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.

# Incidentiemeetkunde

## Which graph properties are not determined by the spectrum?

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. 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 , having a given diameter , admitting a perfect matching  and having a given vertex or edge-connectivity .  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 .

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

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

## A geometrical approach to a combinatorial question

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 :

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].

 N. Alon and B. Mohar, The chromatic number of graph powers, Combin. Probab. Comput. 11  (2002), 1--10.
 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).
 A. Offer and H. Van Maldeghem, Distance-j ovoids and related structures in generalized polygons, Discrete Mathematics 294(1-2) (2005), 147-160.
 J.A. Thas, Ovoids and spreads of finite classical polar spaces, Geom. Dedicata 10(1-4) (1981), 135--143.

## 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.
In preparation, 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 Theo-
rems 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.

## Injecting algebraic graph theory tools into coding theory

The container method (which is a very power tool for upper bounding the number of independent sets,
codes, sum free sets, monotone Boolean functions etc.) has been recently used to count the number of
t-error correcting Lee metric codes . The general idea is to reduce an enumeration problem in coding theory to
evaluating the number of independent sets using graph theory tools, and in particular, the container method.

The main goal of this project is to provide alternative ways of counting the number of t-error correcting
Lee metric codes using other not yet explored graph theory tools which are known to give upper bounds for
independent sets. In particular, we will investigate the use of several eigenvalue bounds for the independence
number, or the Expander Mixing Lemma, among others.

 Anna-Lena Horlemann, Violetta Weger, Nadja Willenborg, Asymptotic Density and Counting Results
for Lee Metric Codes, talk at ALCOCRYPT 2023.

## Zero-Pattern Rank-Metric Codes

Promotor: Alessandro Neri

On the space of mxn matrices over a (finite) field F, one can consider the metric induced by the rank. A linear rank-metric code is a subspace of the space of mxn matrices over F, and its minimum rank distance is the minimum among the ranks of its nonzero matrices. Rank-metric codes have first been studied by Delsarte  in the late 70's and gained interest due to their application in network coding proposed by Silva-Kschischang-Koetter .

Several special families of rank-metric codes with restrictions have been investigated. Among them, of particular interest is the family of zero-pattern rank-metric codes. They are defined as subspaces of matrices whose nonzero entries are only those corresponding to indices from a subset of {1,...,m}x{1,...,n}. They generalize codes in the Hamming metric, which are obtained by only allowing nonzero entries on the diagonal, sum-rank metric codes , where the nonzero entries lie on diagonal blocks and Ferrers diagram codes , where the nonzero entries only appear on a prescribed Ferrers diagram.

The aim of this project is to investigate combinatorial, geometric and algebraic features of zero-pattern rank-metric codes, in analogy to those for Hamming-metric, (sum-)rank-metric and Ferrers diagram codes.

 P. Delsarte. Bilinear forms over a finite field, with applications to coding theory. J. Combin. Theory Ser. A, 25:226–241, 1978.
 T. Etzion and N. Silberstein. Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams. IEEE Trans. Inform. Theory, IT-55:2909–2919, 2009.
 U. Martinez-Penas , Skew and linearized Reed-Solomon codes and maximum sum rank distance codes over any division ring, J. Algebra, 504:587--612, 2018.
 D. Silva, F. R. Kschischang, and R. Koetter. A rank-metric approach to error control in random network coding. IEEE Trans. Inform. Theory, IT-54:3951–3967, 2008.

Alessandro Neri

## Strong Blocking Sets in Coding Theory

Promotoren: Jozefien D'haeseleerAlessandro Neri

A strong blocking set in a finite dimensional projective space over a (finite) field F is a set of points such that its intersection with any hyperplane generates the hyperplane itself. In the last decade they have been shown to have important applications in coding theory. They were first introduced and studied in order to construct covering codes . Recently, they have been shown to be the geometric counterparts of minimal linear codes , a special family of linear codes in F^n in which the supports of their nonzero codewords form a Sperner family in the set {1,...,n}. In addition, over the field with three elements, strong blocking sets correspond also to trifferent codes . Over the last years, constructions of small strong blocking sets have been an active topic of research. Many of these constructions are obtained as union of lines [1,2,4,5,6], a feature that helps in controlling their intersection with hyperplanes. However, it is not clear whether such a restrictive choice can be relaxed in order to obtain infinite families of smaller strong blocking sets.

