Thesisonderwerpen wiskunde bij de Vakgroep Wiskunde: Analyse, Logica en Discrete Wiskunde

Thesissen in de vakgroep Wiskunde: Analyse, Logica en Discrete Wiskunde

Wanneer men een masterscriptie wil maken in de Zuivere Wiskunde kan men uit een waaier van specialisatierichtingen kiezen. Verschillende onderzoekers en onderzoeksgroepen uit de vakgroep Wiskunde: Analyse, Logica en Discrete Wiskunde bieden de mogelijkheid om thesiswerk te verrichten dat aansluit bij hun specialisatiedomein en/of bij hun onderzoek.

Er is geen beperkende lijst van onderwerpen waaruit de studenten moeten kiezen. Bij wijze van voorbeeld vindt men op deze webpagina een aantal concrete onderwerpen. De studenten kunnen ook zelf een voorstel doen, over de richting, aard en karakter van het werk dat ze willen doen, dit wordt zelfs aangemoedigd.

We raden u aan rechtstreeks contact op te nemen met de potentiële promotoren. Zo krijgt u uit eerste hand een goed idee van de inhoud van de verschillende specialisaties en van de mogelijke onderwerpen. Dit kan men bijvoorbeeld doen in de loop van het tweede semester van de 1ste master. Op die manier kan men bijvoorbeeld al 1 of 2 mogelijke richtingen kiezen. Na de examens (rond de proclamatie) zullen de promotoren de onderwerpen meer gedetailleerd toelichten en duidelijk maken wat juist verwacht wordt van de studenten. Dit kan, afhankelijk van de interesse, individueel of in groep gebeuren. Men kan dan een voorlopige keuze maken van bijvoorbeeld 2 of 3 onderwerpen. Aan de hand van de opgegeven literatuur kan men zich beginnen voorbereiden op het eigenlijke thesiswerk en een definitieve keuze maken bij de start van het nieuwe academiejaar.

Lijst van potentiële promotoren:

A. Weiermann, Computeralgebra.
M. Ruzhansky, Analyse en Partiële differentiaalvergelijkingen.
H. Vernaeve, Analyse.
J. Vindas, Analyse en Getaltheorie.
L. Storme, Combinatoriek en incidentiemeetkunde.
F. Pakhomov.


Zie ook …


Incidentiemeetkunde

Which graph properties are not determined by the spectrum?

Promotor: Aida Abiad

Spectral graph theory aims to understand to what extent graphs are characterized by their spectra. Starting from the eigenvalues of a matrix associated to a graph, it seeks to deduce combinatorial properties of the graph. For this, we associate a graph G to a matrix such as the adjacency matrix A or the Laplacian matrix L. Important types of relations are the spectral characterizations. These are conditions in terms of the spectrum of the associated matrix, which are necessary and sufficient for certain graph properties. Two famous examples for the matrices A and L are:

(i) a graph is bipartite if and only if the spectrum of A is invariant under multiplication by -1,
(ii) the number of connected components of a graph is equal to the multiplicity of the eigenvalue 0 of L.

Properties that are characterized by the spectrum for A as well as for L are the number of vertices, the number of edges, and regularity. If a graph is regular, the spectrum of A is determined from the spectrum of L, and vice versa. This implies that for both A and L the properties of being regular and bipartite, and being regular and connected are characterized by the spectrum. If a property is not characterized by the spectrum, then there exist a pair of non-isomorphic graphs with the same spectrum (cospectral graphs). For many graph properties and several types of associated matrices, such pairs of cospectral graphs are not hard to find. Such a pair of regular cospectral graphs has been found for a number of properties. For example: being distance-regular [1], having a given diameter [2], admitting a perfect matching [3] and having a given vertex or edge-connectivity [4].  Motivated by the complexity of properties of graphs that can be characterized by the spectrum of the adjacency matrix of the graph, spectral characterizations of some well-known NP-hard properties have recently been studied [5].

In this project we will investigate other graph properties which cannot be derived from the spectrum of a specific matrix associated to a graph. 

