# Bachelor projects

This page lists some examples of bachelor projects offered this year by teachers of the department. This list is limited to up to two subjects per course. For more subjects or questions, feel free to contact the teacher(s) in question.

# Analysis

##### Differentievergelijkingen

Promotor: Hans Vernaeve

In de cursus Analyse II bekeken we differentiaalvergelijkingen. In dit project bekijken we de parallelle theorie voor de discrete tegenhanger, de zgn. differentievergelijkingen.

Uitgangspunt zijn de eerste hoofdstukken uit het boek "An Introduction to Difference Equations" van S. Elaydi.

##### De discrete Fourier-transformatie

Promotor: Hans Vernaeve

In de cursus Analyse II hebben we periodieke functies ontbonden in Fourierreeksen. In de praktijk zijn signalen digitaal, en daardoor discreet. Om zo'n signaal te ontbinden in een spectrum van frequenties kan men de zgn. discrete Fourier-transformatie gebruiken.

Verschillende aspecten kunnen aan bod komen, o.a. ook het computationele aspect (de zgn. “Fast Fourier transform”).

Uitgangspunt is H.2 van het boek “An introduction to wavelets through linear algebra” van M. Frazier.

## Globale aspecten van krommen en oppervlakken

Promotor: Frederik Broucke

In de cursus differentiaalmeetkunde werd de lokale theorie van krommen en oppervlakken behandeld: gegeven een punt op een kromme of oppervlak, beperkten we ons tot een (voldoende kleine) omgeving van dat punt om de kromme of het oppervlak in de buurt van dat punt te beschrijven. De globale theorie werd nauwelijks behandeld. De bedoeling van deze bachelorproef is om enkele aspecten van de globale theorie te bestuderen. Voorbeelden zijn
• De isoperimetrische ongelijkheid, die een verband geeft tussen de lengte van een gesloten vlakke kromme en de oppervlakte van het ingesloten gebied;
• Oriëntatie van oppervlakken en de karakterizatie van compacte oriënteerbare oppervlakken;
• De Gauss-Bonnetformule, die een verband geeft tussen de kromming van een oppervlak en de topologie (via de Euler-karakteristiek);
• Gedrag van geodeten op oppervlakken en complete oppervlakken.

Als startpunt kunnen we het boek van do Carmo, Differential geometry of curves and surfaces, tweede editie, Dover Publications, New York, 2016 gebruiken.

## De priemgetalstelling

Promotor: Gregory Debruyne

De priemgetalstelling is een van de kroonjuwelen van de analytische getaltheorie. Ze verzekert dat de priemtelfunctie asymptotisch equivalent is met x/log x. Het doel van dit bachelorproject is om een van de vele bewijzen van deze stelling uit te werken en te begrijpen. Naargelang de interesse van de student, kan het gekozen bewijs analytisch van aard zijn (complexe analyse en/of Fourieranalyse) of eerder elementair.

## De scheidingsstelling van Jordan en Brouwer

Promotor: Jasson Vindas Díaz

Begeleider: Michiel Huttener

De stelling van de Jordankromme is een klassieke mijnpaal in de topologie. Dit belangrijke resultaat stelt dat elke Jordankromme het vlak scheidt in twee disjuncte, samenhangende, open gebieden. In het bijzonder kan men (louter topologisch) spreken over punten binnen en buiten  een gesloten kromme.

Het doel van dit project is het bestuderen van een bewijs van de overeenkomstige hogerdimensionale versie veralgemening met behulp van methoden uit de differentiaaltopologie. De versie die bestudeerd zal worden, is de volgende scheidingstelling van Jordan en Brouwer: elk compact, glad hyperoppervlak scheidt R^n in twee disjuncte, samenhangende, open gebieden. Onder andere de volgende theorieën zullen ook bestudeerd en toegepast worden: homotopieën, transversaliteit en intersectietheorie mod 2.

Het uitgangspunt is hoofdstuk 2 van het boek "Differential Topology" door Victor Guillemin en Alan Pollack. De bedoeling is dat je zelf kleine stukjes theorie en voorbeelden uitwerkt die in het boek als oefening vermeld staan.