This project aims to further investigate properties and novel constructions of small strong blocking sets. This will be done by first analyzing concrete examples in small projective spaces.

 G. N. Alfarano, M. Borello, and A. Neri. A geometric characterization of minimal codes and their asymptotic performance. Adv. in Math. Commun., 16(1):115–133, 2022.
 N. Alon, A. Bishnoi, S. Das, and A. Neri. Strong blocking sets and minimal codes from expander graphs. preprint, 2023.
 A. Bishnoi, J. D’haeseleer, D. Gijswijt, and A. Potukuchi. Blocking sets, minimal codes, and trifferent codes. preprint, arXiv:2301.09457,, 2023.
 A. A. Davydov, M. Giulietti, S. Marcugini, and F. Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Adv. Math. Commun., 5(1):119, 2011.
 S. Fancsali and P. Sziklai. Lines in higgledy-piggledy arrangement. Electron. J. Comb., 21, 2014.
 T. Heger and Z. L. Nagy. Short minimal codes and covering codes via strong blocking sets in projective spaces. IEEE Trans. Inf. Theory, 68(2):881–890, 2021.
Alessandro Neri

## 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  for a possible starting point. The student is expected to investigate open problems in this project.

 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  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.

 P. Frankl, R. Graham. Intersection theorems for vector spaces. European Journal of Combinatorics, 6(2):183 – 187 (1985).
 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
 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

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  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  (construction of cospectral graphs with the Kneser graph) extend to the generalized Kneser graphs (see  for their definition).

 E. van Dam and W.H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003), 241–272.
 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.
 J. D'haeseleer, K. Metsch and D. Werner. On the chromatic number of two generalized Kneser graphs. arXiv:2005.05762

Jozefien D'haeseleer

## Sudoku Latin squares and the link with spectral graph theory and finite geometry

A Latin square is an array of order n with n symbols such that each symbol occurs exactly once in each row and column.
We say that two Latin squares L1 and L2 of order n are orthogonal to each other if, given any two symbols λ and μ, there is a unique pair (i, j) such that the entries of L1 and L2 in the i-th row and j-th column are λ and μ, respectively.
Mutually orthogonal Latin squares (MOLS) are a family of Latin squares such that any two of them are orthogonal. One of the most important problems in the area of Latin squares is to determine the maximum number N(n) of MOLS.  It is known that N(n) is at most n-1, and that if there is a set of n-2 MOLS of order n, that this set can be extended in a unique way to a set of n-1 MOLS of order n.

A Sudoku Latin square is a Latin square of order nm such that the array is divided into nm subarrays of size n x m, such that each symbol occurs exactly once in each subarray. For n=m, it is known that the maximum number of mutually orthogonal Sudoku Latin squares (MOSLS) of order n2 is equal to n2-n, but here, it is not always true that every set of n2-n-1 MOSLS can be extended to a set n2-n MOSLS, see for example .

These (Sudoku) Latin squares can also be linked to other combinatorial areas such as finite geometry and graph theory.

In this project, we can investigate the following questions:
• How can we use spectral graph theory to distinguish non-isomorphic MO(S)LS?
• What is known for Sudoku Latin squares with n different from m?
• How can we translate MOSLS to substructures in finite geometries?
 R.A. Bailey, P. J. Cameron, R. Connelly, Sudoku, gerechte designs, resolutions, affine space, spreads, reguli, and Hamming codes. Amer. Math. Monthly 115 (2008), 383–404.
 J. D’haeseleer, K. Metsch, L. Storme, G. Van de Voorde, On the maximality of a set of mutually orthogonal Sudoku Latin squares, Des. Codes Cryptogr. 84 (2017), 143–152.
 S. Kubota, S. Suda, A. Urano, Mutually orthogonal Sudoku Latin squares and their graphs. ArXiv 2111.04992 (2021).
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.

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## 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.

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# 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. Een recent onderwerp behandeld faseovergangen voor Friedmans versie van de stelling van Bolzano Weierstrass.

