Symposium 2023

Learning algorithms are widely utilized capable models of making precise predictions based on

provided distributions to extract information from data in scientific, technological, and economic domains.

In this talk, We will focus on the fundamentals of statistical learning models: linear regression, logistic

regression, k-nearest neighbors, and decision trees, all of which fall within the purview of regression and

classification models.

Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Since then, categorifying the existing mathematical notions proved to be a very valuable technique for gaining additional intuition and for discovering useful generalizations.Categorical logic is the categorification of mathematical logic. It is notable for its connections to theoretical computer science. In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor. This framework provides a rich background for logical and type-theoretic constructions. In this presentation, after a basic introduction to category theory, we will give an introduction to categorical logic through a special type of categories called regular categories.

This presentation provides an overview of Integer Programming (IP) and its application in solving combinatorial optimization problems. It covers the fundamentals of linear programming and explores IP, its relationship with linear programming, and discusses its classification as an NP-complete problemç The presentation also includes a computational trial involving the NMK Daily-Work Assignment problem. It explores the theory and practical implications of IP.

In Dynamical systems there are three paradigms; Elliptic, Parabolic, and Hyperbolic. There is quick introduction to these paradigms. Then since the parabolic dynamics is relatively less known, the focus would be parabolic dynamics. In order to understand parabolic dynamics, there are new concepts. One of these concepts is time changes. Time changes is explained and then be given a theorem about time changes.

Perturbation Methods are ways to obtain analytical approximate solutions to various equations of mathematics.

In this talk, we will explain the notion of a topology on a set. Then, we will focus on Hausdorff and non-Hausdorff topological spaces and make the observation that a sequence in a non-Hausdorff topological space can converge to more than one point. We know from calculus and analysis that this is impossible in real numbers (equipped with the standard topology) and in metric spaces. We will provide an explicit example of a sequence in real numbers which is equipped with finite complement topology converging to infinitely many points.

In this report, our main goal is to introduce the concept of Frattini subgroups and its nilpotency in finite groups. First, we will introduce normal, central and composition series. Then we will define lower and upper central series which will be used to define nilpotent groups and we will define derived series which will be used to define solvable groups. After that we will be dealing with the properties of nilpotent groups and Frattini subgroups.

A Hilbert modular surface X(Γ) is constructed by taking the quotient of H2 by Γ, where Γ is some subgroup of the Hilbert modular group SL2(OF ) for a real quadratic extension F/Q. Hilbert modular forms are 2-dimensional analogues of classical modular forms. X(Γ) is a moduli space of abelian varieties with real multiplication, in particular, Hilbert modular forms can be interpreted as Γ-invariant functions on X(Γ).

The Deutsch-Jozsa algorithm is a pivotal concept in quantum computing. To understand it better we will cover qubits, superposition, and measurement briefly, followed by quantum gates and circuits. The algorithm’s significance in solving function evaluation problems is highlighted. Lastly, we will implement the algorithm using Qiskit, providing a practical glimpse into its execution.

We give a basic overview of topological sets (open, closed, etc.). We investigate the epsilon - delta definitions of functional limits, continuity, and uniform continuity in general metric spaces. Finally, we explore how compactness affects continuity and uniform continuity.

We propose a D-H key exchange scheme that relies on the problem of finding a smooth degree isogeny between two isogenous supersingular elliptic curves which is believed to be hard for quantum computers. We achieve this by a commutative group action, that of the ideal class group, on a set of supersingular elliptic curves, over a finite field of large prime order (kicking in).

In this presentation, we will provide a brief introduction to filters and ultrafilters. We will cover the fundamental properties and theorems concerning ultrafilters. Finally, we will conclude by examining the applications of ultrafilters in topology.

In this project, we aimed to work on Quantum Error Correcting Codes. For this, after giving the basic information about Coding Theory, we presented a few reinforcing examples. Then we talked about Error Correcting Codes. We also talked about Reed-Solomon Codes and Repetition Codes since these codes are examples of Error Correcting Codes. Finally, after giving important definitions about Quantum Codes, we talked about Shor codes, which is an example of Quantum Error Correcting Codes.

