Symposium 2025

In 1834, C.G.J. Jacobi established the formula for the number r_4(n) of representations of a positive integer nas a sum of four integer squares. Jacobi’s original proof was arithmetic in nature. In this work, we present a modern approach using the theory of modular forms, emphasizing its structural and conceptual advantages.

This study focuses on the numerical solution of the one-dimensional diffusion equation, a fundamental model in physics and engineering for describing heat conduction, mass diffusion, and similar transport processes. We examined finite difference schemes such as FTCS, Backward Euler, and the Crank–Nicolson methods to solve the diffusion equation under Dirichlet boundary conditions. Each method is analyzed in terms of stability, accuracy, and computational efficiency, with particular attention to their suitability in practical engineering applications. The Crank–Nicolson method, offering second-order accuracy in both time and space and unconditional stability, is shown to be particularly effective for simulating long-term behavior with higher precision. The results provide valuable insights for engineers and applied scientists who rely on accurate and stable numerical techniques to model the diffusion equation.

Waddington’s landscape provides a powerful and pictorial framework for understanding the behavior of complex multi-stable systems, such as the transition probabilities and paths between attractor states. Such transitions, and their perturbations, play a central role in numerous biological processes, including stem cell differentiation, cancer progression, and immune responses. In this report, we investigate the structure of the quasi-potential landscape to better understand the state transitions in the context of the minimum action path, which is the most probable route between two states. We examine Waddington’s landscape and compare its three realizations: the Helmholtz decomposition, the decomposition based on probability flux, and the normal decomposition. Throughout the report, we present explicit derivations for concise representations of some mathematical processes in the literature. We also aim to provide a clearer theoretical foundation for landscape frameworks by presenting the necessary mathematical background, with particular emphasis on stability theory.

Birdsong analysis plays a crucial role in understanding biodiversity, ecological balance, and the impact of climate change on species distribution. Monitoring bird populations through sound provides valuable insights into migration patterns, habitat health, and environmental change. Motivated by the importance of ornithology and the ecological significance of birds, this project explores automatic bird sound classification using neural networks. Audio recordings are transformed into spectrograms and analyzed with Convolutional Neural Networks (CNN) to identify bird species. This approach shows that neural networks can be used not only for accurately recognizing species, but also for supporting large-scale ecological monitoring and conservation efforts.

Planning in robotics involves converting high-level goals for a robot or robot team into specific sequences of actions that can be executed using lower level control algorithms tailored to the individual robots. In this work, we introduce two fundamental concepts: discrete path planning, which determines a collision-free route in a simplified, grid-like environment, and motion planning, which computes feasible robot movements in complex 2D or 3D spaces filled with obstacles. These techniques form the backbone of many modern applications, from autonomous vehicles and surgical robots to video games and molecular simulations in biology.

A selection of counterexamples to common statements in group theory is presented, illustrating how intuitive claims can fail. The examples address topics such as normality, commutativity, subgroup products, and finite generation, clarifying the conditions under which specific group-theoretic properties hold.

In this presentation, I will start with universal graphs and introduce the Rado graph R. Then, I will talk about random graphs and prove that the infinite random graph is isomorphic to R. Finally, I will discuss the symmetries of random graphs, focusing on the automorphism group Aut(R).

In the first part of my presentation, we will take a general look at what machine learning is and where it is used. Next, we will discuss supervised learning, which plays an important role in machine learning. We will also focus on regression analysis, which helps us predict continuous values. We will also highlight

key points in model selection and generalization, which are key steps in machine learning. 

In the second part, we will discuss parametric methods in machine learning and examine two important approaches: Maximum Likelihood Estimation and Bayesian Estimation. Next, we will discuss the trade-off relationship between variance and bias, which plays a major role in model selection. 

In final section, we will understand the structure of perceptrons in artificial neural networks and examine them in detail in multilayer perceptrons. Later, we will discuss how perceptrons and multilayer perceptrons are trained. Next, we will discuss Autoencoders, which enable us to compress and reproduce data, and Word2vec, a natural language processing method that enables us to represent words in vector space, as well as unsupervised learning techniques. Finally, I will conclude my presentation by discussing the advantages and disadvantages of multilayer perceptrons.