[1] W.H. Haemers, Distance-regularity and the spectrum of graphs, Linear Algebra Appl. 236 (1996), 265--278.
[2] W.H. Haemers and E. Spence, Graphs cospectral with distance-regular graphs, Linear Multilinear Algebra 39 (1995) 91–107.
[3] Z. Blázsik, J. Cummings and W.H. Haemers, Cospectral regular graphs with and without a perfect matching, Discrete Math. 338 (2015), 199--201.
[4] W.H. Haemers, Cospectral pairs of regular graphs with different connectivity, Discussiones Mathematicae Graph Theory 40 (2020), 577--584.
[5]  O. Etesami and W.H. Haemers, On NP-hard graph properties characterized  by the spectrum, Discrete Applied Mathematics 285 (2020),  526--529.

Aida Abiad

A geometrical approach to a combinatorial question

Promotor: Aida Abiad

The k-th power of a graph G =(V, E) is a graph with vertex set V in which two distinct elements of V are joined by an edge if there is a path in G of length at most k between them. The main motivation for this thesis project comes from distance colorings, which have received a lot of attention in the literature. In particular, this project focus on the following question of Alon and Mohar [1]:

Question. What is the largest possible value of the chromatic number of the k-th power graph among all graphs G with maximum degree at most d and girth (length of a shortest cycle contained in G) at least g?

The main challenge in the above question is to provide examples with large distance chromatic number (under the condition of girth and maximum degree). For k = 1, this question was essentially a long-standing problem of Vizing, one that stimulated much of the work on the chromatic number of bounded degree triangle-free graphs, and was eventually settled asymptotically by Johansson by using the probabilistic method. The case k = 2 was considered and settled asymptotically by Alon and Mohar. For larger k, bounds are known, mostly with an extremal and algebraic flavor.

Although much research has been done into trying to solve Alon and Mohar's question, less is known on how  finite geometry can help to prove good bounds. Thus in this project we will investigate the above question from a geometrical point of view. We will do it by investigating further links between the independence (and chromatic) number of the k-th power graph and the distance-j ovoids in incidence geometry [3,4]. We will for instance investigate the bounds from [2, Section 3].

[1] N. Alon and B. Mohar, The chromatic number of graph powers, Combin. Probab. Comput. 11  (2002), 1--10.
[2] A. Bishnoi and F. Ihringer, Some non-existence results for distance-j ovoids in small generalized polygons, Contributions to Discrete Mathematics 12(1) (2000), 157–161. (https://arxiv.org/abs/1606.07288, for longer version).
[3] A. Offer and H. Van Maldeghem, Distance-j ovoids and related structures in generalized polygons, Discrete Mathematics 294(1-2) (2005), 147-160.
[4] J.A. Thas, Ovoids and spreads of finite classical polar spaces, Geom. Dedicata 10(1-4) (1981), 135--143.

Aida Abiad

Expansion and Thresholds in Finite Geometries

Promotor: Ferdinand Ihringer


Current topics in extremal combinatorics and theoretical computer science are results on thresholds and expansion in finite vector spaces. For instance, given two families Y and Z of k-spaces in an n-dimensional vector space, how many elements of Y meet an element of Z in a (k-1)-space? This is closely related to spectral properties of Grassmann graphs.

Goal of the thesis is to study expansion in finite geometries such as projective spaces and polar spaces. See [1] for a possible starting point. The student is expected to investigate open problems in this project.

[1] B. Rossman. Thresholds in the Lattice of Subspaces of $(\mathbb F_q)^n$. https://arxiv.org/abs/1910.00656v1


Rank Bounds in Vector Spaces

Promotor: Ferdinand Ihringer

Bounding families with restricted intersection sizes is an important topic in combinatorics. Here rank bounds are an extremely important tool. Goal of the thesis is to study recent papers on rank bounds for families with restricted intersection sizes thoroughly. See [1] for a classical result and [2,3] for recent developments.

There is the possibility (but no necessity) to conduct research in the framework of this project. For instance, one can investigate special choices for L or one can generalize the work to polar spaces.

