Symposium

In this talk, we will focus on local fields and briefly talk about classifying their extensions, especially the non-ramified ones. We will also state some results from local class field theory which allows us to understand their abelian extensions.

After defining and introducing elementary tools and objects which we will be using in ring theory, we will prove some important results about prime and maximal ideals and by proving a "generalized" case of chinese remainder theorem we will be able to find some connections between number theory and ring theory. We will define radicals and primary ideals. Then we will talk about primary decomposition and its importance.  We will finish our presentation by introducing and proving  Noether-Lasker Theorem.

What is analytic number theory/What does it study? What is Prime Number Theorem? Outline of the proof Some conjectures in analytic number theory (Riemann Hypothesis and Merten's Conjecture) 

In this study , by using the Cayley graph we introduce a New visual of the automorphisms. Moreover we will give techniques to find the degree of an automorphisms via graphs and will show how to multiply that automorphisms.

Two well known facts from elementary number theory are proven by using Bergman spaces.

In this talk we will briefly introduce vector bundles and define Stiefel-Whitney classes. We finish with showing that if n+1 is not a power of 2 then P^n is not parallelizable.

We go over the definition, basic properties and theorems about elliptic curves. We introduce the finite field point counting problem, give the naive approach to solve this problem and then give a brief summary of Schoof's algorithm to solve the problem.

I studied relations of dynamics with surfaces. In the presentation context, dynamics is the iterates of homeomorphism f on a compact metric space. I will give some topological background and define expansivity. My focus will be about the implications of expansivity on compact metric spaces (more specifically closed orientable surfaces). At the final, I will mention Hiraide-Lewowicz Theorem.

In this presentation, we introduce the ideal description problem and the ideal membership problem in the multivariable polynomial ring where the coefficients are from a field. Afterwards, we explain how to solve these problems using gröbner bases.

First, we will define LPPs and explain their geometry. Then we will go over a modeling example in urban transport network design. Second, we will introduce two search algorithms used in optimization problems: Simulated Annealing and Tabu Search. We will lastly review an implementation of SA in solving sudokus.

In this talk, we will provide an introduction to knots and knot invariants, then construct the Seifert surface associated with a knot to compute the knot genus. From then on, we will discuss Dehn surgery to obtain 3-manifolds from knots. To complete the discussion, we shall mention torus knots and lens spaces as well.

First of all, I will talk about Markov and Chebyshev. Afterwards, I will explain simple concentration inequalities and finally I will touch on principal component analysis and finish my presentation.

Every finite group can be seen as a subgroup of a symmetric group. Consequently, symmetric groups have an important role in finite group theory. We will demonstrate how to reveal their linear structure by using representation theory and we will introduce a wonderful combinatorial tool: Young tableaux. This simple tool, tetris-like collection of empty boxes, will enable us to construct irreducible representations of symmetric groups just by placing numbers into the boxes.

In this presentation, we will detect stock market crashes using topological data analysis. We will describe persistent homology and persistent diagrams.

Graph Theory has been very important area of math. Particularly in chemistry it is used to model some specific molecules. One way to analyze graphs more detailed is matrices. Which is  called adjacency matrices. By these matries we can get information about graphs more easily. There are some  different adjacecy matrices which can be defined by using topological indeces. By using topological index we get molecular descriptor which is chemical information encoded with symbols, numbers or result of some standardized experiment. And by these topological indices one can estimate about thermodynamics of molecules. If we need to collect we will need concepts like rank, nullity, eigenvalues, Schur complement and basic graph terms, definitions. So we will focus on these.

In this talk, we aim to explore some applications of Discrete Morse Theory which plays a crucial role on determining the homotopy class of objects. First, we give the necessary background to explain this concept and adress the question of determining the homotopy equivalence between them. We classify this as a pure application and we solve it by giving two theorems that help us to create a homotopy equivalent of a given object. We also provide a very recent real-life application that also shows connections with Information Theory. That is, denoising images using Discrete Morse Theory. We briefly explain this and finish by giving some examples.

We know that Galois groups in number theory are important, so naturally we want to study their representations. A Galois group acts naturally on algebraic varieties which are built from fields and polynomials. This action allows that Galois representations are viewed as automorphisms of certain geometric objects which give us a powerful link between the class field theory and the geometry of algebraic objects. Historically, the first place shown this link was Serre's article in 1972. So the aim of this talk is to explain one of his theorems in the article.

Basic theorems of about sturm Liouville operators and some necessary background materials.

For a nonzero integer n, a set of distinct nonzero integers {a1, a2, . . . , am} such that ai*aj + n is a perfect square for all 1 ≤ i < j ≤ m, is called a Diophantine m-tuple with the property D(n) or D(n)-set. The D(1)-sets are called simply Diophantine m-tuples; and have been studied since ancient times. In this presentation, I give an overview of the results from the literature about Diophantine m-tuples and D(n)-sets. Furthermore, I will also present the paper we studied about strong rational Diophantine D(q)-triples.

A conformal mapping is a complex mapping that has an angle-preserving property. In other words, any analytic complex function is conformal at a point if its first derivative at that point is not equal to zero. Such mapping always exists under certain conditions, and its existence is emphasized in the Riemann mapping theorem. Although Riemann mapping theorem guarantees the existence of conformal mapping, it does not give any way how to find a conformal mapping between regions. There are a few alternative ways how to construct a conformal mapping. In this study, two of them will be stated. One of them cross-ratio theorem requiring three distinct point and their images. The other one is the Schwarz-Christoffel theorem which provides an explicit formula for the derivative of a conformal mapping from the upper half-plane onto a polygonal region. In addition, conformal mappings are widely used in physical applications such as heat-flow, electrostatics, and flow of an ideal fluid.

