Ali Emre Özer
I am an MSc graduate from the Theoretical High Energy Physics (THEP) group at the University of Toronto. Previously, I did my BSc in Theoretical Physics at Imperial College London. Currently I am interested in mathematical finance.
This page contains my contact information, research interests, a collection of writings on various topics (mostly in lecture note format) and shorter article-like pages on my personal interests.
Table of Contents
- Contact Information
- My Research
- Collection of Writings
- Mathematical Finance
- Numerical Analysis and Simulations
- Mathematics
Contact
Email: emreozer2702@gmail.com
Github: https://github.com/emre-ozer
My Research
I am interested in the semiclassical to quantum transition of gravity. Semiclassical gravity refers to a construction where the usual framework of quantum field theory (e.g. the standard model) is placed on a curved spacetime background. This is a very accurate model of systems in which the energies are not too large, or the curvature of spacetime isn't extreme, or there are no black holes. My research focuses on understanding the breakdown of this description.
In the context of AdS/CFT, this breakdown can be formulated as the emergence of spacetime, or a transition from a geometric to a non-geometric phase. From the perspective of von Neumann algebras, this is the series of transitions from Type III to Type II to Type I algebras.
Here are some of my technical summaries on related topics:
Collection of Writings
Various notes I've taken over the years, mostly incomplete and still work in progress. There is no claim of originality, I made sure to reference any sources used. The links open the pdf files.
- Two Dimensional Quantum Gravity and Matrix Integrals (MSc report)
- The Physics of Bose-Einstein Condensation (Expository article)
- A Comparison of Discrete and Continuum Simulations of the Das Sarma-Tamborenea Growth Model (BSc thesis)
- Classical Mechanics
- Mathematical Methods
- Nonlinear Dynamics
- Quantum Mechanics Summary Notes
- Special Relativity
- Basics of Probability
- Thermodynamics
- Variation of Matrix Determinant
Mathematical Finance
Theory
- Black-Scholes equation
- Pricing of vanilla options
- Vanilla Greeks
- American options with trees
- Jump-diffusion models
Computational Methods
- Options with Polygon API
- Option smiles with Polygon API
- Binomial tree: C++ implementation
- Binomial tree: convergence and oscillations
- Jump-diffusion models: C++ implementation
Numerical Analysis & Simulations
These are a series of pages I've written, mostly out of personal interest and as future reference for myself. The structure will always be as follows: pages on theory (including explicit mathematical derivations), followed by my (mostly Python) implementations where I attempt to explain each code block.
Subjects I'm currently working on: fast Fourier transform (Cooley-Tukey algorithm), Runge-Kutte methods for solving systems of coupled ODEs, symplectic integrators, numerical analysis of chaos and Lyapuonv exponents.
Numerical Integration
Mathematics
These are a series of short pieces on various maths topics. They are too short to be included as part of any lecture notes, but are nevertheless interesting to me.
Miscellaneous
- Determinant about identity
- Motivating Shannon entropy
- Variation of operator valued functions
- Principal value of \(1/x\)
Differential geometry
The best book I've read on differential geometry is by far Nakahara's "Geometry Topology and Physics". Here, I intend to give concise definitions for core concepts, doing my best to add personal comments and examples, in particular motivating various definitions which may seem at first to be arbitrary.