## Fractional Calculus

Promotor: Karel Van Bockstal

A fractional derivative is a derivative of any arbitrary order (real or complex). The aim of this project is to give a historical overview on fractional calculus and to study the basic properties of the Riemann-Liouville fractional integral and derivative, and Caputo fractional integral and derivative.

## Fractional Differential Equations

Promotor: Karel Van Bockstal

The goal of this project is to study fractional differential equations (generalization of ordinary differential equations to an arbitrary non-integer order) and methods to solve them (e.g. Laplace transform method).

## Numerical approximations of fractional derivatives

Promotor: Karel Van Bockstal

For most fractional differential equations, it is not possible to provide methods to compute the exact solutions analytically. Therefore, it is necessary to revert to numerical methods. The aim of this project is to study these numerical methods to approximate the Riemann-Liouville fractional derivative and the Caputo fractional derivative.

## Dunkl analysis on Euclidean space

Promotor: Michael Ruzhansky

Supervisor: Vishvesh Kumar

Dunkl theory related to the differential-difference operator, known as the Dunkl operator, is a far-reaching generalization of Fourier analysis on Euclidean space. This theory has a deep connection with algebra, probability, and mathematical physics.  In this project, we will learn some basic concepts of harmonic analysis related to the Dunkl theory. In the due course, based on the interest of the student we can focus on a particular problem. Some of such problems are related to Fourier multipliers, functional inequalities, and partial-differential operators associated with the (rational) Dunkl theory.

References:

 J.-Ph. Anker: An introduction to Dunkl theory and its analytic aspects, 2016. Arxiv: https://doi.org/10.48550/arXiv.1611.08213

 C. F. Dunkl: Differential–difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167–183.

 M. Rösler: Dunkl operators (theory and applications), in Orthogonal polynomials and special functions (Leuven, 2002), E. Koelink & W. Van Assche (eds.), pp. 93–135, Lect. Notes Math. 1817, Springer–Verlag, Berlin, 2003.

 S. Thangavelu and Y. Xu: Convolution operator and maximal function for the Dunkl transform, J. Anal. Math. 97 (2005), 25–55.

## The Black-Scholes equation on almost Abelian groups

Promotor: Michael Ruzhansky

Supervisor: Zhirayr Avetisyan

In mathematical finance, the famous Black-Scholes equation describes the dynamics of the fair price of a financial derivative with a fixed maturity and pay-off, as a function of time and the market prices of the underlying securities. In its classical setting, the Black-Scholes equation assumes that the volatilities and returns are constant, but in real life quant analysts at investment banks have to solve this equation every night in view of the updated market information, in order that the bank prices its derivative products correctly the next morning. While in practice this is done numerically on large computers (because market data are given numerically), for a better qualitative and theoretical understanding it is useful to study hypothetical scenarios with different time-dependent volatilities and returns, especially if they allow for exact solutions.

Imagine now that we are in a world where the market prices of liquid securities are functions on a Lie group G, and so is time (prices and time make up the "price-time" G, similar to the space-time in General Relativity). The returns and volatilities are now variable as well, but so that they are left-invariant with respect to the group action. In other words, the market has certain dynamics, which is governed by given algebraic symmetries. We want to study the behaviour of the fair price of derivatives under such conditions. What happens to the derivatives when the market conditions are highly unstable, e.g., exponential? What if the market conditions are periodic? Are the derivative prices well-defined for very long maturities or less than simplistic (exotic) pay-offs?

More mathematically, the Black-Scholes equation is a linear parabolic equation, and the pay-off is the initial data for the initial value problem. The textbook case with constant returns and volatilities corresponds to a PDE with constant coefficients, which is easily solved by standard methods. In our model this corresponds to the case where the "price-time" G is the commutative group Rn. In this project we want to make a step further and assume that G is an almost Abelian group. The structure of an almost Abelian group is such that (aside from central extensions of the Heisenberg group) there exists a unique co-dimension one normal subgroup, which corresponds to the prices of liquid securities, and a special direction, which corresponds to time. Thus, on an almost Abelian group there is a natural left-invariant Black-Scholes equation, which describes the prices of securities under left-invariant dynamics. The coefficients of this parabolic equation depend on time, and depending on the group G may demonstrate a variety of behaviours from periodic to exponential. The Lie group structure allows to formally solve the equation by separation of variables, reducing the PDE to ODE. Global well-posedness of the initial value problem in different function spaces, estimates on the growth of solutions, Green's function and other questions of PDE theory are among the goals of the project.