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

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
 Intuitionistic linear temporal logicPromotor: Brett McLeanTemporal logic is a type of logic for reasoning with statements that can make reference to the future and/or past. It has applications in computer science, artificial intelligence, and linguistics. Amir Pnueli won the 1996 Turing Prize 'for seminal work introducing temporal logic into computing science and for outstanding contributions to program and systems verification'. The specific framework introduced by Pnueli is called linear temporal logic.Intuitionistic logic is the most well-known constructive logic and is the logic preferred by many popular interactive theorem provers, such as Coq and Lean. One semantics that intuitionistic logic can be given is topological semantics, where each proposition is interpreted as an open set on some topological space.By combining the syntax and semantics of linear temporal logic with propositional intuitionistic logic, we can obtain a framework for speaking and reasoning about dynamical systems (i.e. systems describing how points in a topological space evolve over time). Recent results have shown that we can obtain logics that are reasonably expressive, but nevertheless are decidable .The plan for this project would be to first present background about linear temporal logic and about intuitionistic logic and then to give an exposition of some of the recent results about intuitionistic linear temporal logic, possibly making a contrast with the alternative dynamical topological logic. It may be feasible to prove some novel results modestly extending existing results. It may be possible to fund a short visit to Professor Fernández-Duque at the University of Barcelona.As prerequisites it is necessary to have taken at an introductory course in mathematical logic and to at least have encountered the concept of a topological space. It would be helpful to have taken a course in computability and complexity, or have some familiarity with the basics of these fields. It is not necessary to have any knowledge about dynamical systems.Background reading: The book chapter  is available online and provides an introduction to temporal logic. The book  provides an introduction to modal logic more generally. A relevant introduction to intuitionistic logic is perhaps .References  Guram Bezhanishvili and Wesley H. Holliday, A semantic hierarchy for intuitionistic logic, Indagationes Mathematicae 30 (2019), no. 3, 403–469.  Patrick Blackburn, Maarten de Rijke, and Yde Venema, Modal logic, Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 2001.  David Fernández-Duque, The intuitionistic temporal logic of dynamical systems, Logical Methods in Computer Science 14 (2018).  Yde Venema, Temporal logic, The Blackwell guide to philosophical logic (2017), 203–223.
Brett McLean

## 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.

 Jeff Erickson. Fusible numbers. https://www.mathpuzzle.com/fusible.pdf.
 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 > . 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 . 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.

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

Fedor Pakhomov

# 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

## Rijruimten representaties

Promotor: 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

## Asymptotisch gedrag van distributies

Promotor 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.

Jasson Vindas Diaz

## Fractional Calculus

Promotor: Michael Ruzhansky

Supervisor: Karel Van Bockstal

It has been demonstrated that many systems in science and engineering can be modelled more accurately by fractional-order than integer-order derivatives. Many methods have been developed to solve problems with fractional derivatives. This master's thesis aims to study theoretically (existence and uniqueness of a solution) partial differential equations with fractional order (e.g. constant order or variable order).

Numerical methods to approximate fractional derivatives, inverse problems for fractional partial differential equations, or other topics in this research field can also be studied.

Karel Van Bockstal

## Numerical mathematical study and extension of a 2DH idealized model for the identification and the stability of morphodynamic equilibria in tidal basins

Promotor: Tom De Mulder (EA15, Department of Civil engineering)

Supervisor: Karel Van Bockstal and Tian Qi, MSc (EA15)

You can also contact us if you are interested in an internship or holiday job related to this subject.

Karel Van Bockstal

## A heat transfer problem and a wave propagation problem with Robin type boundary condition in 1D: a case study

Promotor: Michael Ruzhansky

Supervisor: Karel Van Bockstal

A heat and wave problem on a bounded domain in 1D accompanied by Robin-type boundary conditions with real coefficients at the endpoints is considered. The goal of this thesis is to solve these problems by the method of separation of variables. The exact solution to these problems depends on the coefficients appearing in the Robin boundary conditions. The objective is to catch all possible situations in a phase diagram. Another goal is to develop a Python code such that numerical simulations for each situation can be established.

