Selberg sieve and parity problem

Promotor: Bin Chen

Co-supervisor: Jasson Vindas Díaz

Selberg sieve is one of the most powerful elementary tools in analytic number theory for studying the distribution of prime numbers. It was proposed by Norwegian mathematician Selberg in the 1940s. Since then, many outstanding mathematicians have applied and developed this tool to explore core topics in number theory, such as the Goldbach conjecture, the twin prime conjecture, and the gaps between primes. Nowadays, we have realized that the main obstacle to completely solving the above conjecture using the sieve method directly is the parity problem (also first discovered by Selberg). The goal of this project is to learn and master the basic principles of Selberg's sieve method and know what the parity problem is, and finally consider potential applications in Beurling numbers setting.

De Borel-Cantelli lemma's met toepassingen in een wiskundig model voor vermogensevolutie

Promotor: Hans Vernaeve

Het uitgangspunt voor dit project zijn de Borel-Cantelli lemma's in de maattheorie, die vooral in de waarschijnlijkheidsrekening toepassingen hebben. In die context vormen de lemma's een zogenaamde zero-one law, een voldoende voorwaarde waaronder bepaalde gebeurtenissen zich enkel met waarschijnlijkheid 0 of 1 kunnen voordoen.

Voor de toepassing in een wiskundig model - het zgn. yard sale model - voor de evolutie van vermogens is het uitgangspunt het artikel A new probabilistic analysis of the yard-sale model van C. Börgers en C. Greengard, https://arxiv.org/pdf/2308.01485

Naargelang de interesse van de student kan het project verder evolueren naar andere zero-one laws of naar andere analyses van het yard sale model (of verwante modellen).

Tauberstellingen

Szemerédi’s Regularity Lemma, or the Compactness of the Space of Graphons

Promoter: Jasson Vindas Díaz

Supervisor: Paul Meunier

One of the greatest joys for a mathematician is to discover a beautiful understanding of a difficult result. A striking example is the reinterpretation by Lovász and Szegedy of Szemerédi’s Regularity Lemma as the statement that the space of graphons is compact.
Szemerédi originally proved his lemma in the 1970s as a deep technical step in his proof of the Erdős–Turán conjecture on arithmetic progressions. Roughly speaking, it says that any sufficiently large graph can be decomposed into a bounded number of parts that behave regularly with each other—revealing an unexpected structure emerging as graphs grow large. Over the following decades, the lemma found many applications throughout mathematics, from graph theory and Ramsey theory to additive combinatorics and Fourier analysis, and became one of the cornerstones of modern combinatorics.
However, the lemma and its refinements remained highly technical, partly because one could not quite decide whether it was a lemma about graphs—which are fundamentally discrete objects—or about approximations and randomness—which are analytic in flavour. This changed with the introduction of the theory of graphons by Lovász. A graphon can be viewed as a continuous generalisation of a graph, allowing one to use analytic tools to be applied to graph-theoretical problems. As it turned out, these objects provided the exact framework for Szemerédi’s lemma: in the language of graphons, Lovász and Szegedy showed that Szemerédi's Regularity Lemma is equivalent to the statement that the space of graphons is compact.

The aim of this project is to become familiar with the elementary theory of graphons and to understand the proof of this equivalence. A more ambitious second step would be to study in detail the proof of compactness itself. The project is based on Lovász and Szegedy’s article “Szemerédi’s Lemma for the Analyst”, with Chapters 7–9 of Lovász’s textbook Large Networks and Graph Limits providing background on graphon theory.

Fractionele afgeleiden: een theoretische en historische verkenning

Promotor: Karel Van Bockstal

Omschrijving: In dit project bestudeer je het concept van fractionele afgeleiden – afgeleiden van willekeurige reële orde. Fractionele calculus vormt een natuurlijke veralgemening van de klassieke differentiaal- en integraalrekening en heeft toepassingen in onder meer anomalous diffusion, visco-elasticiteit en systeemtheorie.

Het eerste deel van het project bestaat uit een historische analyse van de ontwikkeling van het vakgebied, van de vroege discussies in de 17e eeuw tot de formele definities in de 19e en 20e eeuw. In het tweede, wiskundige deel bestudeer je de fundamentele eigenschappen van de Riemann-Liouville- en Caputo-definities van fractionele integralen en afgeleiden. Je vergelijkt beide benaderingen op vlak van definities, regulariteitseisen, initiële voorwaarden en toepasbaarheid bij het oplossen van (fractionele) differentiaalvergelijkingen.

