Ottilia Andersson, Bachelor's thesis, K21

Date and time: Wednesday 27/8, 9:00
Place: Cramér meeting room, Albano building 1
Student: Ottilia Andersson
Supervisor: Wushi Goldring
Title: "Outer automorphisms of S6"

Abstract

In this thesis, we study inner and outer automorphisms of groups, with a particular focus on the symmetric group. We show that if an automorphism of Sn stabilizes the conjugacy class of transpositions, then it is inner. Using this result, we further show that every automorphism of Sn is inner if n ̸= 6. Finally, a proof of the existence of an outer automorphism of S6 is outlined.

Axel Olsson, Bachelor's thesis, K28

Date and time: Wednesday 27/8, 9:00
Place: Meeting room 25, Albano building 2
Student: Axel Olsson
Supervisor: Yishao Zhou
Title: "Convergence Analysis of Quasi Newton methods and on the relation to Conjugate Gradient"

Abstract

"This thesis investigates the convergence properties of a quasi Newton optimization method called mL-BFGS. A proof of global convergence under the standard assumptions for quasi Newton methods is presented. A new theoretical result is presented that extends the previous known relation between quasi Newton methods and Conjugate Gradient method. It shows in particular that if the memory of L-BFGS is defined to the number of unique eigenvalues of a strongly convex quadratic functions Hessian, and exact line search is used, the L-BFGS have identical iterates as Conjugate Gradient, for a certain initialization. The convergence properties of the mL-BFGS algorithm is analyzed analytically as well as through numerical experiments. The experiments include comparisons with other popular first order methods, Nesterov’s Accelerated Gradient Descent and the Heavy Ball method. The experiments focuses on how the methods compare, especially when the gradients are inexact, where artificial noise has been added to the exact gradient. The results of the experiments shows that momentum has a positive effect on convergence when inexact gradients are used, providing a more robust behavior compared with the other methods. Furthermore, the experiments show that the mL-BFGS is also more stable for problems where the Hessian is ill-conditioned and the gradients are inexact. This shows potential for the mL-BFGS in large scale stochastic optimization, where the gradients are approximated, which introduces noise.

David Sermoneta, Bachelor's thesis, K31

Date and time: Wednesday 27/8, 10:30
Place: Meeting room 25, Albano building 2
Student: David Sermoneta
Supervisor: Salvador Rodriguez-Lopez
Title: "A guided tour of Wavelet theory via the constructions of Multiresolution analyses"

Abstract

We give an exposition of the theory leading up to, and including, the subject of wavelet analysis. Wavelets provide an elegant and practical way to decompose functions in a fashion similar to that of decomposing vectors in R^n , the key difference being that the bases for most function spaces are not finite. The main goal for this thesis is to first develop the necessary theory for what essentially amounts to ”doing linear algebra” in an infinite-dimensional context, and then apply that framework to the study of Wavelet analysis. To start off, we develop a more flexible notion of integration called the Lebesgue integral, and use it to construct normed spaces of functions, such as L^2(R), the inner product space of square-integrable functions. To build towards Wavelet analysis, we define the Fourier transform, a powerful tool in functional analysis that lets us describe functions in terms of complex exponentials. Finally, we introduce the theory of multiresolution analyses, a central object in the study of Wavelets, as they let us construct wavelet bases of L^2(R) by decomposing the space into nested subspaces corresponding to different levels of ”detail”. This provides a framework for analyzing how the components of a function are distributed across these levels. Using this theory, we construct the famous Haar and Shannon wavelets and their corresponding bases.

Ludvig Fagrell, Master's thesis, M7

Date and time: Wednesday 27/8, 12:30
Place: Meeting room 25, Albano building 2
Student: Ludvig Fagrell
Supervisor: Annemarie Luger
Title: "Rational Herglotz-Nevanlinna functions of several variables"

Abstract

 A Herglotz-Nevanlinna function of several variables is a holomorphic function from the poly-upper half-plane to the closed upper half-plane. A recent development shows that a function of this type admits an integral representation determined by a set of constants and a particular positive Borel measure, known as a Nevanlinna measure. The appearance of the representation has sparked further research on Herglotz-Nevanlinna functions, often with emphasis on the class of Nevanlinna measures. 