Political redistricting seeks to divide a geographical region into contiguous districts with nearly equal populations, a computationally challenging task at large scales. Traditional Markov chain methods often suffer from slow mixing, limiting their practical effectiveness. In this talk, we present ReCom (Recombination), a graph-partitioning approach introduced by DeFord et al. (2019) that efficiently explores the space of contiguous district plans. We will also highlight how ReCom compares to traditional methods and why it offers a promising approach for scalable, data-driven redistricting.

In this presentation, we will talk about the necessary definitions of differential geometry, and then we will focus on the special curves such as principal curves, asymptotics, and geodesics. We will mention some prominent theorems and properties about these curves, and their applications to determine the types of curves among these. Moreover, we will see these curves on some examples by taking advantage of the theorems and properties covered before.

This presentation provides an introduction to the concept of Lie algebroids, as studied in Mackenzie – General Theory of Lie Groupoids and Lie Algebroids. We begin by reviewing essential background material, including vector bundles, tangent and cotangent bundles, sections, bundle maps, pullbacks, vector fields, and differential forms, along with operations such as the exterior derivative, interior product, and Lie derivative. We then define Lie algebroids, explain their structural components---the anchor map and Lie bracket---and discuss classifications based on the anchor’s properties. The talk concludes with a detailed example: constructing a Lie algebroid structure on the derivation bundle of a vector bundle. Through this example, we illustrate how the theoretical framework of Lie algebroids emerges naturally from familiar differential-geometric structures.

Every living cell can be seen as a complex network of interacting molecules. Among these, proteins are central: they build cellular structures, transmit information, and catalyze reactions. However, proteins rarely act alone; instead, they interact with one another in intricate ways. Understanding protein–protein interactions (PPIs) is therefore essential for biology and medicine. This report introduces PPIs, explains how graph theory can represent them, and explores their analysis through a combination of network science and machine learning methods such as logistic regression.

In this report, you will find one of the most important theorems in representation theory. The reader is expected to have a brief knowledge of groups, rings, and fields. We also need some additional knowledge about algebras, modules, and quivers that we will discuss. Let us start with the definition of an algebra.

Universal algebra is the study of algebraic structures in a unified setting, focusing on sets with operations and the equations they satisfy. In this presentation, we will first introduce some basic elements of universal algebra, and then discuss Birkhoff’s theorem, highlighting its syntactic and semantic connection.

The Atiyah–Singer Index Theorem establishes a connection between analysis, topology, and geometry. We review the vector bundles, K-theory, and Chern classes. The theorem asserts the equality of the analytical and topological indices, with a precise cohomological interpretation.

We talk about Markov chains, an important type of stochastic process. Markov chains describe systems that move between states at specific times, where the chance of the next state depends only on the current state. Using simple examples, we want to better understand how these systems work and how their states are classified. Markov chains are widely used in real life to model many important phenomena, such as market behavior in economics, prediction of weather patterns, spread and treatment of diseases in healthcare, analysis of communication signals, and optimization in various industries. Their ability to model random changes over time makes them very useful in understanding and solving real-world problems.

This report provides a comprehensive overview of the classification of Coxeter groups. We begin by defining Coxeter groups as a family of groups generated by reflections, emphasizing their fundamental role in representing the symmetries of geometric spaces. The formal definition of a Coxeter group, based on a set of generators and relations, is introduced.

The core of the classification relies on the use of Coxeter diagrams and Dynkin diagrams. We explore how these graphs visually represent the group structure and lead to a powerful classification theorem. The presentation demonstrates that every irreducible Coxeter group falls into one of three distinct categories: finite (or elliptic), affine (or parabolic), and hyperbolic. We also show how any general Coxeter group can be uniquely decomposed into a direct product of these irreducible types. Through detailed diagrams and examples, including crystallographic and non-crystallographic types, this report aims to provide a clear understanding of the complete classification of Coxeter groups.

This report provides a concise overview of asymmetric and code-based cryptography. It begins by introducing the foundational principles of Public-Key Encryption (PKE), Key Encapsulation Mechanisms (KEM), and digital signatures, including advanced non-interactive constructions based on the Fiat-Shamir transform. The main focus then shifts to the core frameworks of code-based cryptography, detailing the McEliece cryptosystem, which utilizes a disguised generator matrix, and the dual Niederreiter framework, which employs a parity-check matrix. The security of these post-quantum systems is shown to rely on the computational hardness of problems such as Syndrome Decoding and distinguishing algebraic codes from random ones.

