Symposium 2024

This report investigates the use of Kato’s semigroup theory to analyze nonlinear evolution equ- ations, with a particular focus on the Camassa-Holm (CH) equation, known for modeling shallow water waves. The study begins with an introduction to the fundamental concepts of semigroup theory, followed by Kato’s semigroup theory. At the end, it gives the problem setting to prove the local well-posedness of the CH equation’s Cauchy problem by applying the theory. This work shows the importance of Kato’s semigroup theory in the mathematical study of partial differential equations, particularly in understanding the dynamic behavior of solutions in nonlinear contexts.

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.

In this paper, we will take Ragers-Ramanujan Identity into consideration. There are a few ad- vanced proofs however there is no a bijective proof. We have stated some statistics on some partitions that can help us to find a bijection between the sides of the identity.

In this talk, we present Diffusion Tensor Imaging (DTI)-based mathematical modeling to predict the anisotropic pathways of glioma invasion. DTI is an imaging technique that measures the anisotropic diffusion of water molecules in a tissue, which can map the pathways of neural fiber tracts and help construct an atlas of the brain’s white matter architecture. On the other hand, glioma is a brain cancer that arises from glial cells in the Central Nervous System. As a methodology, we establish mathematical modeling for glioma invasion and discuss it at both mesoscopic and macroscopic levels. By formulating the transport equation, we present the corresponding macroscopic model for cell invasion. In conclusion, we present some simulations regarding glioma invasion using the proposed mathematical approach.

We study Conformal Field Theory (CFT) in 2 dimensions. This is a well-studied example of a solvable Quantum Field Theory (QFT), due to the rich symmetry structure special to 2d. CFT also has important appearances in pure mathematics, especially in the representation theory of affine Lie algebras. We first give a review of the standard construction of these theories from a physics perspective, and define the Rational Conformal Field Theories (RCFTs), which are special models. We discuss the fusion rules and their diagonalization due to the modular invariance in RCFT, and then study non-invertible symmetries in the context of 2d CFT, which has the structure of a fusion category. We finally discuss a derivation of the asymptotic density of states when non-invertible symmetries are present.

In this presentation we will talk about generative AI models and the underlying technologies. We will take a close look at text2text and text2image models.

This presentation investigates the structure and properties of graphs through the lens of the probabilistic method. This allows us to demonstrate the existence of specific graph structures. We will introduce three methods from this approach, starting with the basics and moving towards the alteration method.

This presentation explores how Markov Chains are used in Machine Learning. We begin by introducing the essential principles of Markov Chains, then build the Hidden Markov Model (HMM). We then explain how to use HMM in Machine Learning and why The Forward and Viterbi Algorithms are needed. Finally, a practical example will be provided to illustrate these concepts, followed by a summary of their significance in the field of machine learning.

Hydrodynamics is one of the most fruitful fields of physics as it paved the way of modern physics and chaos theory. Many mathematicians and physicist have contributed by coming up with better mathematical tools. Bizarre movements of liquids has inspired the wave nature of particles meanwhile its turbulent behavior has led to many ideas in chaos theory starting from Lorenz in 1963. It will be shown that the important Hydrodynamics equations such as Bernoulli Equation and Navier-Stokes Equations can be derived from a Effective Field Theory of Hydrodynamics. The principles that are going to be used will be very general for describing any system of interest with appropriate symmetries.

I will talk about important approximations and ideas that derive Gross-Pitaevskii equation to describe properties of weakly interacting Bose-Einstein condensates.

Introduction to elliptic curves on finite fields, discrete logarithm problem and their usage in cryptographic algorithms.

Definition of Riemann Surfaces followed by some theory considering the holomorphic functions between them and it ends with Riemann Hurwitz Formula.

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 mainly focused on ”Class Group” properties and computing of its generators by the help of Algebraic Number Theory. We covered the extension and embeddings of number fields to compute the signature that provides the number of real and imaginary embeddings. Moreover, we studied on rings of algebraic integers, determining the integral basis, and computing the discriminant of the basis. We used all this knowledge mainly in Dedekind Kummer and Minkowski’s Theorems during our study. We used Minkowski’s bound found by the help of dimension of a number field, signature and the discriminant of the ring of integers to determine which prime numbers are feasible to find the class groups. This bound helps us to put an upper bound for these primes to be factorized. Also, the Dedekind Kummer Theorem is used to factorize ideals, especially the prime ideals determined by the Minkowski’s bound. We applied both of these theorems to class groups to reveal the actual generators of the class groups, which is the main objective of our study.