This thesis is concerned with the class of real-rational functions, which are the rational Herglotz-Nevanlinna functions whose boundary values on R^n are real. The theory on real-rational functions is rather sparse, and many of the results seem to lack formal proofs. This leads to the first of two aims of this thesis, which is to provide a more complete overview of real-rational functions of several variables than what currently exists in the literature. The results include, but are not limited to, a characterization of the real-rational functions in terms of the structure of the functions themselves, a support theorem for Nevanlinna measures representing real-rational functions, an invariance property of real-rational functions and their representing Nevanlinna measures, and an exact description of the relation between any real-rational function of two variables with an affine denominator (possibly after a biholomorphic change of variables) and its representing Nevanlinna measure.

The second aim of this thesis is to present a result on real-rational functions. The result and its context was communicated to the author by the thesis supervisor and, to the best of our knowledge, has not previously appeared in the literature. Specifically, it is shown that a real-rational function of two variables with a denominator that is a product of two affine factors has a nontrivial decomposition into a sum of two real-rational functions.

Filip Ström, Bachelor's thesis, K32

Date and time: Wednesday 27/8, 14:00
Place: Meeting room 41, Albano building 2
Student: Filip Ström
Supervisor: Samuel Lundqvist
Title: "En experimentell studie av termreduktion med Gröbnerbaser och en alternativ algoritm"

Abstract

The purpose of this paper is to compare two methods, in the form of algorithms, for reducing terms. In this context, reduction is described as the process of transforming an expression until it reaches a normal form. A classical example of reduction, commonly introduced early in mathematical education, is Gaussian elimination. The expressions being processed in this study are referred to as terms, which can be described as the product of one or more variables-possibly repeated-multiplied by a coefficient. The first of the two algorithms examined in this project performs term reduction using a reduced Gröbner basis. A Gröbner basis is a special representation of an ideal, structured in such a way that it facilitates determining whether a polynomial belongs to the ideal, and can be computed using Buchberger’s algorithm. The second algorithm, referred to henceforth as the alternative algorithm, is presented in [JLN24] and only applies to ideals of a specific form. The difference between the two algorithms may affect the computation time of the reduction process. This can be measured using a Python program that performs the calculations while counting the number of iteration steps. The results showed that reduction using a reduced Gröbner basis generally required fewer iteration steps than the alternative algorithm, for ideals that were randomly generated with specific properties.

Anya Hanson, Master's thesis, M8

Date and time: Wednesday 27/8, 14:00
Place: Meeting room 25, Albano building 2
Student: Anya Hanson
Supervisor: Salvador Rodriguez Lopez
Title: "Solvability of Dirichlet and Neumann Boundary Value Problems on C^{1,α} Domains"

Abstract

This thesis investigates the solvability of the Dirichlet and Neumann boundary value problems for bounded C^{1,α} domains. Through introducing layer potentials and proving "jump relations" at the boundary, resulting solvability criteria is formulated in terms of operators and thus are investigated using the theory of compact operators and the Fredholm Alternative. The results of the extension of this problem to Lipschitz domains is then introduced and compared to the C^{1,α} case.

More presentations in August

You can find upcoming presentations in the calendar

The following projects will also be presented in August.

Sindre Becker, Bachelor's thesis, K22
Date and time: Tuesday 19/8, 9:00
Place: Meeting room 25, Albano building 2
Student: Sindre Becker
Supervisor: Jakob Reiffenstein
Title: "Primtals satsen genom Newmans Tauberiska sats för Laplace transformationen och Riemanns zetafunktion"

Zacharias Veiksaar, Bachelor's thesis, K33
Date and time: Thursday 28/8, 8:30
Place: Cramér meeting room, Albano building 1
Student: Zacharias Veiksaar
Supervisor: Yishao Zhou
Title: "A Study of Portfolio Optimization in Discrete Time: From Markowitz to Reinforcement Learning"