Indien gewenst, kan het project worden uitgebreid met enkele analytische voorbeelden of een vergelijking met alternatieve definities zoals die van Grünwald-Letnikov.

Voorkennis: Grondige kennis van analyse, inclusief integraalrekening en gewone differentiaalvergelijkingen. Interesse in de structurele en conceptuele uitbreiding van klassieke wiskunde.

Bronnen:
Oldham, K. B., Spanier, J. The fractional calculus. Theory and applications of differentiation and integration to arbitrary order (1974)
Miller, K. S., Ross, B. An introduction to the fractional calculus and fractional differential equations (1993)

Analytische oplossingen van fractionele differentiaalvergelijkingen

Promotor: Karel Van Bockstal

Omschrijving: In dit bachelorproject bestudeer je fractionele differentiaalvergelijkingen – een fascinerende uitbreiding van de klassieke differentiaalvergelijkingen, waarbij de afgeleiden niet langer gehele orde hoeven te hebben, maar ook reële (zelfscomplexe) orde kunnen aannemen. Dit type vergelijkingen vindt steeds vaker toepassingen in o.a. fysica, toegepaste en financiële wiskunde, waar klassieke modellen tekortschieten.

Het doel van het project is om inzicht te krijgen in de theorie achter deze vergelijkingen, om vervolgens analytische oplossingsmethoden te onderzoeken en toe te passen. Een belangrijke techniek die hierbij aan bod komt is de Laplace-transformatie, een krachtig hulpmiddel dat ook in de klassieke analyse een centrale rol speelt.

Afhankelijk van je interesse kan het project theoretisch blijven of uitgebreid worden met concrete toepassingen of numerieke benaderingen.

Voorkennis: Analyse en differentiaalvergelijkingen (basisniveau), interesse in wiskundige modellering en oplossingsmethoden.

Bronnen:
Kilbas, A. A., Srivastava, H. M., Trujillo, J. J. Theory and applications of fractional differential equations (2006)
Diethelm, K. The analysis of fractional differential equations. An application-oriented exposition using differential operators of Caputo type (2010)
Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications (1998)

Numerieke methoden voor fractionele differentiaalvergelijkingen

Promotor: Karel Van Bockstal

Omschrijving: Voor de meeste fractionele differentiaalvergelijkingen (FDE’s) bestaan er geen gesloten analytische oplossingen. Daarom is het gebruik van numerieke methoden essentieel om deze vergelijkingen op te lossen en om relevante grootheden, zoals fractionele afgeleiden en speciale functies, te evalueren.

In dit project bestudeer je numerieke technieken die ontwikkeld zijn voor het oplossen van FDE’s. Mogelijke focussen binnen het project zijn onder meer:

• de numerieke evaluatie van de Riemann-Liouville en Caputo afgeleiden,
• het opstellen en implementeren van algoritmen voor de numerieke oplossing van FDE’s,
• de efficiënte berekening van Mittag-Leffler functies, die een centrale rol spelen als vergalgemening van de exponentiële functie in de theorie van FDE’s.

Afhankelijk van je interesse kan het project zich richten op de theoretische analyse van deze methoden (convergentie, stabiliteit, foutanalyse) of op de praktische implementatie ervan in een programmeertaal zoals Python.

Voorkennis: Analyse, differentiaalvergelijkingen, en basiskennis van numerieke wiskunde. Ervaring met programmeren is een pluspunt maar geen vereiste.

Bronnen (software):
pycaputo: https://pycaputo.readthedocs.io/en/latest/
evaluatie Mittag-Leffler functie: https://nl.mathworks.com/matlabcentral/fileexchange/48154-the-mittag-leffler-function
https://www.dm.uniba.it/it/members/garrappa/software

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: 

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

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

[3] 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.

[4] 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 R^n. 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

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 [1]. Depending on time and motivation we could move on to other type of constructions, perhaps Hodge cohomology or Morse homology.

References: 

[1] 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

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: 

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

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

[3] 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

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.

Topics in Analysis

Promotor: Michael Ruzhansky

If you are interested in some topic in analysis in broad sense, or some combination of analysis and for example the theory of groups (or even in more general mthematics), feel free to contact me and discuss it - maybe we can find an interesting topic to do a project together.

Orthonormal wavelet bases

Promotor: Michael Ruzhansky

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.

Kwantooreliminatie voor algebraïsche theorieën

Promotor: Andreas Weiermann 

Uit de les logica kennen wij al de kwantooreliminatie voor algebraïsche gesloten velden. In dit project wordt kwantooreliminatie voor andere algebraïsche theorien bekeken en wij bestuderen ook sommige toepassingen.