Invariant parabolic equations on almost Abelian groups have not been studied before, and in case of strong results the project may end with a publication. More importantly, the student will be exposed to the techniques of non-commutative analysis in a relatively accessible setting, together with some basic aspects of financial mathematics. And there will be a lot of mathematical fun.

## Heat and wave type equations on different Lie groups

Promotor: Michael Ruzhansky

Supervisor: Joel Restrepo

We will study the solutions of heat and wave type equations on several Lie groups. We will focus in the representations of solutions, time-asymptotic estimes, norm estimates, etc. We will start the analysis from the basis in Rn and then move forward to other group.

## (Co)-Homology Theories for Smooth Manifolds

Promotor: Michael Ruzhansky

Supervisor: Santiago Gómez Cobos

In this project we will study some fundamental (co)-homology theories for smooth manifolds. In short, these are theories in which the goal is to associate an algebraic object to the manifold of study by means of some topological, geometrical or analytical data. Specifically, the objective of the project is to study the famous de Rham theorem which states that singular cohomology is isomorphic to de Rham cohomology. Basic knowledge on topology and differential geometry is expected, and we will study the rest of the theory in detail . Depending on time and motivation we could move on to other type of constructions, perhaps Hodge cohomology or Morse homology.

References:

 Lee, J. Introduction to smooth manifolds. Second. Vol. 218. Graduate Texts in Mathematics. Springer, New York, 2013.

## Quantum-machine-learning the fate of Schrödinger's cat

Promotor: Michael Ruzhansky

Supervisor: Zhirayr Avetisyan

Machine learning, a large subcategory of the vast subject of artificial intelligence, lies in the confluence of mathematics and computer science. Underlying much of it is statistical modelling and prediction. Challenges of this area of research can be broadly grouped into two categorties: modelling and computation. Usual or classical machine learning refers to the situation where both data and algorithms are classical, and the relevant statistics is based on the probability theory. The adjective "quantum" in this context may refer either to the quantum nature of the data and statistics, or the use of quantum algorithms in computations. As a project in pure mathematics geared towards functional analysis, we will be mostly concerned with the conceptual aspects of modelling quantum systems, while the choice of computational methods will be largely irrelevant.

Thus, quantum machine learning is destined to help model and predict quantum systems which cannot be adequately described by classical models. Let us think of Schroedinger's cat which is famously either alive or dead in a non-deterministic manner. Let C^2 be the Hilbert space describing this cat, with the pure states (1,0) and (0,1) corresponding to the living or dead state, respectively. On the other hand, let us think of the pure state (1,1) as representing the half-dead cat in merry oblivion, and the pure state (1,-1) as standing for the cat half-dead in agony. As is normal in predictive modelling, we want to produce a mathematical model that can predict hard-to-observe aspects (output) in terms of more easily observed aspects (input). For instance, imagine that we can somehow measure the extent to which the cat is in oblivion/agony (say, by some sort of scan), and based on that we want to predict the extent to which the cat is alive. Note that if the cat is in perfect oblivion (1,1) or total agony (1,-1) then the state of its livelyhood is maximally uncertain (50/50). This is a manifestation of the quantum nature of the problem: the two parameters, alive-dead and oblivion-agony, are not compatible and cannot be simulatenously decided to perfect precision.