Mathematics is not only a tool for solving equations; it is also used to understand how complex systems work. The human brain, made up of billions of neurons and connections, is one of the most complicated systems in nature. To study these connections, scientists use a mathematical method called graph theory. Graph theory helps us represent brain regions as "nodes" and the connections between them as "edges" in a network. This way, we can study how information flows in the brain, how different regions interact, and even detect patterns related to memory, learning, or mental disorders. In this report, we will first explain the basic ideas of graph theory. Then, we will show how it is applied in neuroscience. Finally, we will discuss what we can learn from this approach and how it can help us understand the structure and function of the brain in a deeper way.

At first look, probability may be seen as very experimental or computational rather than theoretical. Even though this is partly true, Probability and all related topics take their roots from a very abstract theory. This project will take a glance at Probability Theory in an explicit manner. Starting with the definition and properties of σ-algebra and probability measure, concepts which in return construct the probability space. Moving on with these concepts, random variables and their distributions will be defined. In particular, the notion of independent random variables — whose joint distributions factor into the product of their marginals -— will be examined. Using the definition of simple functions, we discover that the expectation of a random variable turns out to be a conjugate of the integral. Building on this, the characteristic functions of random variables will be employed to study the Gaussian distribution which has many applications in various fields, constructing the statement and implications of the Central Limit Theorem.

We present an overview of key concepts in the representation theory of selected Lie groups, focusing on their role in

quantum mechanics. Starting with the rotation group SO(3) and corresponding angular momentum operators, we introduce the double cover Spin(3) and the special unitary group SU(2). We discuss the spin operators and highlight the relationship between SO(3) and SU(2). Finally, we illustrate the rules for spin addition and show how they manifest through character theory.

Representation theory provides a powerful bridge between abstract algebra and linear algebra, allowing us to study groups through their actions on vector spaces. In this presentation, after introducing the fundamental notions about representations, we will see Maschke’s Theorem and Schur’s Lemma. Then we will turn our attention to character theory, which enables us to find numerical invariants that determine representations. Finally, we will apply these ideas to classify the irreducible representations of the symmetric group Sn, using Young diagrams, and outline how to compute their characters using the Frobenius formula. The presentation aims to highlight the elegance of this theory.

This report presents a comparative analysis of heat transfer simulations in a flat plate using two distinct computational methods, Python and COMSOL Multiphysics, incorporating mathematical concepts of fluid mechanics. The simulation models the heat distribution and temperature variations on a flat plate under both steady-state and transient conditions, with a particular focus on key parameters such as temperature gradients, heat flux, and computational time. COMSOL Multiphysics, a finite element method-based simulation software, offers a robust platform for solving complex heat transfer problems, while Python utilizing the finite difference method (FDM) provides an alternative approach for solving partial differential equations that depend on variables such as pressure, density, temperature, and time. This report also explores the application of vector analysis to express fluid characteristics such as flow direction and vorticity, while incorporating heat properties as a significant part of fluid mechanics. The results of both simulations are compared with experimental data and theoretical models to assess their validity and reliability. The findings offer valuable insight into the strengths and limitations of each approach, presenting a comprehensive framework for future heat transfer simulations in engineering applications.

The central theme of this project is the study of fundamental concepts in algebraic geometry, with a particular focus on affine and projective varieties, along with foundational results such as Hilbert’s Basis Theorem and Hilbert’s Nullstellensatz. The report is divided into two main parts. The first part covers essential preliminaries, including affine spaces, algebraic sets, ideals, coordinate rings, rational functions, and discrete valuation rings. The second part delves into projective geometry, exploring the structure and properties of projective spaces, projective algebraic sets, and the connections between affine and projective varieties. A key highlight is the discussion of Bézout’s Theorem and intersection theory for plane curves, including the role of multiplicities, tangent lines, and local rings.

Topological Data Analysis (TDA) offers a powerful framework for extracting information from complex datasets. The report will explore the basic theories, concepts, and foundational ideas that support and explain how Topological Data Analysis works in image processing, specifically focusing on persistent homology as a mathematical tool. We explore the concepts of filtrations and persistence diagrams, which capture topological features of the image, such as connected components and loops, representing the low-dimensional homology groups. We discuss the theoretical properties that ensure the stability of these topological invariants. In short, this work aims to explain the mathematical rigor behind TDA, highlighting its potential as a versatile analytical framework for understanding the intrinsic structure of high-dimensional data, paving the way for broader adoption of machine learning techniques in various scientific domains.