Referenties [1] David Marker. Model theory: An introduction. Springer.

De MacDowell en Specker stelling

Promotor: Andreas Weiermann

In dit project wordt de volgende stelling verder bekeken. Ieder model M van de Peano rekenkunde heeft een echte elementaire eind-extensie N van dezelfde kardinaliteit. We zeggen hierbij dat N een eindextensie van M is als voor elementen a,b in N geldt: als a<b en als b in M dan is ook a in M.

Referenties 

Bell Slomson: Models and Ultraproducts. North-Holland.

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 [1] Thomas Jech. Set Theory. Springer.

De algebraïsche theorieën van binaire relaties en partiële functies

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 [1] gevonden zijn. Het artikel [2] geeft een beschrijving van de mozaïekmethode als gebruikt voor complexiteitsgrenzen en hoe deze kunnen aangepast worden voor verschillende tijdlijnen.

Referenties

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

[2] 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:

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

Quasi-Polish spaces

Promotor: Giovanni Solda

A topological space is quasi-Polish if it is countably based completely quasi-metrizable. The class of quasi-Polish spaces is of particular interest in mathematical logic because, on the one hand, it inheriths many of the good properties that Polish spaces have (to speak freely: Polish and quasi-Polish spaces behave somewhat similarly from the point of view of descriptive set theory), and on the other, it is large enough to include several important non-metrizable spaces that are used in, among others, denotational semantics and generalized computability theory.

The main goal of this project would be to study some basic properties of the class of quasi-Polish spaces. A good goal would be to show that they enjoy the property of Baire, but several other directions can be taken.

References:

[1] Matthew de Brecht, Quasi-Polish spaces, Annals of Pure and Applied Logic, Volume 164, Issue 3, March 2013, pages 356-381

Intuitionistic Modal Logic

Intuitionistic modal logic extends the constructive principles of intuitionistic logic with modal operators that capture forms of necessity and possibility while preserving an intuitionistic semantics. Unlike its classical counterpart, intuitionistic modal logic admits a richer semantic structure—most notably Kripke frames equipped with both an intuitionistic preorder and one or more modal accessibility relations. This combination allows for constructive interpretations of modality relevant to areas such as type theory, program verification, and constructive mathematics. Foundational systems were introduced by Fischer Servi and later developed in depth by Simpson, who provided complete axiomatizations and Kripke semantics for a broad family of intuitionistic modal logics. Because of its ability to model constructive reasoning about knowledge, computation, and obligation, intuitionistic modal logic has become a central topic in contemporary investigations at the interface of logic, semantics, and theoretical computer science.

Standard Reference

Alex Simpson, The Proof Theory and Semantics of Intuitionistic Modal Logic, PhD thesis, University of Edinburgh, 1994.

Constructing graphs with the same spectrum

Promotor: Aida Abiad

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 [1].

Similar methods to this switching seem to have appeared in the literature but in a different context: to construct new Hadamard matrices [2]. A Hadamard matrix is a square matrix whose entries are either 1 or −1 and whose rows are mutually orthogonal. The work in [2] 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

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

Is the spectrum a fingerprint of a graph?

 Promotor: Aida Abiad

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

Promotoren: Aida Abiad, Jozefien D´haeseleer

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.

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

An algebraic approach to the graph integrity

Promotoren: Aida Abiad, Alessandro Neri

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 [3] 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.

[1] N. Alon, A. Bishnoi, S. Das, and A. Neri. Strong blocking sets and minimal codes from expander graphs. arXiv:2305.15297, 2023.
[2] K.S. Bagga, L.W. Beineke, A survey of integrity, Discrete Applied Mathematics 37-38 (1992), 13-28.
[3] C.A. Barefoot, R. Entringer, H.C. Swart, Vulnerability in graphs - A comparative survey, J. Combin. Math. Combin. Comput. 1 (1987), 13-22.
[4] 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.
[5] W. Goddard, H.C. Swart, Integrity in Graphs : Bounds and Basics (2006).
[6] 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.