[1] P. Frankl, R. Graham. Intersection theorems for vector spaces. European Journal of Combinatorics, 6(2):183 – 187 (1985).
[2] J. Q. Liu, S. G. Zhang, J. M. Xiao. A Common Generalization to Theorems on Set Systems with L-intersections. Acta Mathematica Sinica, English Series volume 34, pages 1087--1100 (2018). https://arxiv.org/abs/1707.01715
[3] R. Mathew, T. K. Mishra, R. Ray, S. Srivastava. Modular and fractional L-intersecting families of vector spaces. https://arxiv.org/abs/2004.04937v2

Ferdinand Ihringer
Fedor Pakhomov

Cospectral generalized Kneser graphs

Promotoren: Aida Abiad, Jozefien D´haeseleer

The spectrum of a graph G is the multi-set of eigenvalues of its adjacency matrix. Two graphs are called cospectral if they have the same spectrum. A graph G is determined by spectrum if any graph cospectral to G must be isomorphic to G. Two non-isomorphic graphs that are cospectral are called cospectral mates. An important research area of spectral graph theory is devoted to determining which graphs are determined by their spectra (see [1] for example).
In this project we will investigate whether the generalized Kneser graphs are not determined by their spectra. We will do this by investigating whether the results known for Kneser graphs [2] (construction of cospectral graphs with the Kneser graph) extend to the generalized Kneser graphs (see [3] for their definition).

[1] E. van Dam and W.H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003), 241–272.
[2] W.H. Haemers and F. Ramezani, Graphs cospectral with Kneser graphs, Combinatorics and graphs, 159–164, Contemp. Math., 531, Amer. Math. Soc., Providence, RI, 2010.
[3] J. D'haeseleer, K. Metsch and D. Werner. On the chromatic number of two generalized Kneser graphs. arXiv:2005.05762

Jozefien D'haeseleer

Deelstructuren in eindige projectieve ruimten en eindige klassieke polaire ruimten (Meetkunde)

Promotor: Leo Storme

Binnen eindige projectieve ruimten en eindige klassieke polaire ruimten worden vele verschillende deelstructuren bestudeerd. Dit omvat blokkerende verzamelingen, partiele spreads, en recent ook Cameron-Liebler rechtenverzamelingen, Erdos-Ko-Rado verzamelingen, en tight sets.

Binnen dit onderwerp worden enkele deelstructuren bestudeerd die recent veel aandacht gekregen hebben binnen de eindige projectieve ruimten en/of eindige klassieke polaire ruimten. 

Zo kan er een studie gemaakt worden van partiele k-spreads in eindige projectieve ruimten. Een partiele k-spread in PG(n,q) is een verzameling van paarsgewijs disjuncte k-dimensionale deelruimten in PG(n,q). Een partiele k-spread noemen we maximaal als zij niet bevat is in een grotere partiele k-spread.

Zo kan het recente resultaat van Dr. Maarten De Boeck besproken worden over de ondergrens op de kleinste maximale partiele k-spreads in PG(2k+1,q), alsook andere verwante resultaten over maximale partiele k-spreads in PG(n,q).

 Analoog wordt er op dit ogenblik intensief onderzoek verricht over Erdos-Ko-Rado verzamelingen en Cameron-Liebler verzamelingen in eindige projectieve ruimten. Hier worden er, naast meetkundige, ook  vele andere  technieken gebruikt, zoals matrixtechnieken. Bij een keuze voor de studie van deze deelstructuren kunnen dus verschillende technieken bestudeerd worden.

Leo Storme

Lineaire codes komende van meetkundige structuren (Codeertheorie en meetkunde)

Promotor: Leo Storme

Binnen de codeertheorie worden vele codes bestudeerd die in verband staan met meetkundige structuren. Zo worden in detail de lineaire p-aire codes gedefinieerd door de incidentiematrices van punten met k-ruimten van PG(n,q), q=p^h, p priem, bestudeerd. Analoog worden de duale codes van deze lineaire codes bestudeerd.

Verder zijn er vele verbanden tussen lineaire codes en specifieke deelverzamelingen punten in eindige projectieve ruimten. Dit verband gebeurt heel veel via de kolommen van een generator of pariteitscontrole matrix van deze lineaire codes. Via deze verbanden tussen lineaire codes en specifieke deelverzamelingen punten in eindige projectieve ruimten hebben vele problemen uit de codeertheorie een equivalent meetkundig probleem. Twee concrete voorbeelden zijn het verband tussen lineaire MDS codes en bogen, en tussen lineaire codes die de Griesmer grens bereiken en minihypers.