The proper mathematical setting of quantum statistics is operator algebras. Elements A of a C^* algebra W represent all possible measurements, while the values w(A) of a state w  at an algebra element A is the expectation value of the measurement A in the given quantum state w. The above system of Schroedinger's cat is a very simple example, for it is finite dimensional, but it already demonstrates some of the main features of the subject. Here the C*-algebra W is M(2,C) - the algebra of all 2x2 complex matrices equipped with Hermitian conjugation and operator norm. In the context of supervised learning, the goal is to predict the most likely output measurement A from the input partial measurement B, based on a sequence (B_1,A_1),...,(B_n,A_n) of pairs (B_i,A_i) where B_i is a partial measurement and A_i is the appropriate full measurement to learn from. Note that both inputs B_i and outputs A_i are operators (matrices).

While the examples touched upon above are too simplistic to be considered relevant, they demonstrate the most important features and techniques used in quantum statistics. On their way to understanding the theory behind the model, the student will learn some proper functional analysis and operator theory, as well as certain ramifications thereof in quantum statistical physics.

## Schrödinger equation on hyperbolic surfaces

Promotor: Michael Ruzhansky

Supervisor: Hong-Wei Zhang

It is known that some properties of dispersive equations are more pronounced on negatively curved manifolds. For example, the dispersive inequality and the Strichartz inequality on hyperbolic planes are shown to be stronger than the ones in Euclidean settings. This project aims to understand when and how one can extend these stronger properties from a hyperbolic plane to its corresponding hyperbolic surface.

References:

 N. Burq, C. Guillarmou, and A. Hassell, Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics, Geom. Funct. Anal. 20 (2010), 627–656

 A. Fotiadis, N. Mandouvalos, and M. Marias, Schrödinger equations on locally symmetric spaces, Math. Ann. 371 (2018), 1351–1374

 HW, Zhang, Wave and Klein–Gordon Equations on Certain Locally Symmetric Spaces. J Geom Anal 30, 4386–4406 (2020).

## Homogeneous Lie Groups

Promotor: Michael Ruzhansky

Supervisor: David Rottensteiner

Homogeneous Lie groups play an important role in several fields of analysis. By definition, they are smooth manifolds equipped with a compatible group structure which admit a family of (generally anisotropic) dilations that respects the group and manifold structures. Although as smooth manifolds homogeneous Lie groups are diffeomorphic to Euclidean space, they feature interesting intrinsic geometries. This project will discuss some basic elements of analysis on homogeneous Lie groups such as integration, vector fields, translation-invariant metrics, and function spaces.

## Orthonormal wavelet bases

Promotor: Michael Ruzhansky

Supervisor: David Rottensteiner

Wavelets have been playing a crucial role in many areas of abstract and applied mathematics, in particular in signal analysis and data compression. The aim of this project is to study some fundamental theoretical aspects of multi-resolution analysis, a well-established method for constructing orthonormal wavelet bases for the Hilbert space L2, which often extend to unconditional bases of important Banach function spaces.

## Difference equations on the lattice Zn

Promotor: Marianna Chatzakou

The idea of this project is to investigate how differential equations read when discretized in the setting of the lattice. This analysis can be carried out using the available calculus of pseudo-differential operators on the setting. Numerical analysis methods can also be employed for the approximation of the solution.

## Brascamp-Lieb inequalities

Promotor: Roberto Bramati

Many well-known multilinear integral inequalities in Euclidean analysis, such as Hölder's, Loomis-Whitney, or Young’s convolution inequalities, are instances of a more general family, i.e., Brascamp-Lieb inequalities. These inequalities depend on a family of linear surjective maps and on a set of (Lebesgue) exponents. In a nutshell, they are inequalities for functions defined on quotients of the original Euclidean space in which we try to control, apart from a constant, the average of their pullbacks via the linear maps on the whole space with the product of appropriate Lebesgue norms of the functions. In this project we aim at understanding under what conditions on the maps and on the exponents a Brascamp-Lieb inequality holds, i.e., it has a finite constant. This passes through a heat-flow argument, the analysis of gaussian extremizability and a careful study of the structure of the linear maps involved.

Main reference:
[BCCT08] J. Bennett, A. Carbery, M. Christ, and T. Tao, The Brascamp- Lieb inequalities: finiteness, structure and extremals, Geom. Funct. Anal. 17 (2008), pp. 1343-1415.