In this study, we will give the main informations about impulsive differential equations (IDEs) which are one of the differential equations

In this presentation, we will introduce the concept of reflection in the Euclidean space. After that, we will define root systems which involve specific configurations of vectors in a vector space. We give examples to illustrate the configurations. Then, we will move forward with Dynkin diagrams, which serve as visual representations containing all the information about the root systems. The final part of the presentation is playing a game called Kostant’s Game. Through this game, we will obtain an important result concerning classification of root systems.

In this talk, we will show some topological and metric properties that allow us to calculate the interior angles of a triangle on any surface. The talk will consist of four parts: In the first part, we will use manifold theory for construct the metric space and we call this space as Riemannian manifold. Then second part,the curve and its curvature are defined on this manifold. In the next part these curves will be limited to geodesic ones and polygons will be constructed with geodesic curves. And finally the interior angles of this polygon will be calculated using the Gauss-Bonnet theorem. As a result, we will see that the sum of the interior angles of triangles on surfaces with different curvatures can be different than 180◦.

The classical treatment of the epidemic models assume interactions between every individual. By introducing networks into study of epidemics, we can obtain a more realistic model of epidemic conditions and infer some phenomena about the spread of the epidemic.

The heat equation, despite being a deterministic model, can be studied with a probabilistic point of view. We can imagine that the heat consists of a great number of heat particles, and the heat equation models the movements of these particles. This presentation focuses on this concept of ’movement’.

In this talk, we will give some algebraic definitions about the superalgebra. Then, we will explain the definition of supermatrices and some operations with them. Also, we will talk about the Berezinian which is known as superdeterminant.

This presentation is designed as an introduction for those who are new to this mathematical journey. We’ll start by exploring the basics, gradually navigating through the world of elliptic curves and rational points. Our focus will be on understanding how the weak BSD hypothesis relates to these foundational ideas. Additionally, we’ll highlight the significance of this hypothesis in the realm of mathematics, aiming to present its core concepts in an accessible way.

In this report first we will introduce a mostly topological definition of toric varieties. Then by the help of an example we will see connection of the first definition and the cones. After this we will construct toric varieties with cones. Finally we will give some examples.

Our main topic is elliptic curves in the context of cryptography. In particular, we will focus on elliptic curve discrete logarithm problem (ECDLP), which is the underlying hard problem of a famous key exchange algorithm. To make sense, we will make a quick introduction to cryptography and elliptic curves. Then we will give an example of an attack on ECDLP, where attack is used to mean reducing a problem to an easier one or solving a problem.

I introduce Martingale theory and Optinal Stoppin Theroerem. I gave some useful applications of Martingale in same gambling problmes. Then, I briefly introduced Elephant Random Walk, which is Random Walk with memory. MArtingale theory is effective in ERW, because thus we can deduce the asymptotic behaviour of ERW.

In this talk, we shall cover a seminal result by B. E. Johnson linking abstract harmonic analysis and homological algebra. We will first introduce amenability for locally compact groups, and then go on to define amenability for Banach algebras using a homological construction, first introduced by Hochschild. We will then state the main result, which says that a locally compact group G is amenable if and only if its convolution algebra L^1(G) is amenable, and give the idea of its proof.

In today’s data-driven world, uncovering meaningful patterns in complex datasets poses significant challenges, and Topological Data Analysis (TDA) is an up-and-coming ap- proach to data analysis that focuses on looking of the shape of data, which is a con- venient way to overcome these challenges. This report provides a gentle introduction to the powerful data analysis technique Topological Data Analysis. Also, In this study, sudden changes in time series data will be detected. Before that, to inform basics about Topological Data Analysis, there is an introduction part, a section where fundamental concepts are addressed step by step according to the provided pipeline. This pipeline is based on the sudden changes in time series data case. Then, with an example code of implementation of this technique (used Python), results will be demonstrated. In this report, a basic understanding of what TDA is, how it works, and why it’s important in today’s research is provided.

Core subjects within the Transformers architecture covering attention mechanisms, positional encoding, and embeddings will be explored in this presentation.