Op dit ogenblik is er een groot europees project over random network codes (http://www.network-coding.eu/). Dit is een nieuw type code waarin de codewoorden projectieve deelruimten uit een gegeven eindige projectieve ruimte zijn. Dit impliceert dat vele problemen over random network codes gelijk zijn aan meetkundige problemen. 

Deze masterproef heeft tot doel codes en verbanden met deelstructuren uit projectieve ruimten te bestuderen. Er kan gekozen worden om lineaire codes corresponderend met eindige projectieve ruimten te bestuderen, om een specifiek verband tussen een bepaald probleem uit de codeertheorie met een equivalent meetkundig probleem te bestuderen, of om random network codes te bestuderen.

Leo Storme

Logica

Faseovergangen

Promotor: Andreas Weiermann

Een mogelijke thesis gaat over actueel onderzoek omtrent fasenovergangen voor onafhankelijkheidsstellingen (zoals bijvoorbeeld de stelling  van Kruskal of van Paris en Harrington.

Dit onderwerp sluit direct aan bij de lezing over fasenovergangen en/of bewijstheorie. (Er zijn meerdere thesisprojecten over faseovergangen beschikbaar.)

Andreas Weiermann

Analytische combinatoriek van het oneindige

Promotor: Jasson Vindas en Andreas Weiermann

Een mogelijke thesis gaat over actueel onderzoek omtrent de sterke asymptotiek van telfuncties voor ordinaalgetallen. Daarvoor moet een Tauberstelling van Ingham worden uitgebreid.  Mogelijke toepassingen tot limietwetten voor ordinaalgetallen zullen worden onderzocht.

Andreas Weiermann

Veralgemeende Goodsteinreeksen


Een mogelijke thesis gaat over een veralgemening van het Paris-Kirby-onafhankelijkheidsresultaat voor de Goodsteinreeksen. 
Andreas Weiermann

Logische Limietwetten voor ordinalen

Een mogelijke thesis gaat over eigenschappen van toevallig gekozen ordinalen.We nemen aan dat E een eigenschap van ordinalen is die kan worden uitgedrukt in de logica van eerste orde of in de monadische logica van tweede orde. We nemen verder aan dat een bepaald segment S van ordinalen is gegeven. Hoe groot is de kans dat E waar is voor een ordinaal alpha uit S als we alpha uit S "toevallig" kiezen?

In het project gaat het erover om in een gepaste context eerst een nul een wet voor de logida van eerste orde te bewijzen voor echte deelsegmenten van de ordinalen kleiner dan epsilon_0 en het zou dan worden aangetoond dat voor de monadische logica nog steeds gepaste limietwetten bestaan. 

Andreas Weiermann

Generalizations of Fusible Numbers

Promotor: Fedor Pakhomov

Consider the following mathematical puzzle. We are given fuses that burn out in exactly one minute if ignited from one end. Then we are interested in procedures of measuring intervals of time using only this fuses: In the beginning of a procedure some fuses are ignited (either from one or both ends) and it is allowed to do some further ignitions in future points of time that are the moments when some of the fuses burns out completely.

The set of intervals of time (measured in minutes) that could be measured by some procedure of this kind is called the set of fusible numbers. Notice that if we ignite a fuse from one end at a moment x and another end at a moment y, where |x-y|<1, then the fuse burns out at the moment f(x,y)=(x+y+1)/2. It is easy to see that the set of fusible numbers is the inclusion least set F that contains 0 and for any x,y with f(x,y)>max(x,y) containing f(x,y). An interesting property of the set of fusible numbers that it is a well-ordered set of reals whose order type is the ordinal ε₀. And since the ordinal ε₀ is the characteristic ordinal of fist-order Peano Aritmetic PA, in fact a number of true facts about the set of fusible numbers aren't provable in PA. In particular PA doesn't prove the fact that for any fusible number x there is the smallest fusible number above x.