In this presentation, we will go through the historical background of the Convex Optimization method, significance and importance of it, main mathematical methods and practical applications.

Reinforcement learning (RL) is a machine learning approach where an artificial agent learns by interacting with its environment, similar to how a biological agent does. From accumulated experiences, the agent can optimize specific objectives, expressed as cumulative rewards. In this work fundamentals of reinforcement learning for a Markow Decision Process are discussed, including most notably the Q-learning. Since these traditional techniques become insufficient in high-dimensional environments, we introduced Deep Q-Learning, which integrates Q-learning with deep neural networks. Lastly, an application involving drone control, where performance surpasses that of human experts, is presented.

This presentation will explore the construction and relationship between derived, triangulated, and infinity categories. By delving into the computation of homotopy colimits within derived category set- tings, we will provide an alternative definition of (co)homology of topological spaces. Additionally, we will demonstrate how these methods can be applied to prove the existence of certain spaces with specific singular cohomology(existence of Eilenberg–MacLane space K(G,n) for Abelian groups ).

My presentation presents a comprehensive study of various mathematical models used to analyze physical phenomena, including wave propagation, heat diffusion, and the application of the separation of variables method. Through a combination of analytical and numerical approaches, we explore how these models describe the behavior of physical systems under different conditions. The study emphasizes the im- portance of mathematical modeling in understanding complex systems. One simulation and its visualization are included to demonstrate the practical applications of these models.

Firstly, we will look at definitions of a representation, the character and irreducible representation. Then some basic properties, for example; inner product of two characters, the kernel of the character X... After that; we will define character relations of two groups (H subgroup of G), Frobenius Reciprocity, conjugate, Inertia Subgroup. We will finally define Clifford’s Theorem which is our main purpose and Theorem (Ito).

In this presentation, we will explore some intriguing results in complex geometry. We’ll begin by discussing complex structures and differential forms, and then move on to complex manifolds. This will lead us to K ̈ahler manifolds, a type of complex manifold with a rich structure. We’ll see that the operators in Hodge theory on K ̈ahler manifolds lead to significant theorems, such as the Hodge Decomposition and the Hard Lefschetz Theorem. Finally, we’ll conclude with a discussion of the Hodge Conjecture and its significance.

This presentation will discuss the concept of integrability and the conditions under which integ- rability arises. Subsequently, the fundamental motivation behind Painlev ́e Integrability will be introduced, followed by the presentation of the necessary conditions for a solution to be of P-type. To ensure these conditions, Painlev ́e Analysis will be explored, with examples provided on both ODEs and PDEs.

Some topics related to algebraic geometry and algebraic number theory. Bezout’s Theorem in algebraic geometry and structure of prime numbers of the form p = x2 +n2y and quadratic forms in algebraic number theory.

Complex networks, which are network structures with non-trivial topologies and node or vertex properties, are utilized to model systems of interest across many disciplines. Such models range from physical mappings, such as the network-like structure of the nervous system, to functionally equivalent examples such as neural networks and maps of social interactions. In turn, complex networks of interest can be modelled and analyzed using the framework of statistical mechanics, which was originally developed to analyze general behaviour of systems of many constituents. This report aims to investigate methods to analyze complex ne- tworks using statistical mechanical models; the main model of interest will be the Ising Model treatment of networks, also including spin glass-type network structures. Other models such as the Potts Model and XY Model will also be briefly explored as alternative models in the same framework. These models provide pre- dictions of the behaviour and properties of a given network topology, especially near and during phenomena classified as critical phase transitions (alternatively, critical phenomena). Various properties and behaviours significant to complex networks, such as scaling and scale-free structures, percolation, synchronization and condensation can be treated as critical phenomena in the statistical mechanical framework; this relationship will be explored as part of the report. Finally, various applications of network models, as exemplified above, will be introduced briefly and some of the utility of the statistical mechanical methods in these applications will be discussed.