Erd˝os–Ko–Rado problemen

Promotoren: Jozefien D'haeseleer en Robin Simoens

Gegeven een verzameling van n elementen, hoeveel deelverzamelingen van grootte k kan men maximaal vinden zodat elke twee deelverzamelingen een element gemeen hebben? 
Een mogelijkheid is om alle C(n-1,k-1) deelverzamelingen te selecteren die een vast element bevatten, met C(n-1,k-1) het aantal combinaties van k-1 elementen uit n-1 elementen. Erd˝os, Ko en Rado bewezen dat, als n > 2k, dit de beste manier is [1]. Met andere woorden, elke collectie van deelverzamelingen van grootte k zodat elk paar een nietlege doorsnede heeft, bestaat uit hoogstens C(n-1,k-1) deelverzamelingen.
Deze stelling, afgekort de EKR-stelling, kan veralgemeend worden naar andere wiskundige structuren waarvoor een doorsnede gedefinieerd is. Voorbeelden zijn multisets, permutaties en vectorruimten (over eindige velden). Er bestaan ook varianten waarbij de doorsnede een gegeven getal moet zijn (niet zomaar een niet-nul getal).
De stelling van Hilton-Milner zegt dat de maximale grootte van een collectie deelverzamelingen van grootte k die niet allemaal eenzelfde element bevatten, direct een pak kleiner is dan het extreme geval, nl. C(n-1,k-1)-C(n-k-1, k-1)+1. Dit soort resultaten worden stabiliteitsresultaten genoemd.

In deze bachelorproef is het de bedoeling om enkele varianten van de EKR-stelling te bestuderen. Een goed startpunt is het boek [3], waar de bewijzen vooral steunen op algebraïsche technieken. Er kan ook gekeken worden naar recentere, alternatieve manieren om de EKR-stelling en gerelateerde stabiliteitsresultaten zoals de stelling van Hilton-Milner te bewijzen [2].

Bibliografie:

[1] P. Erd˝os, C. Ko en R. Rado. Intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. 2 (1961), 12:313–320. 
[2] P. Frankl. A simple proof of the Hilton–Milner theorem. Mosc. J. Comb. Number Theory 8 (2) (2019), 97–101.
[3] C. Godsil en K. Meagher. Erdos-Ko-Rado theorems: algebraic approaches. Vol. 149. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2016).
[4] A. J. W. Hilton en E. C. Milner. Some intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. 2 (1967), 18:369–384. 

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.

De vriendschapsstelling

Promotoren: Leo Storme en  Robin Simoens

Als op een feestje elke twee personen juist één gemeenschappelijke vriend hebben, dan is er steeds iemand die bevriend is met iedereen. Deze eigenschap is een gevolg van de zogenaamde vriendschapsstelling (friendship theorem) die bewezen werd door Erdos, Renyi en Sós [ERS]. De stelling zegt dat een graaf waarin elke twee toppen juist één gemeenschappelijke buur hebben, isomorf is met een vriendschapsgraaf: een ``unie'' van 3-cykels die een top delen. De stelling kent meerdere elegante bewijzen. Het originele bewijs [ERS] maakt zelfs deel uit van de zogenaamde ``proofs from THE BOOK''. 

Een P_k-graaf is een graaf met de eigenschap dat er tussen elk paar toppen juist één pad van lengte k bestaat. De vriendschapsstelling zegt dat elke P_2-graaf isomorf is met een vriendschapsgraaf. Een vermoeden van Kotzig zegt dat P_k-grafen niet bestaan als k>2. Het vermoeden is bewezen voor k>19 door Kostochka in [K]. Het algemene geval is nog steeds een open probleem (ondanks enkele foutieve bewijzen in de literatuur). In datzelfde artikel claimt Kostochka dat een analoge redenering als in [ERS] ook toelaat om het vermoeden te bewijzen als k>29, maar een bewijs ontbreekt.

Recent is er een analogon bewezen voor de vriendschapsstelling in het geval van gerichte grafen [CCK]. Hierin spelen projectieve vlakken een rol.

In deze bachelorproef is het de bedoeling om de vriendschapsstelling en haar bewijs (of meerdere bewijzen) te bestuderen. Er kan ook gekeken worden naar veralgemeningen van de stelling, zoals de versie voor gerichte grafen, en hoe het zit met het vermoeden van Kotzig voor deze varianten.

Bibliografie:

[CCK] M. Choi, H. Chu, S.-R. Kim, A digraph version of the Friendship Theorem, Discrete Math. 348 (2025), #114238.

[ERS] P. Erdos, A. Rényi en V. Sós, On a problem of graph theory, Studia Sci. Math. Hungar. 1 (1966), 215--235.

[K] A.V. Kostochka, The nonexistence of certain generalized friendship graphs. In Combinatorics (Eger, 1987). Vol. 52 (1988). North-Holland, Amsterdam, 341--356.

Quantum Latijnse vierkanten

Promotoren: Leo Storme en Robin Simoens

Voor deze bachelorproef is het aangeraden om het vak ``Kwantummechanica 2'' gevolgd te hebben. Het uitgangspunt voor dit project is het artikel [Q36].