For  any monotone function f(x₁,..,xₙ) on reals analogously to the set of fusible numbers we define the set F(f) that is the inclusion least set consisting of 0 and together with any x₁,..,xₙ containing f(x₁,..,xₙ) as long as f(x₁,..,xₙ)>max(x₁,..,xₙ). In follows from Kruskal's tree theorem that the set F(f) is always well-ordered. This project is about studying the properties of F(f) for some specific functions f.

[1] Jeff Erickson. Fusible numbers. https://www.mathpuzzle.com/fusible.pdf.
[2] Jeff Erickson, Gabriel Nivasch, and Junyan Xu. Fusible numbers and Peano arithmetic. In Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 2021.

Fedor Pakhomov

Fast-Growing Functors

Promotor: Fedor Pakhomov


For countable ordinals Λ equipped with certain additional structure one could define hierarchies of fast-growing functions < F_α:ℕ→ℕ|α<Λ > [1]. The fast-growing hierarchies are important in proof theory since they allow to characterize the provably total computable functions of fist order theories: In particular a computable function is provably total in first-order Peano arithmetic iff for some α<ε₀ the function is computable in time bounded by F_α. Functions from fast growing hierarchy could be naturally extended to functions on the class of ordinals F_α:On→On [2]. In fact this extension is achieved by showing that those functions could be naturally treated as functors on the category of well-orders and strictly monotone functions between them. The goal of this project is to study the properties of the functors F_α:On→On for particular ordinals α≤ε₀.

[1] Rose, H.E., 1984. Subrecursion: functions and hierarchies.
[2] Aguilera, J. P., F. Pakhomov, and A. Weiermann. "Functorial Fast-Growing Hierarchies." arXiv preprint arXiv:2201.04536 (2022).

Fedor Pakhomov

The Continuum Hypothesis, Forcing, and the Real Numbers

Promotor: Juan Aguilera

Cantor's Continuum Hypothesis asserts that there is no set of real numbers whose cardinality is strictly between those of the set of all natural numbers and the set of all real numbers. By theorems of Kurt Gödel and Paul Cohen, the Continuum Hypothesis is neither provable nor refutable from the Zermelo-Fraenkel axioms of set theory.

Cohen's proof introduced the technique of "forcing." Essentially, it is a way of extending a set-theoretic universe into a larger one with different properties, with the goal of "forcing" a certain statement to become true or false. This thesis will explore the basics of forcing and some of its applications to the theory of the real numbers. 

Problems that could be explored include: definability ("must there exist real numbers which cannot be defined?"), cardinal characteristics ("how many Lebesgue-null can we join while ensuring that the union is still null?"), the Axiom of Choice and its weakenings which involve the real numbers, and more.
jaduiler

Reverse Mathematics and Infinite Games

Promotor: Juan Aguilera

Reverse Mathematics is the branch of logic that studies which axioms are needed to prove which theorems in mathematics. Classical results include (i) the Bolzano-Weierstrass theorem cannot be proved without using the full strength of mathematical induction on the natural numbers; (ii) the perfect set theorem cannot be proved without using the full strength of transfinite recursion for sets of natural numbers; and more.

This thesis would study the main axiomatic systems that show up in reverse mathematics and some theorems from algebra or analysis that they correspond to. Its goal would be to study the reverse mathematics of theorems from infinite game theory which assert that various two-player infinite games are determined. Depending on the precise direction pursued, the thesis will involve aspects of set theory, proof theory, and/or computability theory. 

It might be possible for the thesis to explore open problems.
jaduiler

Complete Ultrafilters and the Topology of Provability

Promotor: Juan Aguilera

Provability logic is an extension of propositional logic which allows expressing the structural properties of formal provability. These algebraic properties also happen to hold of other objects, such as certain algebras or topological spaces.

The purpose of this thesis is to attack an open problem on whether a certain topological space is completely described by this logic. This is a space on ordinal numbers obtained from countably complete ultrafilters. Due to some recent advances, a solution to the problem should be within reach now. The thesis involves aspects of set theory, general topology, and modal logic.
jaduiler

Analyse

Infinitesimale Analyse en Topologische Vectorruimten

Promotor: Hans Vernaeve

In de cursus Infinitesimale analyse behandelen we distributietheorie, en kort ook algemene topologie, met infinitesimalen (d.m.v. een nietstandaard model). Klassiek worden resultaten over distributies gekaderd binnen de (zeer uitgebreid bestudeerde) algemene theorie van de topologische vectorruimten. In deze thesis willen we

1) de bestaande nietstandaard theorie van topologische vectorruimten beter leren kennen

2) een aantal verdere stellingen over distributietheorie binnen het infinitesimale framework uitwerken.