ISimon’s problem is a computational problem that is proven to be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm solving Simon’s problem, usually called Simon’s algorithm. To understand it better we will cover qubits, superposition, and measurement briefly, followed by quantum gates and circuits.

Isogeny-based cryptography uses the difficulty of finding isogenies between elliptic curves for post-quantum security. The KLPT algorithm is essential for computing endomorphisms in this context.

This report explores key topics in functional analysis, including normed spaces, dual spaces, and the Hahn-Banach theorem, which are important for understanding the properties of mathematical structures in both finite and infinite dimensions. It also examines the characterization of ideal decompositions, with a focus on how the continuum hypothesis applies to operator ideals in infinite-dimensional Hilbert spaces. The goal is to present these ideas clearly, highlighting their connections and significance in both theory and practice.

In this presentation, first we will discuss the use of Copula Functions as a more effective alterna- tive to traditional correlation measures in time-series analysis. Specifically, we will utilize Gumbel copulas to capture upper tail dependence, which is crucial for assessing the probability of extreme events occurring together. By leveraging the copula’s ability to model the dependency structure between marginal and joint distributions, copulas provide deeper insights into the relationships within time-series data that conventional correlation methods often miss. Furthermore, we will integrate conformal prediction methods to enhance the reliability of forecasts by constructing prediction sets with guaranteed coverage probabilities. This combina- tion allows for precise modeling of extreme risks and also the identification of high-confidence data points in time-series predictions, offering a significant improvement over traditional correlation-based methods.

This talk will introduce complex manifolds and vector bundles, explaining their fundamental roles and how they help us understand geometric properties. We will start by explaining what complex manifolds are, focusing on their structure as differentiable manifolds with complex coordinates. Then, we will discuss vector bundles over these manifolds and their importance in studying geometric and topological features. Additionally, the talk will cover cohomological tools that are essential in analyzing these geometric structures. We will introduce de Rham cohomology, which uses differential forms to study topological properties, and Dolbeault cohomology, which is particularly useful for complex manifolds and their associated complex structures. A significant portion of the discussion will center on Hermitian metrics.

Markov Equation is a Diophantine equation with connections to rational approximation theory and cluster algebras. In this talk, we’ll define the equation and show that its solutions have the structure of a binary tree. We will talk about the still unsolved Uniqueness Conjecture and the combinatorial structure of the solutions.

We will talk about theory of Diophantine approximation and Lagrange spectrum, define Markov equation and talk about the relationship between Lagrange spectrum and Markov equation.

In this talk, We will quickly introduce the very basic notions of ∞-categories and then move on to Higher Algebra, which is the study monoids in Higher Categories. We will then move on to the definition and necessity of a En algebra and finish by constructing steenrod Operations in a higher-algebraic way and discuss some basic relations enjoyed by these operations.

Edge ideals of graphs lie in the center of algebra and combinatorics and these ideals are well- known and studied by many researchers. In this talk, we will give a brief introduction to the edge ideals of weighted oriented graphs and we will investigate their associated primes.

I will be giving all relevent definitions and a quick summary of why this theorem matters first so that people may understand the theorem. After presenting the (most important part of) the Cartan Hadamard Theorem, I will prove the necessary technicalities in the theorem and then prove three lemmas and a corollary generalizing the exponential map (the riemannian geometric one) to metric spaces along the way, proving certain regularity results for a particularly important convering space and race to the finish line of the proof of Cartan Hadamard.

Edward Witten, 'A New Proof of the Positive Energy Theorem’' başlıklı makalesinde, genel göreliliğn temel sonuçlarından biri olan Pozitif Enerji Teoremi’ne yeni bir yaklaşım sunmaktadır. Bu teorem, izole bir gravitasyonel sistemin toplam enerjisinin negatif olamayacağını ve sıfır enerjinin yalnızca düz Minkowski uzayında mümkün olduğunu ifade eder.

I am going to briefly introduce probability space and talk about the things I learned and think surprising about independence, variance and covariance. Then I present dimensionality reduction techniques used for classifiers on Palmer Data Set and soccer match probabilities and compare their success.