We will provide a foundational overview of operator algebras, beginning with the prerequisite analytical and algebraic structures. We first establish the concept of Banach algebras, and then delve into elementary spectral theory. We will define the spectrum and spectral radius of an element and explore fundamental results, including the Spectral Mapping Theorem and the Gelfand Theorem on the non-emptiness of the spectrum. Building on this, we will present the Gelfand Representation Theorem, a crucial tool for understanding commutative Banach algebras. Then, we will focus on the more specialized structure of C∗-algebras, defined by the pivotal C∗-condition. We will finish by looking at the Gelfand Representation Theorem, which demonstrates that every commutative C∗-algebra is isometrically ∗-isomorphic to an algebra of continuous functions, and the subsequent development of the functional calculus for normal elements.

Grassmann algebra is a fundamental structure in mathematics with wide-ranging applications across multiple areas of mathematics and physics. Most notably, it serves as the foundation for differential geometry, by constituting the natural setting which differential forms reside. This paper begins with presenting the defining properties of Grassmann Algebra, outlining the working principles of the key mechanism of the algebra, wedge product. Following that, we give an exposition of formal construction of Grassmann algebra from free associative algebra with the goal of emphasizing how these properties are imposed in the structure of the algebra. The intrinsic relationship between the exterior product and the determinant is explored in Section 4. Finally, we investigate invariant subalgebras, one of the primary focuses of this paper. Here, we present a novel classification of invariant subalgebras. 

In this paper, to understand the differential cryptanalysis described in the Heys article in more detail, we first studied the fundamentals of cryptanalysis, block cipher primitives, and the general logic of differential cryptanalysis. This report provides a practical theoretical explanation of differential cryptanalysis given as an example in Heys article.

In this presentation, we briefly review basic topological definitions and introduce the concepts of cells, complexes, and triangulations. We will focus on the classification theorem for compact connected surfaces. In particular, we will show that every compact connected surface is homeomorphic to a sphere, a connected sum of n tori, or a connected sum of n projective planes. Finally, we will outline a seven-step proof of the theorem, following the approach given in Kinsey’s Topology of Surfaces.

This presentation explores the Collaborative Vehicle Routing Problem, focusing on the challenge of fair cost allocation among logistics partners engaged in horizontal collaboration. We begin with the transition from the classical Traveling Salesman Problem (TSP) to the Vehicle Routing Problem (VRP), highlighting its economic significance and real-world applications. The mathematical foundations are presented using graph-based VRP formulations, including the Miller–Tucker–Zemlin (MTZ) subtour elimination constraints. We then introduce the concept of collaboration in VRP, distinguishing between horizontal and vertical cooperation, and comparing centralized and decentralized cooperation models. Special emphasis is placed on the need for fair cost allocation, formalized through cooperative game theory concepts such as the core and the Shapley value. The Shapley value’s role in attributing marginal contributions to each partner is explained, along with an illustrative example. The study also incorporates computational experiments using OR-Tools in Python to solve VRP instances. These experiments simulate three companies operating from a single depot, evaluating routes for individual companies and various coalitions. Coalition costs and Shapley value allocations are computed to assess fairness and stability. The results demonstrate both the efficiency gains from collaboration and the importance of fair allocation methods to sustain cooperation.

We investigate the mathematical structure of neural networks through the lens of quiver representation theory. This framework encodes both the architecture and computation of a network in algebraic terms, providing a rigorous alternative to the traditional functional viewpoint. We show that neural networks can be formalized as thin quiver representations equipped with activation functions, and that network morphisms correspond to quiver representation morphisms. A key consequence is that isomorphic networks compute identical functions, explaining invariances such as the positive scale invariance of ReLU networks. We further introduce data-induced representations, which associate to each input a thin quiver representation of the delooped network quiver, thereby transforming nonlinear forward passes into linear computations. This construction reveals that datasets naturally embed into representation spaces, bridging neural networks with moduli spaces of quiver representations. Our results demonstrate that methods from representation theory and algebraic geometry can be systematically applied to the study of neural networks, opening avenues for structural insights into capacity, symmetries, and generalization.

