Kurdish Mathematical Society Seminar (K. M. S. )�Convexity and Linear Distortion

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By

Mohsen Hashemi

Supervisor:

Distinguished Professor Gaven J. Martin

Co-supervisor:

Associate Professor Winston Sweatman

New Zealand Institute for Advanced Study (NZIAS)

Massey University of Albany

New Zealand

22 January 2022

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WHERE DOES LINEAR DISTORTION APPLY?

Linear distortion is applicable in a wide range of areas, such as

• Electronics
• Material Sciences
• Arts
• Photography
• Communications
• Energy
• Crystallisations
• Many other subjects.

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Some applications of distortion

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In communications and electronics, distortion means the alteration of the waveform of an information-bearing signal, such as an audio signal representing sound or a video signal representing images, in an electronic device or communication channel. Distortion is usually unwanted, and so engineers strive to eliminate or minimize it.

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Some other art applications of distortion

Frequently artists use distortion or abstraction to convey feelings and a particular mood, because often things can be expressed more successfully in forms that are personalized, rather than through the use of realism.

The Scream

Edvard Munch

1893-1910

The Starry-Night

Vincent Van Gogh

1889

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Fun applications

You can use distortion for fun with some applications in computers or mobile phones:

Ming

Mohsen

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Convexity

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Generalization of Convexity

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Generalisation of convexity

Open problem

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Real Analysis

Homeomorphism

Diffeomorphism

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Weak Derivative

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Weak Derivative

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Two Examples

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Weak Derivative

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Example

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Sobolev Space

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Example of a Sobolev Function

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Example of a Sobolev Function

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Example of a Non-Sobolev Function

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Quasiconformal Mappings

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Quasiconformal mappings are generalizations of conformal mappings. They can be considered not only on

Riemann surfaces, but also on Riemannian manifolds in all dimensions, and even on arbitrary metric spaces.

The importance of quasiconformal mappings in the complex analysis was realized by Ahlfors and Teichmuller

in the 1930s. Ahlfors used quasiconformal mappings in his geometric approach to Nevanlinna’s value

distribution theory. He also coined the term quasiconformal in his 1935 work that earned him one of the first

two Fields Medals. Teichmuller used quasiconformal mappings to measure a distance between two conformally

in equivalent compact Riemann surfaces, starting what is now called Teichmuller theory.

Ahlfors (1907 – 1996)

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Quasiconformal Mappings

There are three main definitions for quasiconformal mappings in Euclidean spaces:

1- Geometric definition

By using the modulus curve family

2- Analytic definition

3- Metric definition

By using the linear distortion

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Definition of Linear Distortion

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f

L_f

ℓ_f

r

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Two important maps

Conformal map

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Visualisation of conformal mappings

Source: https://youtu.be/pR5Z-pEt6Uk

Conformal mapping

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Quasiconformal mappings

Source: Computing Quasiconformal Maps on Riemann Surface using Discrete Flow

By

Zeng, Lui, Luo, Liu, Chan, Yau and Gu

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Visualisation of quasiconformal mappings

By:

Shahar Z. Kovalsky, Noam Aigerman, Ronen Basri , Yaron Lipman

Weizmann Institute of Science

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Analysis Background

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The failure of lower semicontinuity

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Iwaniec Example

Lower semicontinity property

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Tadeusz Iwaniec (born October 9, 1947) is a Polish-American mathematician, and since 1996 John Raymond French Distinguished Professor of Mathematics at Syracuse University. He and mathematician Henryk Iwaniec are twin brothers.�

Iwaniec’s Paper

A book written by Tadeuse and Gaven

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Geometric Analysis and Related Topics�In honour of Tadeusz Iwaniec’s 70th birthday

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Iwaniec’s Research

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My MSc Research

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Graphs

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My MSc research

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My PhD Research

t=0

t=0

t=0

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My PhD Research

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My PhD Research

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My Research Graphs

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The Answer of Problem 1

The best rank-one direction and the minimum of quadratic coefficient are:

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Solution and Example

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Solution and Example

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My PhD Research

Question 2: The second problem is about some conjectures by Gehring and Iwaniec concerning the gap between the linear distortion and the other finite distortions with regard to limit processes.

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My PhD Research

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My PhD Research

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Our New Results

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Our New Results

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Our New Results

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Our New Results

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Amazing Geometric Intuition in Higher Dimensions

R

0.9 R

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Don’t peel your fruits at higher dimensions

If you can travel to higher dimensional spaces, don’t peel your fruits. Because there is nothing left to eat.

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�Application in Material Sciences and Crystallisations�

A

B

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Some Applications of Distortion

Source : Interface and hysteresis in solid phase transformations

By : Sir John Ball

Distortion or deformation can occur during welding as a result of the non-uniform expansion and contraction of the weld and base metal during the heating and cooling cycle.

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Questions?

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