Other references:
[BrL] H.J. Brascamp, E.H. Lieb, Best constants in Young’s inequality, its converse, and its generalization to more than three functions, Adv. Math. 20 (1976), 151–173.
[CLL] E.A. Carlen, E.H. Lieb, M. Loss, A sharp analog of Young’s inequality on S^N and related entropy inequalities, J. Geom. Anal. 14 (2004), 487– 520.
[L] E.H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math. 102 (1990), 179–208.

## Resonances of the Laplacian on the upper half plane

Promotor: Roberto Bramati

Resonances are fundamental spectral objects, historically arising from quantum mechanics. They are the poles of the meromorphic continuation of the resolvent of a differential (or Schrödinger) operator across its continuous spectrum and they can be thought of as a replacement for eigenvalues, when they are absent. After reviewing the main definitions and Fourier-analytic tools, in this project we will study the basic case of the Laplacian on Euclidean spaces. Then we will move on to the case of the Laplacian on Riemannian symmetric spaces of the noncompact type of rank one, and in particular to the model case of SL(2,R)/SO(2), developing all the necessary tools to use the analog of the Fourier transform for this setting. In this context, when performing the meromorphic continuation of the resolvent, we will have to face the difficulties coming from the singular Plancherel density. Depending on the time, we can also study how the resonances and associated residue operators are related to the representations of the underlying group SL(2,R).

References:
[HP] J. Hilgert, A. Pasquale, Resonances and residue operators for symmetric spaces of rank one, J. Math. Pures Appl. (9) 91 (2009), no. 5, 495-507.
[H] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.
[L] S. Lang, SL(2,R), Graduate Texts in Mathematics, 105. Springer-Verlag, New York, 1985.

# Logic

## Grote kardinaalgetallen

Promotor: Andreas Weiermann

In dit project worden oneindige verzamelingen gebestudeerd die zo groot zijn dat men hun bestaan niet meer in de gewone wiskunde bewijzen. Het gaat erom de "small large cardinals" te kaderen en met elkaar te vergelijken. Daarvoor zou de theorie uit het boek van Jech uitgewerkt worden.

Omdat logica in semester zes valt wordt van de student wat zelfstudie over verzamelingenleer verwacht.

## Intuitionistische logica

Promotor: Andreas Weiermann

In dit project bekijken we de basisbegrippen van de constructieve wiskunde.

De bedoeling is de volledigheidsbewijs voor de intuïtionistische propositielogica door middel

van Kripke modellen te bestuderen en in het vervolg het Rieger Nishimura tralie te onderzoeken.

Referenties

Troelstra, van Dalen Introdution to constructive mathematics.

## Borel en analytische verzamelingen

Promotor: Andreas Weiermann

In dit project gaat het erom basisresultaten uit de descriptieve verzamelingenleer te bespreken. Een verzameling in een poolse ruimte is Borels als zij element is van de kleinste sigma-algebra die de open verzamelingen bevat. Een analytische verzameling is gegeven als beeld van een continuë functie van de Baire ruimte naar een poolse ruimte. Zulke verzamelingen hebben de eigenschap van Baire en zij zijn Lebesgue meetbaar. Een versameling is Borels als en slechts als zij en haar complement analytisch zijn.

## Referenties  Thomas Jech. Set Theory. Springer.

De algebraïsche theorieën van binaire relaties en partiële functies
Promotor: Brett McLean

De bestudering van de algebraïsche eigenschappen van verzamelingen begon met het werk van Boole. Met een algebraïsche eigenschap bedoelen we een eigenschap van operaties op verzamelingen uitdrukbaar zonder de elementen van de verzamelingen te noemen, bijvoorbeeld de distributieve wet A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) of de wet van De Morgan (A ∪ B)c = Ac ∩ Bc.

Men kan vergelijkbare bestuderingen van binaire relaties en van (partiële) functies maken. De algebraïsche bestudering van binaire relaties is gestart door De Morgan en gevorderd met name door Tarski. Het kan gezegd worden dat de algebraïsche bestudering van functies is gestart door Anton Sushkevich, en een bekende naam die op dit gebied heeft gepubliceerd is Karl Menger.

