Math colloquium

**Fyrirlesari: **Yadollah Zare, Galatasaray University

**Titill: Center conditions: pull-back of differential equations.**

#### Ágrip:

The space of polynomial differential equations of a fixed degree with a center singularity has many irreducible components. We prove that pull-back differential equations form an irreducible component of such a space. This method is inspired by Ilyashenko and Movasati’s method. The main concepts are the Picard-Lefschetz theory of a polynomial in two variables with complex coefficients, the Dynkin diagram of the polynomial and the iterated integral.

Staðsetning: V-158, VR-II

Tími: Föstudag 12.Júní kl.10:00

**Notice that is recommended to keep two meters apart from other attendants.**

Math colloquium

**Fyrirlesari: **María Óskarsdóttir, Háskólinn í Reykjavík

**Titill: **Ranking nodes relative to influence with the Personalized PageRank algorithm applied to fraud detection and credit risk measurement

Staðsetning: VRII-258

Tími: Fimmtudagur 12.mars kl. 10:50

#### Ágrip:

Various phenomena in both the physical and the digital world can be represented with networks, that is, entities that are connected in some way, for example communication, computer, financial and social networks. A central theme in the analysis of networks is finding the most important nodes in a network. The PageRank algorithm was developed to rank webpages in search engines, to find the most important webpages on the internet, but has been applied in numerous others applications. The ranking can be personalized so that nodes which are important relative (or close) to a predefined set of nodes are ranked higher. This approach has been used to identify certain behavior in networks where there is a strong social effect, for example fraud and churn. In this presentation we show how the personalized PageRank algorithm can be extended for two specific types of networks. First, we look at a bipartite network which consists of claims and the involved parties, i.e. policyholders and brokers, with the goal of finding fraudulent insurance claims. Then we consider multiplex networks, in which each node can be connected to another node by more than one type of edge, such as two different networks connecting the same individuals. They arise naturally in lending, as two borrowers can be connected by geographical location, economic activity, and many other relationships. We present a methodology to leverage multiplex networks by a novel multiplex Personalized PageRank algorithm, which we subsequently apply to credit risk assessment.

Math colloquium

**Fyrirlesari:
**Sangmin Lee, Seoul
National University

**Titill: **Complete 1st post-Minkowskian potential from scattering amplitudes.

Staðsetning: VHV-007 (Veröld)

Tími: Mánudagur 24.Janúar kl.10:00

#### Ágrip:

Building upon recent progress in applying amplitude techniques to perturbative general relativity, we propose a closed-form formula for the conservative Hamiltonian of a spinning binary system at the 1st post-Minkowskian order. It is applicable for general spinning bodies with arbitrary spin multipole moments. It is linear in gravitational constant by definition, but exact to all orders in momentum and spin expansions. At each spin order, our formula implies that the spin-dependence and momentum dependence factorize completely. We compare our formula to a similar one derived in 2017 from a spinning test-body near a Kerr black hole and find perfect agreement.

Math colloquium

**Fyrirlesari: **Ragnar Sigurðsson, University of Iceland

**Titill: **Norms on complexifications of real vector
spaces.

Staðsetning: VRII-258

Tími: Fimmtudagur 6.febrúar kl. 10:50

#### Ágrip:

The subject of this lecture is of general interest and it only requires knowledge of elementary linear algebra.

The complexification V_C of a real vector space

V is the smallest complex vector space which contains V

as a real subspace. If V is a normed space, then it is

of interest to know how norms may extend from V to V_C.

I will look at a real normed space V and give formulas

for the smallest and largest extension of a general norm

on V to a norm on V_C. These formulas are not explicit

so it is of interest to find explicit formulas in particular

examples. This is possible for extentions of norms induced

by inner products. The Lie norm is the largest

extension of the Euclidean norm on R^n to a complex norm

on C^n.

In complex analysis we deal a lot with plurisubharmonic

functions and an important source for examples are

functions of the form log||f||, where f is a holomorphic

map from a complex manifold into C^n and ||.|| is a norm

on C^n. In his thesis, Auðunn Skúta Snæbjarnarson, studied

the Lie norm on C^n and calculated interesting formulas for

the so called Monge-Ampere measure of log||f||, which is

indeed not an easy task.