This report explores the concept of cyclic functions with in Hardy spaces H2(D), focusing on the important result that a function is cyclic if and only if it is an outer function. Mathematical tools such as the Poisson kernel, Blaschke products, and the distinction between inner and outer functions are introduced to build a foundation for understanding this result. The discussion includes how certain properties, like the presence of zeros within the unit disk, can influence a function’s ability to be cyclic. Additionally, the report highlights the implications of Beurling’s Theorem in identifying when a function can generate the entire Hardy space through polynomial multiplication. The aim is to present these concepts clearly and effectively, connecting foundational ideas to more advanced topics in complex analysis.

Some theorems and definitons of curves, isometries, surfaces, and then theorema egregium and its applications.

Gödel’s Completeness Theorem is a cornerstone of model theory, demonstrating that the semantic notion of ”truth” can be fully captured by the syntactic notion of ”proof” in first-order logic. In this talk, we will present the Henkin construction proof for the Completeness Theorem. Using this technique, we will also prove that the set of sentences true in all decidable theories is not recursively enumerable.

In this presentation we give an overvew of important category theoretic concepts and examine the idea of category equivalence. We conclude with investigating an example of this construction in the context of abelian categories and prove Mitchell’s Embedding Theorem which is a fundamental result in category theory.

There will be a summary of homology theory required to prove the Lefschetz fixed-point theorem. Then, the sketch of the proof of the theorem will be presented. Finally, I will end the presentation with giving examples where the theorem can be used.

In this talk, we will provide an overview of Picard-Lindel ̈of theorem, which addresses the existence and uniqueness of the solutions to ordinary differential equations. We will focus on the theorem’s statement and outline the ideas behind its proof. Finally, we will present some illustrative examples.

This presentation aims to discuss basic concepts of public-key cryptography. To do this, firstly we will mention fundamental concepts via secret-key cryptography. While defining these concepts we observe keystones of building cryptographic systems and definitions. We will see the structure of attack games which will help us understand descriptions of various protocols. Also, essential public-key exchange protocols will be discussed.

In this presentation, I will begin by explaining the concepts of topology and topological spaces. I will then discuss the ideas of product topology and compactness. Afterward, I will introduce the Tychonoff Theorem and conclude by exploring a few of its applications.

We introduce the concept of 2-categories and mention important results. After examining the need for higher levels of abstraction, we give different constructions of infinity categories and show their equivalence. Finally we will introduce the infinity category of spaces and investigate its role in the setting of higher category theory.

This study examines the non-differentiability of the Weierstrass function. The Weierstrass func- tion is considered a significant milestone in 19th-century mathematics; it is continuous everywhere but differentiable nowhere. The study presents the mathematical background and theorems necessary to prove the non-differentiability of this function. Initially, the convergence of function series and differentiability are discussed, followed by a proof of the Weierstrass function’s non-differentiability under specific conditions. This result includes a detailed analysis of the conditions that ensure the function’s non-differentiability.

Bu sunumda, sayılar teorisinin  önemli kavramlarından biri olan p-sel sayılardan bahsedeceğiz, p-sel rasyonel sayılarının inşasına ve uygulamalarına değineceğiz.

Coding theory, a critical component of information theory, plays an important role in enhancing data transmission accuracy and efficiency across digital communication systems. Codes are studied by va- rious scientific disciplines—such as information theory, electrical engineering, mathematics, linguistics, and computer science for the purpose of designing efficient and reliable data transmission methods. Since the theoretical structure of codes is based on areas such as algebra, geometry etc. in mathematics, the work of mathematicians in this field plays an important role. In this project, we discussed what a code is, how it is defined, and the main parameters of a code. Our main aim is able to provide an overview of coding the- ory. Additionally, we give some basic information of linear codes, including their generator and parity-check matrices and also we try to understand the definition of the dual codes of any codes family. Finally, in this direction we give an example to explain the relationship between the binary Hamming code family and the Reed-Muller code family via the dual code, known in the literature.

The 3-body problem has intrigued many since the 18th century due to its chaotic nature. In this talk, we will focus on the Euler and Lagrange solutions to the 3-body problem and explore the possible trajectories of celestial bodies. We will then discuss further readings and the significance of this problem in the context of space missions.

Using Gröbner Basis from Computational Algebraic Geometry to prove well-know Parametre Continuation Theorem