Het doel van dit project zal zijn om enkele resultaten te beschrijven die aantonen dat de algebraïsche eigenschappen van binaire relaties logisch ingewikkelder zijn dan de de algebraïsche eigenschappen van functies. Concreet is het objectief om aan te tonen dat:

Zelfs met de redelijk weinig expressieve verzameling {compositie, domein} van operaties, zijn de eigenschappen van binaire relaties niet eindig axiomatiseerbaar met de eerste-orde-predicatenlogica , terwijl de eigenschappen van functies dat wel zijn .

Een inleiding tot de bestudering van binaire relaties en partiële functies is te vinden in .

Referenties

 Robin Hirsch and Szabolcs Mikulás, Axiomatizability of representable domain algebras, The Journal of Logic and Algebraic Programming 80 (2011), no. 2, 75–91.

 Marcel Jackson and Tim Stokes, An invitation to C-semigroups, Semigroup Forum 62 (2001), no. 2, 279–310.

 Brett McLean, Algebras of partial functions, Ph.D. thesis, University College London, June 2018.

De mozaïekmethode voor temporele logica’s

Promotor: Brett McLean

Temporele logica is een soort logica voor rekening over uitspraken die kunnen verwijzen naar de toekomst en/of het verleden. Temporele logica heeft toepassingen in de informatica, kunstmatige intelligentie en taalkunde. Amir Pnueli heeft de 1996 Turing Prize gewonnen 'for seminal work introducing temporal logic into computing science and for outstanding contributions to program and systems verification'.

Voor elke specifieke temporele logica is het belangrijk om te weten hoe rekenkundig kostbaar het is om te beslissen of een uitdrukking van de logica geldig is. De mozaïekmethode is één manier om algoritmen te produceren die geldigheid kunnen beslissen. De mozaïekmethode kan ook gebruikt worden om complexiteitsgrenzen voor de geldigheidsproblemen vast te stellen.

Het doel van dit project is om een bewijs via de mozaïekmethode van de computationele complexiteit van een specifieke temporele logica op en specifieke tijdlijn te presenteren. Bijvoorbeeld de basic temporal logic van Arthur Prior (ook bekend als tense logic) op de gehele getallen heeft NP-volledig complexiteit.

Een inleiding tot de syntax en semantiek van basic temporal logic (Hoofdstuk 1), computationele complexiteitstheorie (Appendix C) en de mozaïekmethode (Sectie 6.4) kan in het boek  gevonden zijn. Het artikel  geeft een beschrijving van de mozaïekmethode als gebruikt voor complexiteitsgrenzen en hoe deze kunnen aangepast worden voor verschillende tijdlijnen.

Referenties

 Patrick Blackburn, Maarten de Rijke, and Yde Venema, Modal logic, Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 2001.

 Maarten Marx, Szabolcs Mikulás, and Mark Reynolds, The mosaic method for temporal logics, Automated Reasoning with Analytic Tableaux and Related Methods (Roy Dyckhoff, ed.), Springer Berlin Heidelberg, 2000, pp. 324–340.

### Second Incompleteness Theorem and Provability Logics

Promotor: Fedor Pakhomov

Gödel's Second Incompleteness Theorem is a classical result in mathematical logic stating that no consistent theory can prove its own consistency. Later, Solovay isolated the propositional logic of the provability of formal arithmetic: the collection of all formulas built from Boolean connectives and the special unary connective □ (□φ is read as "φ is provable") that are universally valid laws about provability. Essentially, the result was that all these laws follow from relativizations of the Second Incompleteness Theorem.

The goal of this project is to study these results and then investigate some alternative, provability-like interpretations of □.

References:

 Boolos, G. 1993. The Logic of Provability. New York and Cambridge: Cambridge University Press.
 Hájek, P., and Pudlák, P. Metamathematics of First-Order Arithmetic. Vol. 3. Cambridge University Press, 2017.