Math
Phys seminar

**Fyrirlesari: ****Jakob Björnberg****,
****Chalmers University of Technology**

**Titill: **Random permutations and Heisenberg models.

#### Ágrip:

We discuss probabilistic representations of
certain quantum spin systems, including the ferromagnetic Heisenberg model, in
terms of random permutations. Properties of the cycle structure of the random
permutations are connected with phase transitions in the spin-system. In particular, it is expected that the cycle
structure converges to a distribution known as Poisson–Dirichlet, in the limit
of large systems. This problem is open
but we present some partial progress.

Math
colloquium

**Fyrirlesari: Lukas
Schneiderbauer, Háskóli Íslands**

**Titill: **Non-Commutative Geometry: An introduction.

#### Ágrip:

This is my attempt to introduce non-commutative
geometry to mathematicians. After putting forward the main ideas and main theorem(s),
I will concentrate on the construction of simple examples in the context of
fuzzy spaces (special cases of non-commutative geometries). In case time still
allows it, I shall tell you about my past research in this area.

Math
colloquium

**Fyrirlesari: Anna Helga Jónsdóttir og Benedikt Steinar Magnússon, Háskóli Íslands**

**Titill 1: ** Student evaluations of teaching at the University of Iceland – analysis of data from 2013 – 2017

#### Ágrip 1:

Student evaluations of teaching (ísl. kennslukönnun) is administrated at the end of each and every course at the University of Iceland with the purpose of improving teaching and learning. In the talk, analysis of data from student evaluations from 2013 to 2017 at the UI will be presented. Mixed effect models were used to investigate possible relationships between the grades students give courses and several variables, such as the age and gender of the student and the teacher, number of students taking the course and the average final grade in the course.

**Titill 2: **Online course notes in Edbook and the role of the textbook

#### Ágrip 2:

In the last years I, with the help of many good people, have been developing a platform for online course notes called Edbook (http://edbook.hi.is). It consists of using Sphinx, which was developed for and in Python, along with specialized extension suited for teaching material in Mathematics. We have been using these notes in a few courses, mostly big calculus courses. The students have overall been very happy with them but in the spring of 2019 I had students in Mathematical Analysis II (STÆ205G) answer a more detailed survey about their usage of the teaching material they used. I will introduce the Edbook platform and the results of the survey which raises some questions about the role of the textbook today.

Staðsetning: HB5 (Háskólabíó)

Tími: Föstudag 15.nóvember kl. 11:40

Room: HB5 (Háskólabíó)

Time: Friday 22^{th} November, 11:40hrs

Math
colloquium

**Fyrirlesari: ****Sigurður Freyr Hafstein****, **Háskóli
Íslands

**Titill: **Lyapunov functions for stochastic differential equations and their
computation

Staðsetning:
HB5 (Háskólabíó)

Tími: Föstudag 15.nóvember kl. 11:40

#### Ágrip:

Attractors and their basins of attraction in deterministic dynamical systems are most commonly studied using the Lyapunov stability theory. Its centerpiece is the Lyapunov function, which is an energy-like function from the state-space that is decreasing along all solution trajectories.

The Lyapunov stability theory for stochastic differential equations is much less developed and, in particular, numerical methods for the construction of Lyapunov functions for such systems are few and far between. We discuss the general problem and present some novel numerical methods.

Math colloquium

**Fyrirlesari: Iman Mehrabinezhad, **Háskóli Íslands

**Titill: **A new method for computation and verification of contraction
metrics

Staðsetning: HB5 (Háskólabíó)

Tími: Föstudag 8.nóvember kl. 11:40

#### Ágrip:

The determination of exponentially stable equilibria and their basin of attraction for a dynamical system given by a general autonomous ordinary differential equation can be achieved by means of a contraction metric. A contraction metric is a Riemannian metric with respect to which the distance between adjacent solutions decreases as time increases.

The Riemannian metric can be expressed by a matrix-valued function on the phase space.The determination of a contraction metric can be achieved by approximately solving a matrix-valued partial differential equation by mesh-free collocation using Radial Basis Functions (RBF).

Then, we combine the RBF method (to compute a contraction metric) with the CPA method to rigorously verify it. In particular, the computed contraction metric is interpolated by a continuous piecewise affine (CPA) metric at the vertices of a fixed triangulation, and by checking finitely many inequalities, we can verify that the interpolation is a contraction metric. Moreover, we show that, using sufficiently dense collocation points and a sufficiently fine triangulation, we always succeed with the construction and verification.

This presentation is based on a joint work with Prof. Sigurdur Hafstein (University of Iceland), and Prof. Peter Giesl (University of Sussex, UK).