In this study, we investigate the passive cable equation which describes the signal propagation along neurons, using finite-difference methods. A compartmental model is used to obtain a linear, constant-coefficient partial differential equation. Using explicit and Crank-Nicolson schemes, two-level finite formulas are obtained to solve the cable equation numerically. Von Neumann’s method is discussed to investigate and compare the stability of the schemes. Lax’s equivalence theorem is given to connect the concepts of stability, consistency, and convergence for well-posed initial-boundary problems and linear finite-difference approximations. Consistency is examined for the explicit scheme of the cable equation to ensure the convergence of the scheme.

The Kempf-Ness theorem establishes a deep correspondence between two seemingly different notions of stability for group orbits: the algebraic stability (where stability is defined via closedness of orbits in the Zariski topology) and the analytic stability of differential geometry (where stability is tied to the existence of

critical or minimal-norm points). In this talk, we give a proof of the theorem for symmetric subgroups of GL(n,C). We outline the role of the Hilbert-Mumford criterion in detecting unstable orbits, explain the equivalence between minimal-norm and critical points, and present the core argument that links these geometric and algebraic perspectives. Finally, (if time permits) we explore an application to quantum information theory, where the theorem provides an explicit, computable condition (via partial traces of squared density matrices) for the stability of multipartite quantum states under local operations.

Topological Data Analysis (TDA) is a tool that captures the topological features of data, and is especially useful due to its resistance to noise and ability to work with limited data. This report gives an introduction to TDA by including some information about simplexes, simplicial complexes, homology groups, and persistent homology, and gives an example of how TDA can be helpful by combining with a Machine Learning(ML) algorithm. In the shown example it is easy to see that the topological features of the data helps us to make more efficient classification and it helps us to ignore noise. At the end, the results of the model with TDA application is compared to an ML classification model without TDA features to see the difference. The report includes some interesting facts and theorems which are not very essential for TDA application, but are very important for grasping the ideas behind TDA.

Code-based cryptography is one of the main candidates of post-quantum cryptography, that is, to prevent the quantum computers from solving an integer factorization or the discrete logarithm problem over an elliptic curve or over a finite field. It uses hard problems from algebraic coding theory. The problems usually contain decoding a random linear code that is NP-hard. Our aim is to equip the reader with a general knowledge about the algebraic coding theory and the mathematical basis of code-based cryptography mostly focusing on Hamming-metric codes and mentioning rank-metric codes. This section introduces general terminology and important varieties of codes.

Group actions are a way to permute the elements of a set X in a way that respects the structure of the acting group G. In this presentation I’ll show that this allows us to realize groups as the symmetries of some object (This is how groups were originally invented, as transformations. The usual group axioms as we know them would come later). I’ll show some examples of Group Actions, some applications to Counting Problems, and finally end with an overview of the proof of the Sylow Theorems.

In this study, we demonstrate the application of topological methods to the analysis of time series data. First, we discuss the different types of time series, their components, and the techniques used in time series analysis. Next, we use persistent homology to find topological features in the time series that have been embedded in higher dimensions. Finally, we apply these methods to two datasets: one generated from a synthetic time series and the other from a real-world time series. The resulting topological features are interpreted and compared with the results of conventional time series analysis methods.

We investigate the representation of primes in the form $p= x^2+ny^2$, where $p$ is prime and $n$ is a natural number. Beginning with the classical cases n = 1,2,3, we present elementary proofs of the relevant characterization theorems. We then introduce the theory of binary quadratic forms and genus theory, applying these tools to determine, for finitely many values of n, which primes admit such representations. Special attention is given to Euler’s conjectures for $p= x^2 + 27y^2$ and $p= x^2 + 64y^2$. Finally, we offer a brief excursion into Hilbert class field theory, highlighting how it resolves the problem for infinitely many, but not all values of n.

In this report, we learn about Roman Domination in Graphs. A Roman dominating function on a graph G = (V ; E) is a function f from the vertex set to the set of three elements 0,1,2 with the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function is the value f(V ) = u ∈ V f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. In this report, we study the graph-theoretic properties of this variant of the domination number of a graph.

During the DRP 2025 program, we studied the Zariski Cancellation Problem (ZCP), which for integers $n \geq 3$ remains open in characteristic zero. This conjecture is still one of the most challenging problems in algebraic geometry. We examined various tools to approach this problem and also focused on the study of the Koras-Russell ring which has recently been considered as a candidate for approaching the ZCP. In this regard, the study of automorphisms of the Russell cubic and their classification is very important. One of the methods for finding such automorphisms is through the study of derivations.