# Discrete mathematics

## Constructing graphs with the same spectrum

Spectral graph theory studies the relation between structural properties of the graph and the eigenvalues of
associated matrices. Graphs are often studied by their adjacency matrix. Two graphs with the same spectrum (eigenvalues) for some type of matrix are called cospectral with respect to the corresponding matrix. The spectrum contains a lot of information of the graph, but in general it does not determine the graph (up to isomorphism). That motivates why is interesting to construct cospectral graphs: they help us understand the weaknesses in identifying structures only using the spectrum. Several tools for constructing cospectral graphs are known to exist, the most used one is a switching method .

Similar methods to this switching seem to have appeared in the literature but in a different context: to construct new Hadamard matrices . A Hadamard matrix is a square matrix whose entries are either 1 or −1 and whose rows are mutually orthogonal. The work in  presents several operations that switch substructures of Hadamard matrices with the aim to produce new, generally inequivalent, Hadamard matrices. These operations have applications to the enumeration and classification of Hadamard matrices.

This project aims to provide an overview of the methods presented in [1,2], which have been used independent of each other, and to find new relationships between them.

Referenties

 C. D. Godsil and B. D. McKay. Constructing cospectral graphs. Aequationes Math. 25 (1982), 257–268.
 W.P. Orrick. Switching Operations for Hadamard Matrices. SIAM J. Discrete Math. 22(1), 31-50.

## Is the spectrum a fingerprint of a graph?

In this project we look at the spectrum (that is, the set of eigenvalues) of the adjacency matrix (and some variations) of a graph, and ask the following fundamental question:

Given a graph and the spectrum of a matrix associated to the graph, what does the spectrum tell about a certain graph property?

Graph properties such as connectedness, diameter, independence number, chromatic number and regularity, are also known to be related to the spectrum of a graph. If a property is not characterized by the spectrum, then there exist a pair of non-isomorphic graphs with the same spectrum. Motivated by the complexity of properties of graphs that can be characterized by the spectrum of a matrix associated to a graph, spectral characterizations of some well-known NP-hard properties have been studied in the literature. In this project we will investigate other NP-hard graph parameters that do not follow from the spectrum of a graph.

## A graph theory approach to generalized Kneser and Johnson graphs

While plenty of combinatorial properties and graph parameters are known for Kneser graphs (see for example [2, 4] and references herein), much less is known for their extension to subspaces, that is, for generalized Kneser graphs [1, 5, 3]. The same applies to Johnson graphs and generalised Johnson graphs. This thesis aims to complete this gap by extending known results for Kneser/Johnson graphs to their generalised versions.

 J. D’haeseleer, K. Metsch, D. Werner, On the chromatic number of two generalized Kneser graphs,
European J. Combin. (2022)
 P. Frankl and Z. Füredi, Extremal problems concerning Kneser graphs, J. Combin. Theory Ser. B, 40
(1986), 270-284.
 K. Liu, M. Cao, M. Lu, Treewidth of the generalized Kneser graphs, arXiv:2011.12725.
 L. Lovasz, Kneser’s conjecture, chromatic number and homotopy, J. Combin. Theory Ser. A, 25 (1978),
319-324.
 R. Simoens, Cospectral generalized Kneser graphs, Master thesis UGent, 2022.

## An algebraic approach to the graph integrity

The integrity of a graph G = (V, E) is defined as I(G) = min{|S| + m(G − S) : S ⊆ V (G)}, where m(G − X) denotes the order of the largest component in the graph G − X. An I-set of G is any set S for which the minimum is attained. This parameter was introduced by  as an alternative measure of the vulnerability of graphs to disruption aused by the removal of vertices. The motivation was that, in some respects, connectivity is oversensitive to local weaknesses and does not reflect the overall vulnerability.

Some eigenvalue bounds are known for both the vertex and edge-version of this graph parameter, see e.g. [6, 1]. This project aims to find new algebraic bounds for the graph integrity, and to study its value for some specific classes of graphs.

