# KMS Seminar Series

## Seminars 2024 - 2025

Seminar #11:

# Geometry and Integers

Speaker: Prof. Caucher Birkar

Tsinghua University & The University of Cambridge

Date: Thursday, September 5th, 2024

Time: 16:00 - 17:00 Erbil time (GMT+3)

MS Teams: A meeting link will be provided to registered participants.

Download the seminar flyer and share it with your colleagues

## Abstract

In this talk we will discuss some recent interesting connections between properties of sets of nonnegative integers with different kinds of geometries. Starting with a primitive integer vector, we examine certain associated functions and relate the setting to statements in convex and algebraic geometries.

Available Material: Report

Seminar #10:

# The Generic Failure of Lower-semicontinuity for the Linear Distortion Functional

Speaker: Dr. Mohsen Hashemi

Massey University, Auckland, New Zealand

Date: Thursday, August 8th, 2024

Time: 13:00 - 13:45 Erbil, Kurdistan

Zoom Link: Zoom link will be provided to registered participants

The seminar is based on a joint work with Gaven J. Martin (Click on this link to download the full paper)

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## Abstract

We consider the convexity properties of distortion functionals, particularly the linear distortion, defined for homeomorphisms of domains in Euclidean n-spaces, n ≥ 3. The inner and outer distortion functionals are lower semi-continuous in all dimensions and so for the curve modulus or analytic definitions of quasiconformality it follows that if {fn}∞ n=1 is a sequence of K-quasiconformal mappings (here K depends on the particular distortion functional but is the same for every element of the sequence) which converges locally uniformly to a mapping f, then this limit function is also K-quasiconformal. Despite a widespread belief that this was also true for the geometric definition of quasiconformality (defined through the linear distortion H(fn)), T. Iwaniec gave a specific and surprising example to show that the linear distortion functional is not always lower semicontinuous on uniformly converging sequences of quasiconformal mappings. Here we show that this failure of lower semicontinuity is common, perhaps generic in the sense that under mild restrictions on a quasiconformal f, there is a sequence {fn}∞ n=1 with fn → f locally uniformly and with lim supn→∞ H(fn) < H(f). Our main result shows this is true for affine mappings. Addressing conjectures of F.W. Gehring and Iwaniec, we show the jump up in the limit can be arbitrarily large and give conjecturally sharp bounds: for each α < √ 2 there is fn → f locally uniformly with f affine and α lim sup n→∞ H(fn) < H(f) We conjecture √ 2 to be best possible.

## Seminars 2022 - 2023

Seminar #9:

Existence and Stability of -harmonic Maps

- Speaker: Seyed Mehdi Kazemi Torbaghan

- University of Bojnord, Iran

- Time: October 9th, 2022

## Abstract

In this seminar, we first recap the α-energy functional, Euler-Lagrange operator and α stress-energy tensor. Second, it is shown that the critical points of the α-energy functional are explicitly related to harmonic maps through conformal deformation. In addition, an α-harmonic map is constructed from any smooth map between Riemannian manifolds under certain assumptions. Next, we determine the conditions under which the fibers of horizontally conformal α-harmonic maps are minimal submanifolds. Then, the stability of any α-harmonic map from a Riemannian manifold to a Riemannian manifold with non-positive Riemannian curvature is demonstrated. Finally, the instability of α-harmonic maps from a compact manifold to a standard unit sphere is investigated.

Seminar #8:

Non-associative algebras and applications to digital data transmission

- Speaker: Susanne Pumpluen

- The University of Nottingham, UK

- Time: June 7th 2022

## Abstract

We look at a class of nonassociative algebras that can be defined using skew polynomials, and discuss their structure and applications. These algebras can be seen as canonical generalizations of central simple algebras, and some behave surprisingly similar to their classical "cousins". They are employed to build codes used for wireless digital data transmission, e.g. in mobile phones, laptops or portable TVs. To make the talk accessible to a broad audience, we will explain the main ideas using the example of Hamilton's quaternion algebra. We will explain how to generalize the construction of Hamilton's quaternion algebra to nonassociative four-dimensional algebras, and how to use them in space-time block coding.

Seminar #7:

(p,q) arithmetic Klienian Group

- Speaker: Kaywan Salehi

- Massey University, New Zealand

- Time: May 7th, 2022.

## Abstract

Arithmetic Kleinian group is an important branch of low dimensional geometry and topology. One of the significant side of it, is two generating arithmetic groups. In our research, we will complete Maclaclan and Gaven martin work for all possible Arithmetic groups which generated by two elements $(f; g)$ of order $(p; q)$. They calculated $(p; q)$- cases for $p; q < 6$. But for $p; q < 6$, cases still have remainded. Here we try to introduce some Theorems and Methods to complete our work.

