try to define everything

from scratch, but it probably will help to have already seen the basics

of manifolds, differential forms, and Riemann surfaces.

The talk is taken from my recent paper

“Quantum field theories of extended objects”, arXiv:1605.03279 [hep-th]

which is a mixture of speculative quantum field theory and speculative

mathematics. In the talk, the speculative mathematics will be presented

on its own, without the motivations from quantum field theory.

Below is the abstract from a note I am presently writing to try to

interest mathematicians in looking at this structure:

Continue reading 'Daniel Friedan (26/08/16)'»

Math Colloquium

### Speaker: Brittany A. Erickson

Title: A Finite Difference Method for Plastic Response with an Application to the Earthquake Cycle

Location: V-157, VR-II.

Time: Monday, August 8 at 13:20.

### Abstract:

We are developing an efficient, computational framework for simulating multiple earthquake cycles with off-fault plastic response. Both rate-independent and viscoplasticity are considered, where stresses are constrained by a Drucker-Prager yield condition. The constitutive theory furnishes a nonlinear elliptic partial differential equation which must be solved through an iterative procedure. A frictional fault lies at an interface in the domain. The off-fault volume is discretized using finite differences satisfying a summation-by-parts rule and interseismic loading is accounted for at the remote boundaries through weak enforcement of boundary conditions. Time-stepping is done through an incremental solution process which makes use of an elastoplastic tangent stiffness tensor and the return-mapping algorithm to obtain stresses consistent with the constitutive theory. Solutions are verified by convergence tests along with comparison to a finite element solution. I will conclude with some application problems related to earthquake cycle modeling.

Math Colloquium

### Speaker: Anthony Bonato

Title: Conjectures on Cops and Robbers Games on Graphs

Location: TG-227 (Tæknigarður, 2nd floor)

Time: Thursday, June 30 at 13:20.

### Abstract:

The game of Cops and Robbers gives rise to a rich set of conjectures, mainly associated with the cop number of a graph. Arguably the most important such conjecture is Meyniel’s, which posits a \(O(n^{1/2})\) upper bound on the cop number of a connected graph of order n. We discuss the state-of-the-art on Meyniel’s conjecture, and explore other conjectures on cop number ranging from topics within computational, probabilistic, and topological graph theory.

Math Colloquium

### Speaker: Sylvain Arguillère

Title: Constrained Shape Analysis Through Flows of Diffeomorphisms

Location: TG-227 (Tæknigarður, 2nd floor)

Time: Friday, June 24 at 13:20.

### Abstract:

The general purpose of shape analysis is to compare different shapes in a way that takes into account their geometric properties, such as smoothness, number of self-intersection points, convexity… One way to do this is to find a flow of diffeomorphisms that brings one (template) shape as close as possible to the other (target) shape while minimizing a certain energy. This is the so-called LDDMM method (Large Deformation Diffeomorphic Metric Matching).

Finding this minimizing flow requires solving an optimal control problem that can be seen as looking for (sub-)Riemannian geodesics on the infinite dimensional group of diffeomorphisms with respect to a right-invariant (sub-)Riemannian structure, creating a framework reminiscent of fluid mechanics, and opening the door to some new and exciting infinite dimensional geometries.

In this talk, I will introduce all these concepts, and give the geodesic equations for such structures. Then, I will extend this framework to the case where constraints are added to the shape, in order to better describe the objects they represent, and give some applications in computational anatomy.

Math Colloquium

### Speaker: Adam Timar, Renyi Institute, Budapest

Title: Allocation rules for the Poisson point process

Location: Árnagarður 101.

Time: Friday, June 3, at 13:20-14:20.

### Abstract:

Consider the Poisson point process in Euclidean space. We are interested in functions on this random point set whose value in each configuration point is given by some “local” rule (no “central planning”). One example is the so-called allocation problem, where we want to partition R^d to sets of measure 1 and match them with the point process, in a translation equivariant way. We want to make the allocated set optimal in some sense (e.g., the distribution of the diameter shows fast decay). We will present some allocation schemes, among them one with an optimal tail, which is joint work with R. Marko.

Math Colloquium

### Speaker: Alessandra Cagnazzo

Title: Anomalous dimension in 2d non-linear sigma models.

Location: TG-227, (tæknistofan, Tæknigarður 2nd floor)

Time: Friday, May 20 at 13:20.

### Abstract:

2d non-linear sigma models are widely used in high and low energy physics. In my talk I will present how to compute the one-loop anomalous dimension for fields in some cases in which the background is a coset space. I will review the formula for the case in which the background is a symmetric coset (that admits a \(Z_2\) automorphism) and present the new result for \(Z_4\)-cosets. The last result is applicable to the string described by the pure spinor formalism on \(\mathrm{AdS}_5\times S^5\) and it is relevant for AdS/CFT.