Een Latijns vierkant is een n*n rooster met n verschillende symbolen die juist één keer voorkomen in elke rij en elke kolom. Een sudoku is een voorbeeld van een Latijns vierkant. Latijnse vierkanten (meer bepaald, onderling orthogonale Latijnse vierkanten) hebben toepassingen in codeertheorie.

Met de snelle opkomst van quantum codeertheorie voelt het natuurlijk om Latijnse vierkanten te veralgemenen naar de quantum wereld. In [QLS] introduceren Musto en Vicary het concept van quantum Latijnse vierkanten. Recent werd bewezen dat deze eigenschappen hebben die klassiek onmogelijk zijn: het beroemde 36 officieren probleem van Euler heeft wél een quantummechanische oplossing [Q36].

Het doel van deze bachelorproef is om zowel de klassieke als de quantum versie van Eulers 36 officieren te begrijpen. Als extra uitdaging kan eventueel gekeken worden naar manieren om een quantum projectief vlak te definiëren aan de hand van onderling orthogonale quantum Latijnse vierkanten.

Bibiliografie:

[QLS] B. Musto en J. Vicary, Quantum Latin squares and unitary error bases. Quantum Inf. Comput. 16 (2015), 1318--1332.

[Q36] K. Życzkowski et al, 9 × 4 = 6 × 6: Understanding the Quantum Solution to Euler’s Problem of 36 Officers. J. Phys.: Conf. Ser. 2448 (2023), 012003.

Propagation processes on graphs

Promotoren: Aida Abiad en Robin Simoens

The burning and forcing processes are both instances of propagation processes on graphs that are commonly used to model real-world spreading phenomena. For instance, they are used for a wide range of network-related problems such as recommender systems, electric power networks and the control of quantum systems.

Burning on a graph can be understood as a process where in each step, we set a vertex on fire. Fire has the property that it propagates from every vertex to its neighbours in each step. It can be compared to the spread of gossip among a group of friends. The burning number is the least number of steps that one needs to set the whole graph on fire.

Zero forcing is a process of colouring the vertices of a graph according to the rule "if a coloured vertex has a unique white neighbor, color that neighbour". A subset of vertices is a zero forcing set if it eventually colours all vertices. Otherwise its complement is a zero blocking set. The minimum size of such sets are called the zero forcing number and zero blocking number, respectively. 

In general, both the zero forcing number and the burning number difficult to compute (in fact both are NP-hard), but for certain graph classes they can be determined explicitly, and also bounds can be obtained. The goal of this project is to explore bounds or exact values of the zero forcing and the burning number for some particular classes of graphs coming from geometries: Johnson graphs, Hamming graphs and their generalisations, see [1-4]. This project aims to obtain similar results for related graph classes.

Bibliografie:

[1] A. Abiad, R. Simoens and S. Zeijlemaker, On the diameter and zero forcing number of some graph classes in the Johnson, Grassmann and Hamming association scheme, Discrete Appl. Math. 348 (2024) 221-230.

[2] F. Afzali, A.H. Ghodrati and H.R. Maimani, Failed zero forcing numbers of Kneser graphs, Johnson graphs, and hypercubes. J. Appl. Math. Comput. 70(3) (2024) 2665-2675.

[3] H. Lin, W. Lin, W. and G.J. Chang, The zero blocking numbers of generalized Kneser graphs and generalized Johnson graphs. arXiv preprint (2025), arXiv:2508.02125.

[4] H. Tanaka and N. Tokushige, Burning numbers via eigenpolytopes - Hamming graphs, Johnson graphs, and halved cubes. arXiv preprint (2025), arXiv:2508.17559.

Multicolor Ramsey Numbers

Promoter: Vladislav Taranchuk

Take any six birds in Ghent, did you know that it has to be the case that either three of them have all sat on the same tree before (not necessarily at the same time), or three of them have never sat on a common tree? A quick argument can prove that this has to be the case. This statement is a classical case of a problem in graph theory called a Ramsey problem. In general, the question can be formulated as follows: Let F be a graph and r, n be positive integers. Is it possible to color the edges of the complete graph F so that we cannot find a monochromatic copy of F?

For this bachelor thesis we will look at some classical cases of this problem and see how finite geometry has been of use to provide some surprisingly powerful answers to this question. We will mostly restrict ourselves to cases where F is a cycle, a complete graph, or complete bipartite graph. We will also look at some generalizations and see how finite geometry has been used in recent years to tackle the generalizations as well.