Available Material: Slides, Report, Video

Seminar #6:

Study the notions of ϕ-biflatness

ϕ-biprojectivity for ϴ-Lau products

- Speaker: Eghbal Ghadri

- University of Kurdistan, Iran

- Time: April 27th 2022

## Abstract

For two Banach algebras A and B and a nonzero multiplicative linear functional ϴ on B, Monfared introduced the ϴ-Lau product B. In this paper, we investigate and study the notions of ϕ-biprojectivity, ϕ-biflatness and ϕ-Johnson amenability of B and their relations with A and B. As an application, we characterize ϕ-biflatness and ϕ-biprojectivity for ϴ-Lau product of Banach algebras related to locally compact groups and discrete semigroups.

Seminar #5:

Post-Quantam Secure Communications

- Speaker: Wrya Karim Kadir

- University of Bergen, Norway.

- Time: March 9th, 2022

## Abstract

Imagine a mature enough quantum computer is available in the near future. What can it do? There are several tasks for which a quantum computer will be useful. It can also be a threat to our privacy. The one that is mentioned most frequently is that quantum computers will be able to read secret messages communicated over the internet using the current cryptographic schemes. Which simply means that none of us would be able to securely and privately communicate via internet if we stay with our current online devices. In this talk we will address this issue. We will also describe the actions that we can do in advance in order to stay secure in the world with quantum computers.

Seminar #4:

Density of the "quasi r-rank Artin problem"

- Speaker: Andam Mustafa

- Salahaddin University, Kurdistan, &

Roma Tre University, Italy

- Time: February 27th 2022

## Abstract

For a given finitely generated multiplicative subgroup of the rationals which possibly contain negative numbers, we will demonstrate formulas for the densities of primes subject to GRH for which the index of the reduction group has a given value. Moreover, we will completely classify the cases of rank one, torsion groups for which the density vanishes and the set of primes for which the index of the reduction group has a given value, is finite. For higher rank groups we will mention some partial results.

Seminar #3:

Transferring Topological Problems into Algebraic Problems

- Speaker: Haval Mahmood

- University of Soran, Kurdistan.

- Time: February 19th 2022.

## Abstract

Transferring geometric problems into the languages of permutations groups. Our goal is to describe the geometric problem about spaces and then transfer it into the theory of permutation groups by using the fundamental groups. The Riemann existence theorem provides a one to one correspondence between them. As a result, we can deal with finite primitive groups instead of indecomposable meromorphic functions.

Available Material: Slides, Report, Video

Seminar #2:

The parallel chaotic system

- Speaker: Mohsen Hashemi

- Time: January 22nd 2022

## Abstract

This research is primarily concerned with the convexity properties of distortion functionals -particularly the linear distortion - defined on quasiconformal homeomorphisms of domains in Euclidean n-spaces, though we will mainly stick to three-dimensions. The principal application is in identifying the lower semi-continuity of distortion on uniformly convergent limits of sequence of quasiconformal mappings. For example, given the curve family or analytic definitions of quasiconformality – discussed in this research – it is known that if {𝒇𝑛} is a sequence of K-quasiconformal mappings (and here K depends on the particular distortion but is the same for every element of the sequence) which converges to a function 𝒇, then the limit function is also K-quasiconformal.

Despite a widespread belief that this was also true the geometric definition of quasiconformal-ity (via the linear distortion 𝐻(𝒇) defined below) Tadeusz Iwaniec gave a specific surprising example to show that the linear distortion function is not lower semicontinuous. The main aim of this research is to show that this failure of lower semicontinuity is actually far more common. Perhaps even generic in the sense that it might be true under mild restrictions on a quasiconformal 𝒇, there may be a sequence if {𝒇𝑛} with 𝒇𝑛→𝒇 uniformly and with the below property

lim𝑛→∞sup𝐻(𝒇𝑛)→𝐻(𝒇).

The main result in my thesis is to show this is true for wide class of linear mapping.

Available Material: Slides, Report, Video

Seminar #1:

The parallel chaotic system

- Speaker: Sohrab Karimi

- Ferdowsi University, Iran

- Time: January 15th 2022

## Abstract

This work presented some novel systems for chaotic systems, the so-called parallel systems. Indeed we will define a parallel system for a chaotic system where the attractor obtained from the new system exhibits complex chaotic dynamics as well. The application of these systems is discussed. For instance, It is shown with the help of these systems, the synchronisation problem between two systems can be transformed into an optimal control problem.

Available Material: Slides, Report, Video