Abstract: Random forests enhance decision trees by reducing their weaknesses through ensemble methods. This seminar explores their mathematical foundations from a probabilistic viewpoint, focusing on decision trees as recursive partitioners and the role of probabilistic splits in variance reduction. We also discuss bootstrap resampling, which creates diverse training subsets and robust aggregation. Through theory and examples, we reveal how these elements combine to improve generalization and justify the effectiveness of random forests.

Advisor: Asst. Prof. Naci Saldı

Abstract: The Yang-Baxter equation is a key tool in both knot theory and quantum groups. In this talk, we will explore how it appears in the study of knots, leading to invariants like the Jones polynomial. We will also discuss its connection to Hopf algebras, which play a central role in quantum groups. Through examples, we will see how these ideas come together and why the Yang-Baxter equation is so important in mathematical physics and algebra.

Advisor: Asst. Prof. Cihan Okay

Abstract: Representation Theory studies algebraic structures through their manifestations in other mathematical objects. In the first half of this talk, we explore the philosophy of the theory through a variety of examples, including group representations in Harmonic Analysis, Quantum Physics and Number Theory. We then shift our focus to the study of finite group representations, discussing historical developments and emerging functorial methods within the theory.

Advisor: Assoc. Prof. Laurence Barker

Abstract: We will consider the observability of networks under linear dynamics. Specifically, we will investigate when the observability breaks down. In this investigation, we encounter and define a class of vertex pairs that induce this breakdown. We will then consider their defining properties and give some constructive methods to obtain them.

Advisor: Prof. Dr. Fatihcan Atay

Abstract: A sequence of vector spaces, each of them mapped to the next space by a linear operator is called exact if the range of each mapping is the kernel of the subsequent one. An exact sequence is called short when there are only three non-trivial vector spaces in the sequence, in which case there are two operators, where one is an injection and the other a surjection. The sequence is said to split if the surjection has a right inverse, or equivalently the injection has a left inverse. This language, developed by Vogt, in the context of Frechet-Hilbert spaces is a useful tool to study subspaces and quotient spaces of some important function algebras. In this talk, we will go over the theory and some open problems regarding the more general, and the more specific settings.

Advisor: Assoc. Prof. Alexander Goncharov

Abstract: In this talk, we will explore various definitions of the cohomology of (small) categories, depending upon the coefficient system, such as traditional category cohomology (1965), Buaes-Wirsching Cohomology (1985), and Thomason Cohomology (2013). After introducing fundamental concepts and frameworks, we give a detailed examination of cohomology computations for simple examples of categories. Lastly, we will finish by introducing the functoriality problem. This presentation is aimed at an audience ranging from undergraduates to researchers, and no prior knowledge is necessary.

Advisor: Prof. Dr. Ergün Yalçın

Abstract: The Dedekind eta function, a fundamental object in the theory of modular forms, plays a key role in number theory, elliptic functions, and mathematical physics. In this seminar, we will introduce the eta function, explore its transformation properties under the modular group, and prove a functional equation. Our discussion will highlight the significance of modularity and its implications in broader mathematical contexts. No prior knowledge of modular forms will be assumed, making this an accessible introduction to the topic.

Advisor: Assoc. Prof. Hamza Yeşilyurt