Presentations of independent projects in mathematics

Matilda Colarieti Tosti, Bachelor's thesis, K2

Date and time: Monday 2/2, 8:30
Place: Cramér meeting room, Albano building 1
Student: Matilda Colarieti Tosti
Supervisor: Alan Sola
Title: "Convergence of random series"

Abstract

This thesis studies the convergence of random series, where the terms are given by random variables rather than fixed numbers. The main fokus is the development and application of martingale methods, particularly in L^2, to reduce convergence questions to tractable conditions such as boundedness of the underlying random variances. After reviewing foundational concepts from real analysis and probability, we present key martingale properties and demonstrate how they yield general criteria for almost sure convergence of random series.

Carlotta Kvitberg, Independent projects for mathematics teachers

Date and time: Monday 2/2, 9:00
Place: Mittag-Leffler meeting room, Albano building 1
Student: Carlotta Kvitberg
Supervisor: Per Alexandersson
Title: "S-Kalaha: Solving a Swedish variant of Kalaha"

Abstract

S-Kalaha is a Swedish variant of the Mancala game Kalah. Using concepts prevalent in combinatorial game theory, like game trees and strategies, this Bachelor’s thesis studies S-Kalaha as a finite combinatorial game. It also offers a presentation of S-Kalaha, listing its rules, ways of modifying its structure, and even an algorithm for playing S-Kalaha without a physical board. The thesis delves into some programming by estimating game complexity walues for S-Kalaha, and weakly solves S-Kalaha for specific configuration of the board. It also finds a relationship between first-player losses and the number of pits on the board.

Melvin Segerman, Bachelor's thesis, K1

Date and time: Monday 2/2, 10:30
Place: Mittag-Leffler meeting room, Albano building 1
Student: Melvin Segerman
Supervisor: Yishao Zhou
Title: "Visibility Graphs för tidsserier: Matematiska egenskaper och tillämpningar"

Abstract

"This paper presents the mathematical foundations of visibility-based graph representations of time series, focusing primarily on Natural Visibility Graphs and Horizontal Visibility Graphs, while also introducing Invisibility Graphs as a complementary construction. Key theoretical properties are formulated and proven, including invariance under strictly monotone transformations and the exact degree distribution of Horizontal VIsibility Graphs for i.i.d. stochastic processes. These results are then used to contrast theoretical behaviour of periodic, stochastic and chaotic time series, with empirical focus on stochastic, chaotic and financial data.

Building on this theory, the paper develops local visibility- and invisibility-based indicators designed to quantify concave and convex price dynamics within rolling windows. The methodology is applied to daily SPY (SP 500 ETF) price data and compared to simulated reference series generated from white noise and the logistic map in a chaotic regime. The empirical findings show that SPY exhibits graph characteristics that differ from both purely stochastic and low-dimensional chaotic systems, with convex indicators in particular highlighting periods of rapid market stress.

The results show that the visibility graphs grasp structural asymmetries in financial time series while being a descriptive tool of trend shape analysis. Possible extensions involve assessing predictability and embedding the proposed indicators within trading or risk-management schemes."

Anders Lindberg, Independent projects for mathematics teachers, L3

Date and time: Monday 2/2, 12:00
Place: Mittag-Leffler meeting room, Albano building 1
Student: Anders Lindberg
Supervisor: Rikard Bögvad
Title: "Transcendenta tal"

Abstract

"This thesis deals with transcendental numbers, focusing on their definition, historical development, and the central results that form the foundation of the theory of these numbers. The thesis begins with a presentation of algebraic and transcendental numbers, as well as fundamental concepts in set theory and Diophantine approximation, which are important building blocks for the subsequent discussion.
Next, Cantor’s argument on enumerability is examined, followed by Liouville’s construction of the first explicitly known transcendental numbers and Hermite’s and Lindemann’s proofs of the transcendence of e and π, respectively. Furthermore, the Gelfond–Schneider theorem and its significance for the development of the theory are presented. Finally, further problems and open questions in the field, such as transcendence measures and Schanuel’s conjecture, are discussed, illustrating the ongoing nature of research in this area.
The aim of the thesis is to provide a comprehensive, yet detailed, introduction to the most important results and ideas concerning transcendental numbers.
I would like to express my sincere gratitude to my supervisor, Rikard Bögvad, for his constructive guidance during this work."

Niklas Hellberg, Independent projects for mathematics teachers, L4

Date and time: Monday 2/2, 12:00
Place: Cramér meeting room, Albano building 1
Student: Niklas Hellberg
Supervisor: Boris Shapiro
Title: "Finite Fields: An Introduction"

Abstract

This paper presents the foundational theory of finite fields through several algebraic perspectives. Our aim is to develop a clear understanding of finite field structure and to illustrate its applications in a pedagogically accessible way. We begin with a historical overview of finite fields, followed by an introduction to the core algebraic concepts. A group-theoretic approach is then used to analyze the cyclic and symmetric properties of finite fields. We subsequently examine the Frobenius map and cyclotomic cosets, emphasizing their role in describing the internal symmetries of finite fields. The theoretical basis connecting finite fields to polynomials is introduced as a basis for computational methods, while linear-algebraic viewpoints connect finite fields to vector space structures and provide the foundation for modern linear coding theory. Finally, we discuss key applications building on these theoretical frameworks and connect the material to the broader research literature.

Presentations later in the week

There are some more presentations later in the week. Abstracts can be found in the calendar article for each day.

Calendar for the Department of Mathematics

Felix Nordgren Odhner, Independent projects for mathematics teachers, L5

Date and time: Tuesday 3/2, 10:00
Place: Meeting room 25, Albano building 2
Student: Felix Nordgren Odhner
Supervisor: Sofia Tirabassi
Title: "Polynomials of Degree 3 and 4: Classical Solution Methods and Their Significance"

Jessica Ramström, Independent projects for mathematics teachers, L6

Date and time: Thursday 5/2, 8:00
Place: Meeting room 25, Albano building 2
Student: Jessica Ramström
Supervisor: Pavel Kurasov
Title: "Sturm-Liouville theory"

Gülhan Sariismailoglu, Bachelor's thesis, K3

Date and time: Thursday 5/2, 14:00
Place: Meeting room 9, Albano building 1
Student: Gülhan Sariismailoglu
Supervisor: Gregory Arone
Title: "Picks sats"

Amanda Borg, Independent projects for mathematics teachers, L1

Date and time: Friday 6/2, 12:00
Place: Cramér meeting room, Albano building 1
Student: Amanda Borg
Supervisor: Per Alexandersson
Title: "Backtracking och n-queens: hur radordning påverkar sökningens effektivitet"

Fabian Lukas Grubmüller, Master's thesis, M1

Date and time: Friday 6/2, 13:30
Place: Cramér meeting room, Albano building 1
Student: Fabian Lukas Grubmüller
Supervisor: Anders Mörtberg
Title: "The Category of Iterative Sets in Cubical Agda"