Hans Vernaeve

Veralgemeende functies in de infinitesimale analyse

Promotor: Hans Vernaeve

in de cursus infinitesimale analyse behandelden we veralgemeende functies (Schwartz distributies) met infinitesimalen. In deze thesis willen we het gebruik van veralgemeende functies onderzoeken die buiten de distributie-theorie vallen. Dit houdt aan de ene kant een studie in van hoe in de (standaard) analyse veralgemeende functies gebruikt worden die buiten het kader van Schwartz vallen, en aan de anderen kant hoe we dit intuïtief kunnen behandelen via infinitesimalen.

Hans Vernaeve

Asymptotische verdeling van veralgemeende priemgetallen en veralgemeende gehelen

Promotor: Jasson Vindas

De theorie van de veralgemeende priemgetallen stelt zich als doel de verzameling van de gewone priemgetallen te vervangen door een tamelijk willekeurige stijgende rij van positieve reële getallen (veralgemeende priemgetallen). Ze bestudeert de multiplicatieve groep die hierdoor voortgebracht wordt (veralgemeende gehelen) en het verband tussen asymptotische eigenschappen van de verdeling van de veralgemeende gehelen en priemgetallen. Dit probleem werd voor het eerst bestudeerd in 1937 door Beurling, die in deze context abstracte versies vond van de priemgetallenstelling.

Het doel van de thesis is klassieke en recente resultaten in dit gebied te bestuderen en een dieper begrip te verkrijgen over een aantal nuttige technieken uit de analyse die hier gebruikt worden (o.a. uit de Fourier-analyse en complexe analyse).

Jasson Vindas Diaz

Analytische voorstellingen en randwaarden

Promotor: Jasson Vindas

Een idee dat in de analyse zeer nuttig gebleken is, is het bestuderen van reële objecten d.m.v. complexe objecten. Zo kan men eigenschappen van een functie van een reële veranderlijke beter begrijpen door over te gaan op het complexe vlak en haar te beschouwen als de sprong die twee analytische functies maken wanneer men de reële as doorkruist. Dit idee leidt tot het concept analytische voorstelling. Het eerste doel van de thesis is een aantal technieken te bestuderen om analytische voorstellingen te construeren. Het omgekeerde probleem is ook interessant: de randwaarden van een analytische functie geven ook waardevolle informatie over de functie zelf. Het tweede doel van de thesis is om randwaarden te bestuderen van analytische functies die behoren tot bepaalde ruimten van functies van reële veranderlijken.

Jasson Vindas Diaz

Rijruimten representaties

Promotoren: Andreas Debrouwere en Jasson Vindas

Verscheidene functie- en distributieruimten kunnen gerepresenteerd worden als Köthe rijruimten. Naast hun inherente elegantie, zijn zulke representaties ook belangrijk voor de isomorfisme classificatie en de topologische eigenschappen van zulke ruimten. Deze representaties worden vaak bewezen met behulp van diepe stellingen uit functionaalanalyse omtrent de structuur van de Köthe rijruimten. Het doel van deze thesis is om vertrouwd te geraken met deze technieken en enkele van deze isomorfismen te bewijzen.

References

R. Meise, D.Vogt, Introduction to functional analysis, Clarendon Press, 1997.

D. Vogt, Sequence space representations of test functions and distributions, in: Functional analysis, holomorphy and approximation theory, Proc. Semin., Rio de Janeiro 1979, Lect. Notes Pure Appl. Math. 83 (1983, 405-443.

Jasson Vindas Diaz

Holomologische algebra technieken in de analyse

Promotoren: Andreas Debrouwere en Jasson Vindas

Verscheidene stellingen in de analyse worden impliciet bewezen met behulp van technieken uit de homologische algebra. Zulke techieken worden systematisch bestudeerd in de theorie omtrent de zogenaamde afgeleide projectieve limiet functor.

