Symposium

In this study, we will introduce the definition of a manifold and give some non-trivial examples. Additionally, we will cover some basic notions (tangent spaces, differential forms, curvatures) and operations (differentiation, integration) on manifolds. Then we will state the Stokes’ and the Gauss - Bonnet theorems on manifolds.

At the beginning I’ll talk about cryptography terminology. Then I will summarize the parts we’ve covered in the book Understanding Cryptography by Paar. I will share the codes for some certain cryptosystems we’ve written during our studies. Unfortunately I won’t be able to run them during the presentation.

We will present rich interactions amongst low-dimensional objects such as knots, surfaces, and 3-manifolds by introducing and discussing topological notions of Heegaard splitting and Dehn surgery.

This will be an introductory talk on topological quantum field theories with the main purpose being to formulate M. Atiyah’s axioms of TQFTs in a more elegant way using category theory. In order to do so, we will first discuss cobordisms and M. Atiyah’s axioms then go on to talk about some basic vocabulary from category theory to reach our desired result.

After briefly introducing the definition and the theory of elliptic curves, we will delve deeper into isogenies and how these mappings between two elliptic curves are used in one of the five candidates for quantum-safe cryptographic schemes, namely Supersingular Diffie Hellman Key Exchange. We'll lastly demonstrate all this theory and the hard problem of our cryptographic scheme on Isogeny Graphs.

In this talk, we introduce the so-called p-adic norm on the field of rational numbers for a prime number p. This then leads to the definition of the p-adic numbers which form a complete field. In the meantime, we will also classify all the possible norms on the field of rationals and give a proof of the Ostrowski’s theorem.

In this talk, we concentrate on the Banach Fixed Point Theorem (Banach Contraction Theorem) which is one of the fundamental theorems in functional analysis. We start with some necessary prerequisites of the theorem then focus on what the theorem is. Finally, we stand on the application areas of the theorem and prove the existence and uniqueness of the solution with the help of the theorem over a few of them.

In this talk, we present an outline of cyclic codes. To do this, we discuss the origin of coding theory, linear codes and cyclic codes respectively. Finally, we will talk on open problems in algebraic coding theory.

In the first part of the talk I give some intuitive explanations of concepts like expectation, variance, and probability density function. Then I talk about the history and the importance of the normal distribution and the story of Gauss and Ceres that gave rise to the normal distribution, followed by an illustration of central limit theorem and its statement. Then I continue by giving some background for central limit theorem, try to give some intuition as to why it works, and finally outline the proof of it which uses characteristic functions and Lévy's continuity theorem. 

In my talk, I will outline how one computes the knot group, fundamental group of the knot complement, using Wirtinger presentation and compare it with how one can compute the same group from grid diagrams. I will conclude my talk by highlighting some computations I have completed for torus knots and some future directions that will follow this work.

In this talk, we will talk about two different combinatorial objects; friezes and triangulations and then we will see how they are related to each other.

First, we will introduce what is the optimization and what is the structure of an optimization algorithm. Then we will talk about two major methods of non-linear programming named line search methods and trust region methods.

We will investigate two beautiful techniques: one from the proof of Mordell’s theorem, the other from classifying torsion groups over quadratic fields.

In this presentation, we will talk about the Quadratic Reciprocity law. It has various proofs that are published by many different mathematicians. The Quadratic Reciprocity was initially claimed by Euler and Lagrange; however, it was first proven by Gauss. We will specifically discuss the proof of the law given by Eisenstein. The theorem helps us determine if a number is a quadratic residue. Furthermore, we will define the Legendre Symbol and Gauss’s Criterion to explain the Eisenstein’s proof.

Main theorems of class field theory describe the Galois groups of the abelian extensions of local/global fields in terms of their arithmetics. I will try to explain what the statements of these theorems say. In order to do that, I will talk about some basic notions from algebraic number theory and Galois theory.

Compactness Theorem is one of the most fundamental tools in Model Theory and it has lots of applications outside of logic. In this talk, we will go over the terminology of model theory, G ̈odel’s Completeness Theorem and Compactness Theorem. We will see Compactness Theorem in action and talk about applications of the Compactness Theorem outside logic. Finally, we will see how Compactness Theorem in logic is related to Compactness in Topology.

In this presentation we will start by constructing a symplectic geometry, then further discuss properties of a symplectic manifold. After this discussion some preliminaries will be provided about classical mechanics and classical mechanics will be constructed using symplectic geometry.

In this talk, we first introduce some basic definitions, facts and some standard examples about convex polytopes. Then, we state and prove the Gale’s Evenness Condition which is a well known theorem that characterizes the faces of a cyclic polytopes.

I will be answering 4 questions on the structure of polynomial ideals. Some well-known theorems that I will talk about are Hilbert’s Basis Theorem, Buchberger’s Criterion and Buchberger’s Algorithm.

In 1828, botanist Robert Brown observed the random motions of particles suspended in a liquid. Only after 1920, Norbert Wiener rigorously studied this phenomenon which is now called Brownian Motion or Wiener Process. In this talk, we will shortly answer the following questions: What is a random function or a stochastic process? In particular, what is the Brownian motion? Historically, when has the idea of Brownian motion first appeared in the literature and how has the theory developed? How can one prove the existence of such a process using simple walks? What are the characteristic properties of Brownian motion and why are some of those properties are counter-intuitive? Why does the modern theory of integration fail to define integration with respect to Brownian motion and what are the ways to overcome this problem? Can Brownian motion be used to keep the value of our money safe and how do economists receive Nobel Prize by using Brownian motion?

After introducing some basic notions in knot theory, we will define the Jones polynomial and discuss some of its applications. Then we will talk about Seifert surfaces and matrices for links and using those we will define the Arf invariant of knots and links. Finally, we end our talk proving a theorem that reveals a surprising relation between the Jones polynomial and the Arf invariant.

With the developments in the imaging technologies over the recent decades, there was a big data explosion in neuroscience and biology, but the lack of mathematical tools showed a gap in our mathematical knowledge. The need for analyzing this data led to the birth of network theory. Network theory allows us to make a variety of models and inferences using graphs, a mathematical construct encoding of the interactions between entities via nodes and edges. Network theory used for complex systems has been an effective method for us to understand the data and predict future outcomes. Brain as a complex system is no exception to that. As a result of the studies, it was revealed that complex brain networks show small world topology characterized by high clustering and short path length. Herein, we used a famous dataset, C-elegans, to test these hypothesis. We performed a simple graph theoretical analysis with this dataset and showed that the C-elegans synapse network has the characteristics of small world network topology.

In this talk, we will prove the mean value property of harmonic functions and give some applications of this property.

It is experimentally observed that when a strong magnetic field is applied to a 2D electron system at low temperatures, the transverse conductivity of the system always takes quantized values. This rich phenomenon is referred to as the Quantum Hall Effect. The integers that appear in the conductance of these systems are examples of topological quantum numbers and are closely related to the first Chern numbers which are well studied in Mathematics. In this short talk, we will see an example of a quantum mechanical electron system that possesses the Integer Quantum Hall Effect, after going through some concepts including Landau levels, the Berry Phase and the Kubo formula.

I will characterize all representations of sl(2;C) which will give all representations of SU(2). Representations will give us all linear actions of SU(2).

In this talk, we will talk about perturbation methods and asymptotic expansions.

In this presentation, we will talk about definitions of quiver, modul, path algebra and examples. We will also present some ideas in the proofs.

In this presentation, we will state genus class number one problem which is still open problem posed by Euler and Gauss.