Nikolina Bellon, Master's thesis, M5
Date and time: Thursday 28/8, 9:00
Place: Meeting room 25, Albano building 2
Student: Nikolina Bellon
Supervisor: Wushi Goldring
Title: "Semisimple Lie algebras and root systems"

Emre Kaplaner, Bachelor's thesis, K25
Date and time: Thursday 28/8, 10:00
Place: Cramér meeting room, Albano building 1
Student: Emre Kaplaner
Supervisor: Yishao Zhou
Title: "Dynamic Mean–Variance Portfolio Choice: Markowitz Foundations, Time Inconsistency, and Game-Theoretic Equilibria"

Johannes Erixon, Bachelor's thesis, K24
Date and time: Thursday 28/8, 10:30
Place: Meeting room 41, Albano building 2
Student: Johannes Erixon
Supervisor: Wushi Goldring
Title: "Jordan Normal Form"

Anton Christenson, Master's thesis, M6
Date and time: Thursday 28/8, 10:30
Place: Meeting room 25, Albano building 2
Student: Anton Christenson
Supervisor: Peter LeFanu Lumsdaine
Title: "Functorial Semantics for Fragments of First-Order Logic"

Lucas Dixon Leijon, Bachelor's thesis, K26
Date and time: Thursday 28/8, 12:30
Place: Meeting room 41, Albano building 2
Student: Lucas Dixon Leijon
Supervisor: Rikard Bögvad
Title: "Hyperbolic Groups"

Olli Pettersson, Independent projects for mathematics teachers, L14
Date and time: Thursday 28/8, 13:00
Place: Meeting room 9, Albano building 1
Student: Olli Pettersson
Supervisor: Per Alexandersson
Title: "The Geometry of a Good Cut: Moser, Pizza, and Beyond"

Rubina Parvin, Bachelor's thesis, K30
Date and time: Thursday 28/8, 14:00
Place: Meeting room 25, Albano building 2
Student: Rubina Parvin
Supervisor: Sofia Tirabassi
Title: "Generating functions and their applications to counting"

Yue Su, Independent projects for mathematics teachers, L15
Date and time: Friday 29/8, 9:00
Place: Cramér meeting room, Albano building 1
Student: Yue Su
Supervisor: Per Alexandersson
Title: "Avgörbara och oavgörbara problem-en analys utifrån strategiska spel"

Titti Westlin, Bachelor's thesis, K34
Date and time: Friday 29/8, 10:30
Place: Cramér meeting room, Albano building 1
Student: Titti Westlin
Supervisor: Per Alexandersson
Title: "Bevis av Lindström-Gessel-Viennots lemma samt några tillämpningar."

Erika Berger, Bachelor's thesis, K23
Date and time: Friday 29/8, 11:00
Place: Meeting room 25, Albano building 2
Student: Erika Berger
Supervisor: Wushi Goldring
Title: "From Abstraction to Action: Exploring Symmetry through Group and Representation Theory"

Minna Litzén, Bachelor's thesis, K27
Date and time: Friday 29/8, 13:00
Place: Cramér meeting room, Albano building 1
Student: Minna Litzén
Supervisor: Håkan Granath
Title: "Fibonacci modulo m"

Dennis Partanen, Bachelor's thesis, K29
Date and time: Friday 29/8, 13:00
Place: Meeting room 25, Albano building 2
Student: Dennis Partanen
Supervisor: Pavel Kurasov
Title: "Solving the Schrödinger equation by example"

Jorge Martín, Master's thesis, M9
Date and time: Friday 29/8, 13:30
Place: Zoom
Student: Jorge Martín
Supervisor: Alexander Berglund
Title: "The Halperin conjecture"

Marcus Ibrahim, Independent projects for mathematics teachers, L13
Date and time: Friday 29/8, 14:30
Place: Cramér meeting room, Albano building 1
Student: Marcus Ibrahim
Supervisor: Håkan Granath
Title: "Permutationsgrupper"