Het doel van deze thesis is om vertrouwd te geraken met deze abstracte theorie en hem vervolgens toe te passen op verscheidene concrete problemen uit complexe analyse en functionaalanalyse zoals de stelling van Mittag-Lefler, stelling van Cousin, karakterisatie van surjectieve differentiaaloperatoren, ... .

References:

J. Wengenroth, Derived functors in functional analysis, Springer-Verslag, Berling, 2003.

A. Debrouwere, J. Vindas, Solution to the first Cousin problem for vector-valued quasianalytic functions, Ann. Mat. Pura Appl. 196 (2017), 1983-2003.

Jasson Vindas Diaz

Asymptotisch gedrag van distributies

Promotoren: Lenny Neyt en Jasson Vindas

De studie van het asymptotisch gedrag van distributies is een zeer divers en bruisend onderwerp in de functionaalanalyse, en biedt een uniforme benadering voor verschillende aspecten uit de asymptotische analyse en haar toepassingen. Door middel van algemeen structureel onderzoek is men vaak in staat nieuwe resultaten te bekomen en klassieke stellingen efficiënt te bewijzen, zoals bijvoorbeeld de priemgetalstelling. Het doel van deze thesis is om vertrouwd te geraken met de achterliggende theorie, alsook deze toe te passen in een specifieke context.

Referenties

R. Estrada, R. Kanwal, A distributional approach to asymptotics. Theory and applications, Birkhäuser Boston, Boston, MA, 2002.

S. Pilipovic, B. Stankovic, J. Vindas, Asymptotic behavior of generalized functions, Series on Analysis, Applications and Computation, 5, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012.

J. Vindas, R. Estrada, A quick distributional way to the prime number theorem, Indag. Math. (N.S.) 20(1) (2009), 159-165.

Lenny Neyt

Partial differential equations with singularities

Promotor: Michael Ruzhansky

In this project we will deal with different models of partial differential equations with coefficients exhibiting singular behaviour. It is well known than the classical theory of distributions does not apply in the case of strong singularities, however, recently new approaches have emerged based on the so-called very weak solutions. We will investigate properties of such solutions in several fundamental models from points of view of both pure and applied mathematics.

References:

Garetto C., Ruzhansky M., Hyperbolic second order equations with non-regular time dependent coefficients, Arch. Ration. Mech. Anal., 217 (2015), 113-154.

Ruzhansky M., Tokmagambetov N., Wave equation for operators with discrete spectrum and irregular propagation speed, Arch. Ration. Mech. Anal., 226 (2017), 1161-1207.

Ruzhansky M., Tokmagambetov N., Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field, Lett. Math. Phys., 107 (2017), 591-618.
Michael Ruzhansky

Functional inequalities and applications

Promotor: Michael Ruzhansky

We will deal with a fascinating area of mathematical analysis devoted to functional inequalities associated to operators with different geometries. This is an internationally very active area of research that witnessed a big boost during the last years. In this project we will aim at deriving new inequalities and at linking them to several problems of geometry and physics, as well as the analysis on groups and the calculus of variations.

References:

Ruzhansky M., Suragan D., Yessirkegenov N., Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability and superweights for Lp-weighted Hardy inequalities, Trans. Amer. Math. Soc. Ser. B, 5 (2018), 32-62.

Ruzhansky M., Suragan D., Yessirkegenov N., Sobolev type inequalities, Euler-Hilbert-Sobolev and Sobolev-Lorentz-Zygmund spaces on homogeneous groups, Integral Equations Operator Theory, 90 (2018), no. 1, 90:10.

Ruzhansky M., Suragan D., Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups, Adv. Math., 317 (2017), 799-822.
Michael Ruzhansky

Subelliptic harmonic analysis

Promotor: Michael Ruzhansky

In this project we will investigate the analytic objects appearing in the setting of non-Riemannian geometry by applying suitable versions of the Fourier analysis. We start with the analysis on compact manifolds (and compact Lie groups) where we introduce spaces related to sums of squares of vector fields. Such spaces can be well described in terms of their behaviour on the Fourier side which has to be adapted to the geometry of the space. The applications of such research will include new trends in harmonic analysis and new estimates for solutions of associated partial differential equations.