In this talk, we will give some basic definitions about Poisson and symplectic manifolds. Then we will explain the relation between the Poisson and symplectic manifolds with Weinstein splitting theorem.

In this talk we will mainly work on stating the Seifert-Van Kampen theorem. The fundamental group and some necessary categorical language will roughly be introduced to do that. At the end, we will compute the fundamental group of a few well-known manifolds such as klein-bottle and torus with the help of Seifert-Van Kampen theorem.

Code-based cryptogtaphy is the area of research that focuses on the study of the cryptosystems based on error-correcting codes. Our first aim is to understand basic concepts in coding theory, especially linear codes. The second goal is to understand the Goppa codes which is linear since is used in the McEliece cryptosystem. The last part of the study is to work McEliece and Niederreiter cryptosystems and their equivalence when they are constructed based on the same algebraic error-correcting codes.

It is known that Selman Akbulut knot bounds two different discs in $B^4$ which are not smoothly isotopic relative to boundary to each other. Recently, Kyle Hayden and Isaac Sundberg gave an alternative proof using Khovanov Homology. During the talk we will present the recent proof.

In this presentation, first we give a definition of completions using inverse limit and then talk briefly on their basic properties. Our main aim will be to introduce Hensel’s Lemma to make feel the listeners the usefulness of completions and then we end our concentration on Hensel’s Lemma by introducing the idea of Lifting idempotents which will be treated as a corollary of Hensel’s Lemma. If there will be enough time, we will introduce another way of defining completions using krull’s topology and Cauchy Sequences.

Nowadays, it is quite difficult to go through a day and not interact with machine learning: face recognition on our phones, smart search algorithms, travel time estimation, product recommendations and many more. It turns out that machine learning is a purely mathematical concept that lies at the intersection of linear algebra, real analysis and statistics. We will give an overview of how and why neural networks work via one of the cutest machine learning applications: building a program that recognizes cat pictures!

To find a root for a function f(x)=0 , the following methods are used: Bisection Method, Fixed point iteration, Secant Method, Newton Method.

In this talk, we will talk about quivers, path algebras and their representations. Then, we talk about the equivalence between their representation categories.

In this talk we will introduce matrix Lie groups and examine some of their properties, using the example of O(2,R), the matrix Lie group formed by 2 × 2 orthogonal matrices. We will also talk about the corresponding matrix Lie algebras.

What is a virtual knot and how did it come about? What is the difference from the classic knot? Reidemeister movements in virtual knots.What is gauss code and how to use gauss code in virtual node.

I will talk about why Fundamental theorem of Galois theory does not work for infinite extensions, history of this theorem, and how infinite Galois theory related with profinite groups, projective limits and Krull topology.

In this talk, we will state the inverse Galois problem, and then develop some useful and delicate tools, which are of independent interests, in algebraic number theory and the theory of elliptic curves to present some partial solutions for certain groups.

In this talk, I am going to present Monte Carlo statistical methods with an emphasis on Markov Chain Monte Carlo (MCMC) methods. Monte Carlo methods constitute the backbone of modern statistical inference and have a great use in optimization, machine learning and natural sciences. I will briefly talk about random variable generation, accept reject methods, Monte Carlo integration and then continue with Markov chains. I will end my talk with the discussion of Metropolis-Hastings algorithm.

Although randomness concept is not directly associated with Graph Theory, random graphs are emerged from a bounding problem for Ramsey numbers. There are several different random graph sample spaces according to which property of the graphs is randomized, but in this presentation, we will be focusing on certain sample space where edges of samples are randomized. Then, use of expectation and variance will be discussed with some examples. After that, for a fixed graph, a uniform random walk on that graph will be explained. Also, we will try to connect this concept with Markov chains and it's applications.

We studied the classical groups particularly the special orthogonal group, special unitary group, and symplectic group, and their topological properties primarily compactness, connectedness, and path connectedness. Then, we continued with the definition of homotopy and, discussed the intuition behind it. In order to construct the fundamental groups, we defined concatenation and discussed the simply connectedness. Constructing and understanding the fundamental groups, lead us to our main goal.  We aim to find the fundamental groups of the classical groups. It helps us to understand these certain groups better. However, these were not enough for us to finalize our process. To be able to find the fundamental groups, we need more tools. These tools contain covering maps, path lifting, homotopy lifting properties, a few other theorems, and especially fiber bundles!

We will introduce Model Theory than learn some applications.

Homology is a topological invariant that enables us to count the holes of a shape. In this talk, after introducing the concept of cells and simplicial complexes, we briefly look into the chain groups. Finally, after defining cycle and boundary groups of a chain complex, we define the homology groups as their quotient. We finish the talk by computing the homology groups of the Klein bottle.

Topological phases of matter are phases of matter that violate Landau’s symmetry breaking paradigm. These are different phases of matter that have the same symmetry but are characterized with different topological invariants. Toric Code, first introduced by Alexei Kitaev, is a 2-D spin model which is one of the simplest examples of a topological phase. In this short talk, I will explain the Toric Code and present its fundamental properties. I will also talk about how this model can be used to store quantum information.