In our presentation topological order is explained and the need of topological order is demonstrated by giving examples that cannot be explained with existing theories in 80’s. Later, we will focus on a basic model for topologically ordered systems, toric code. After defining the toric code, examples will be solved to show that properties of the topological order is sustained.

The General Theory of Relativity allows us to comprehend the phenomenon of gravitation. In order to understand the theory, it is essential to delve into the underlying mathematics. This paper has the objective of presenting the core mathematical framework that underpins the theory. Additionally, the Einstein equation will be presented, along with outcomes from two of its solutions, namely black holes and the big bang.

I will start by giving examples of simple algorithms to classify documents and move on to neural networks.

We discuss an obstruction to a knot being smoothly slice that comes from minimum-genus bounds on smoothly embedded surfaces in definite 4-manifolds. This method was used to give an alternate proof of the non-sliceness of the (2,1)-cable of the figure-eight knot by Aceto, Castro, Miller, Stipsicz, and Park in 2023. We present the basic ideas from knot theory and 4-manifold topology, leading up to the proof of the said theorem. We also discuss, if time permits, establishing similar bounds using minimum-genus functions of indefinite 4-manifolds.

It is common to define dimension - like concepts in algebraic (and geometric) structures. In order to understand vector spaces, noetherian rings, algebraic varieties, algebraic and Lie groups, algebras and some other objects better, we use ”dimension”. But the definition of that varies from structure to structure. Model theorists have tried to generalize this concept and introduce ”Morley Rank”. Morley rank behaves like dimension (it is defined to do so) and can be applied to a much wider class of structures. In this talk, we focus on groups of finite Morley rank and their classification problem (algebraicity or Cherlin-Zilber conjecture). Methods are (surprisingly) similar to classification of finite groups. We also present an important theorem which connects representations of algebraic groups and groups of finite Morley rank. (rk= 0 means finite groups already.) So looking at their representations is natural.

The Cartan–Hadamard theorem is a proposition within Riemannian geometry that addresses the configuration of complete Riemannian manifolds exhibiting sectional curvature that is not greater than or equal to zero. This theorem asserts that the universal covering of such a manifold is diffeomorphic to a Euclidean space using the exponential mapping at any given point.

The Kempf-Ness Theorem gives a way to identify critical elements and explores the relation between critical elements and closed orbits. It has broad applications from symplectic reduction to complexity theory. In my presentation, I will give the preliminary necessary to understand the theorem and prove the theorem itself.

Over the last two centuries Set Theory and Mathematical Logic have been extensively studied. A product of this is the exploration of non-standard models of different mathematical structures. These non-standard models bring with them a variety of interesting properties and results. We review a few of these in this DRP Project.

We consider the edge and cover ideals of a finite simple graph with which we will transform vertex coloring problems to problems in commutative algebra. Moreover, we investigate the associated primes of powers of cover ideals. When the induced subgraph of G is critically (s+1) chromatic we obtain a characterization of these associated primes. Finally we will study the minimal free resolutions of edge ideals and introduce Fröberg’s theorem.

In this talk, firstly we explain what the Poincar ́e group is. Then we desribe its unitary representation on the physical Hilbert space and examine the importance of this representation for quantum field theory. Lastly we state the Lie algebra of the Poincaré group.

We introduce some key ideas in Homotopy Type Theory, including the univalence axiom and the homotopy interpretation.

I will give a brief historical background on classical vs contructive mathematics. Then I will introduce the Pier Martin-L ̈of Type Theory and give reasons to study it. Lastly, I will demonstrate the embedding of logic into type theory and prove some logic propositions in Agda.

Fundamental Theorem of Algebra is a widely used theorem in mathematics. There are several proofs of the FTA that uses different branches of mathematics. In this talk, we will present an almost algebraic proof of it by using Galois theory and field extension after mentioning some basic concepts in ring theory.

There is an algebraic correspondence between affine algebraic sets and projective algebraic sets. But when we ask a question like “Is closure in Pn of an arbitrary algebraic set is a projective algebraic set?” the answer cannot be reached by this algebraic correspondence. Thus we need a new tool to handle this problem which is called an analytic set. In this presentation, we will define analytic sets and we will prove the Chow’s theorem which gives the connection between analytic sets in Pn and projective algebraic sets.