This paper analyses the concept of plane wallpaper groups, which are patterns that tile the plane infinitely without gaps or overlaps. In our work, we employ group theory definitions and methods such as operations of the Euclidean group in matrix notation. We investigate symmetries of infinite patterns, propose classification for them, and provide a correspondence between all cyclic and dihedral point groups and shapes of patterns. We also prove the existence of exactly 17 distinct wallpaper groups, up to isomorphism. The classification is illustrated with examples of simple geometrical shapes. In conclusion, we look at ways of generalizing the usage of the theory to higher-order cases. The study of symmetries of infinite tilings connects art to mathematics and finds many applications in the finite real world.

In Model Theory, a major theme is the study of dividing lines, properties that separate model-theoretic structures into distinct categories based on the complexity or “tameness” of their behavior. NIP is one such dividing line. In Machine Learning, PAC learnability provides criteria for when a hypothesis class admits a hypothesis with minimal error. Both NIP and PAC learnability can be characterized in terms of VC-dimension. This connection allows us to prove theorems in Model Theory using tools from Machine Learning and vice versa. In this talk, we will explain this relationship and present an example of its application.

This work presents a systematic development of category theory as a foundational framework for mathematical structures and their relationships. Beginning with elementary categorical constructions, we proceed through functorial correspondences and natural transformations to establish the machinery of universal properties. The exposition culminates in an examination of monoidal categories and their application to Topological Quantum Field Theory (TQFT), demonstrating a connection between categorical abstraction and mathematical physics.

This talk explores the interplay between algebraic geometry and phylogenetic models, with a focus on the Jukes–Cantor (JC69) model for the tripod tree. We begin by introducing the necessary foundations of algebraic geometry, including affine varieties, polynomial ideals, Gröbner bases, and coordinate transformations, to establish the mathematical framework required for phylogenetic analysis. In this part, we also illustrate Gröbner basis computations using Macaulay2. Building on these preliminaries, we present the tripod phylogenetic tree and describe how the JC69 substitution model can be expressed as a parameterization map from model parameters to joint probabilities at the leaves. We discuss the role of Fourier/Hadamard transforms in simplifying model coordinates and highlight the resulting phylogenetic invariants that define the algebraic variety of the model. The presentation concludes with explicit computations for the JC69 model on the tripod tree, illustrating how algebraic geometry tools reveal the model’s structural properties and potential applications in statistical inference.

This report provides a detailed exposition of nonparametric methods, focusing on Reproducing Kernel Hilbert Spaces (RKHS) and their application in learning preference distributions. It elaborates on two distinct problems: learning user preference distributions from distance queries and from pairwise comparison queries. The report covers foundational concepts such as vector spaces, RKHS properties, and various kernel functions. It delves into the formulation of these learning problems as linear systems, analyzes conditions for identifiability in both noiseless and noisy settings, and discusses recovery guarantees through constrained optimization and regularization techniques, including graph regularization. Key insights from Rademacher complexity, optimal transport metrics, and universal approximation theorems are integrated to provide a comprehensive understanding of the theoretical underpinnings and practical implications of these data science methodologies.

Representation theory, the study of abstract algebraic structures via their realizations as linear transformations or matrices, provides a powerful bridge between group theory and linear algebra. Beginning with the formal definition of vector spaces (V,+,·) over a field F, we illustrate fundamental concepts through examples and outline their key properties. Linear transformations T : V →V are then introduced, followed by group actions and various types of representations ρ : G →GL(V), each accompanied by illustrative cases. The relationship between group actions and representations is explored in detail. Finally, we establish the correspondence between linear representations and their associated matrix representations with respect to a chosen basis of V.

This report provides a summary of key concepts in deep learning, based on the material covered in chapter 12 of Introduction to Machine Learning by Ethem Alpaydın I have read during DRP Turkiye 2025. My mentor during the program was Dr. Banu Baydil.

The report begins with the foundational Multilayer Perceptron (MLP), explaining its role in automatic feature learning and clarifying the transition to Deep Neural Networks through the addition of multiple hidden layers. It then explore the practical challenges of training these deep architectures, covering the generalization of the backpropagation algorithm for multiple hidden layers and methods for improving training convergence. Subsequently, the report examines the critical role of regularization in preventing overfitting. The focus then shifts to specialized architectures, detailing the function of Convolutional Layers, before concluding with an advanced application in Generative Adversarial Networks (GANs) and their unique adversarial training process.