References

Delgado J., Ruzhansky M., Schatten classes and traces on compact groups, Math. Res. Lett., 24 (2017), 979-1003.

Cardona D., Ruzhansky M., Multipliers for Besov spaces on graded Lie groups, C. R. Acad. Sci. Paris, 355 (2017), 400-405.

Nursultanov E., Ruzhansky M., Tikhonov S., Nikolskii inequality and Besov, Triebel-Lizorkin, Wiener and Beurling spaces on compact homogeneous manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci., 16 (2016), 981-1017.
Michael Ruzhansky

Multiplication of distributions

Promotor: Michael Ruzhansky

It is well known that in general it is not possible to have well defined product in the space of distributions. In some cases it is impossible to give a meaning to a product, while in some other cases it is possible to define a product, but we lose some good properties. As a simple example one can consider Heaviside function H, which obviously can be multiplied by itself (Hn=H, for all positive integer n), but when differentiate equality H2=H3, using Leibniz rule, we arrive to contradiction, i.e. multiplication is not compatible with the Leibniz rule. There are other difficulties when we want to study nonlinear operations within distributions, but there are several cases when we can multiply them.

The main objectives of thesis are to master distribution theory, to explore difficulties and possible definitions for product of distributions, and to deal with examples of distributions whose products are of interests for applications in analysis and mathematical physics.

References:

Friedlander, F. G. and Joshy M. Introduction to the Theory of Distribution, Cambridge University press, Cambridge, 1998.
Hoskins, R. F. and Sousa Pinto, J. Theories of generalized functions, Distributions, ultradistributions and other generalized functions, Woodhead publishing, Cambridge, 1994.
Oberguggenberger, M. Multiplication of distributions and applications to partial differential equations, Longman Scientific and Technical, Harlow, 1992.



Michael Ruzhansky

Approximation theory on harmonic manifolds of purely exponential volume growth

Promotor: Michael Ruzhansky

Supervisor: Vishvesh Kumar

Description: Approximation theory is significantly useful in different branches of science and engineering such as big data analysis. In this project, we will develop some important results in approximation theory using the newly developed Fourier-Helgason analysis on harmonic manifolds of purely exponential volume growth. Examples of such manifolds are hyperbolic spaces and harmonic NA groups. This project will be using basic knowledge from different areas of mathematics such as special functions, analysis on manifolds, Fourier analysis, measure theory, functional analysis, and approximation theory. In this project, the student will have a chance to contribute to further important developments in the Fourier analysis on harmonic manifolds as well as its application to approximation theory. If the student is interested we can also explore its applications to big data analysis and related areas.

References:

[1] K. Biswas, G. Knieper, and N. Peyerimhoff, The Fourier transform on harmonic manifolds of purely exponential volume growth, The Journal of Geometric Analysis 31.1 (2021): 126-163. [2] V. Kumar and M. Ruzhansky, A note on K-functional, Modulus of smoothness, Jackson theorem and Bernstein-Nikolskii-Stechkin inequality on Damek-Ricci spaces, J. Approx. Theory, 264, (2021), 105537. [3] D. Gorbachev and S. Tikhonov, Moduli of smoothness and growth properties of Fourier transforms: two-sided estimates. J. Approx. Theory 164 (2012), no. 9, 1283–1312. [4] I. Pesenson, Bernstein-Nikolskii and Plancherel-Polya inequalities in Lp-norms on non-compact symmetric spaces. Math. Nachr. 282 (2009), no. 2, 253–269.
Michael Ruzhansky

The Black-Scholes equation on almost Abelian groups

Promotor: Michael Ruzhansky

Supervisor: Zhirayr Avetisyan

Description: 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.  

Michael Ruzhansky

Partial differential equations on Lie groups

Promotor: Michael Ruzhansky

Supervisor: Joel Restrepo

Description: We will consider several equations of different type on Lie groups. We will study, for instance, heat equations, wave equations, Schrodinger equations, etc. We use classical methods to find a local or global solution. We will discuss the representations of solutions, time-asymptotic estimes, norm estimates, among other properties. We first go throught the basis in the most classical Lie groups like euclidean, compact and graded case.

Michael Ruzhansky