 N. Alon, A. Bishnoi, S. Das, and A. Neri. Strong blocking sets and minimal codes from expander graphs. arXiv:2305.15297, 2023.
 K.S. Bagga, L.W. Beineke, A survey of integrity, Discrete Applied Mathematics 37-38 (1992), 13-28.
 C.A. Barefoot, R. Entringer, H.C. Swart, Vulnerability in graphs - A comparative survey, J. Combin. Math. Combin. Comput. 1 (1987), 13-22.
 D. Benko, C. Ernst, D. Lanphier, Asymptotic Bounds on the Integrity of Graphs and Separator Theorems for Graphs, SIAM Journal on Discrete Mathematics 23(1) (2009), 265-277.
 W. Goddard, H.C. Swart, Integrity in Graphs : Bounds and Basics (2006).
 Yinkui Li, Yongtang Shi, Xiaofeng Gu, Spectrum bounds for the scattering number, integrity, tenacity of regular graphs, Future Generation Computer Systems 83 (2018), 450-453.

## Combinatorische onderzoeksgebieden met verbanden met projectieve meetkunde (Projectieve meetkunde, Discrete wiskunde I en II)

Promotor: Leo Storme

Binnen de combinatoriek zijn er verschillende domeinen die verbanden hebben met projectieve meetkunde. Zo zijn verschillende problemen binnen de codeertheorie equivalent aan meetkundige problemen binnen de projectieve meetkunde. Analoog worden er voorbeelden van grafen geconstrueerd via meetkundige structuren uit projectieve ruimten. Zo worden er bijvoorbeeld extremale voorbeelden van grafen gevonden.

Binnen dit bachelorproject is het de bedoeling om een verband tussen projectieve meetkunde en een andere domein uit de combinatoriek te bespreken. Dit omvat de beschrijving van het verband en ook een bespreking van hoe, via meetkundige technieken, deelstructuren uit projectieve meetkunden helpen om nieuwe resultaten in dit ander domein te vinden.

## Studie van verzamelingen deelruimten in vectorruimten over eindige velden (Projectieve meetkunde, Discrete wiskunde I en II)

Promotor: Leo Storme

Binnen verschillende domeinen binnen de wiskunde worden verzamelingen deelruimten van de vectorruimte V(n,q) van dimensie n over het eindig veld GF(q) bestudeerd.

Zo worden binnen de vectorruimte V(n,q) verzamelingen deelruimten van dimensie k, die paarsgewijs snijden in (k-t)-dimensionale deelruimten snijden, bestudeerd voor de transmissie van informatie door wireless netwerken.

Binnen dit bachelorproject is het de bedoeling om bepaalde verzamelingen deelruimten die aan vooropgegeven voorwaarden voldoen, te bestuderen, en hun eigenschappen te bestuderen. Vele van de bewijzen van de eigenschappen van deze verzamelingen deelruimten zijn gebaseerd op eigenschappen uit lineaire algebra, maar ook andere technieken worden gebruikt om deze eigenschappen te bewijzen.

## Saturerende verzamelingen in eindige projectieve ruimten (Projectieve meetkunde, Discrete wiskunde I en II)

Promotor: Leo Storme

Begeleider: Lins Denaux

Binnen eindige projectieve ruimten worden saturerende verzamelingen bestudeerd. Een rho-saturerende verzameling S in PG(n,q) is een verzameling punten zodat elk punt van PG(n,q) linear afhankelijk is van hoogstens rho+1 punten van S.

Zo is bijvoorbeeld een 1-saturerende verzameling S in PG(n,q) een verzameling punten zodat elk punt van PG(n,q) tot minstens 1 rechte van PG(n,q) behoort die minstens twee punten van S bevat.

Binnen de theorie van de saturerende verzamelingen wordt vooral gezocht naar zo klein mogelijke saturerende verzamelingen.

Er worden bijzondere deelverzamelingen punten in PG(n,q) gebruikt, zoals kegelsneden, om kleine saturerende verzamelingen te construeren. Er worden inductieve methodes gebruikt.

Binnen dit bachelorproject worden er enkele voorbeelden van saturerende verzamelingen en enkele technieken om kleine saturerende verzamelingen te construeren, bestudeerd.