Karel Van Bockstal

## The implementation of flow through porous media

Promotor: Michael Ruzhansky

Supervisor: Karel Van Bockstal

There are numerous applications in which it is of interest to study the flow of multiple fluids through porous media. Some examples of these include the transport of dissolved nutrients through biological tissue and the extraction of petroleum from underground deposits. In many practical situations, the physics of such flows are characterised by coupled governing equations that are nonlinear and transient, and the problem is further complicated by material heterogeneity and irregular geometries.

This thesis aims to study (e.g. the derivation) and implement existing numerical algorithms for such systems of equations (Darcy's law, porous medium equation, Navier-Stokes equation). The code will be implemented using the FEniCS Project, which is based on the variational framework and the finite element method. There are different demos available for solving these types of equations. Furthermore, a computer can be provided for performing the simulations.

Karel Van Bockstal

## Inverse problems in science and engineering

Promotor: Michael Ruzhansky

Supervisor: Karel Van Bockstal

In most of the course notes (partial) differential equations were only depicted as a tool for modelling physical phenomena. If some universal law is translated into mathematical language, then this model can be used to predict the behaviour of the quantity of interest, and it is often called the forward or direct problem. Inverse problems (IPs) -as the name suggests- do the opposite. They induce the reason which led to the result from the observed data.

An example of an inverse problem is determining the cause of a disease based on the results of a medical examination. It is easy to make a mistake when solving inverse problems. For example, symptoms associated with an HIV infection look like symptoms of other illnesses. It is thus impossible to tell, exclusively based on symptoms, whether the problem is related to HIV or another medical condition. Therefore, the problem of determining the cause of a disease is called ill-posed, i.e. there is no unique cause (or solution). Additional medical investigations (measurements) are required to determine the correct cause. Similar issues are encountered when studying inverse problems for partial differential equations.

Inverse problems arise in many mathematical physics areas, and applications rapidly expand to geophysics, chemistry, medicine and engineering. A typical example is computed axial tomography (CAT or CT scan). CT provides clinically relevant anatomic and functional information, is relatively noninvasive, and has low short- and long-term risks.

The topics of the master dissertation range from the mathematical modelling and the theoretical analysis of inverse problems for partial differential equations where some parameters (right-hand side (heat/load source), kernel, diffusion coefficient, etc.), unknown boundary condition(s) or portion of the boundary are to be found, to the development of efficient numerical schemes and their practical application in sciences, engineering (e.g. diffusion equation, beam equation, Maxwell's equations) and finance.

The exact topic of the master dissertation is worked out in consultation with the thesis advisors. For each problem under consideration, the most important questions are:

• Which additional measurement is required for the unique reconstruction of the solution?
• How can the solution be reconstructed?

Karel Van Bockstal

## Regularisation methods for inverse problems

Promotor: Michael Ruzhansky

Supervisor: Karel Van Bockstal

Inverse problems often lead to mathematical problems that are not well-posed in the sense of Hadamard, i.e. to ill-posed problems. This means especially that their solution is unstable under data perturbations. Numerical methods that can cope with this problem are so-called regularization methods. This thesis is devoted to the study of these regularization methods. Click here for a more detailed description.

## Reference:

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, vol. 375. Dordrecht: Kluwer Academic Publishers, 1996.

Karel Van Bockstal

## The finite element method for Maxwell's equations

Promotor: Michael Ruzhansky

Supervisor: Karel Van Bockstal

Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism. Maxwell's equations describe how charges, currents, and changes of the fields generate electric and magnetic fields. This thesis aims to learn how numerical solutions to (practical) problems related to these fundamental equations can be obtained. The goal of the dissertation is

• To introduce the physical laws that describe the phenomena of electromagnetism, leading to Maxwell's equations in their classical form;
• To study Sobolev spaces for vector functions (definition/trace theorem/Green's theorem);
• To study finite elements on tetrahedra;
• To study and implement (using the finite element library DOLFIN of the FEniCS project) a particular problem derived from Maxwell equations (curl-curl formulation in terms of the magnetic or electric field, scattering problem,...).
The exact topic of the master dissertation is worked out in consultation with the thesis advisors. The Maxwell system is an example of a coupled system of partial differential equations. Another example of a coupled system arises in thermoelasticity. Also, in this research field, similar research questions can be stated. Mixed systems can also be studied in a general (abstract) framework (including bilinear forms).

## Reference:

P. Monk, Finite Element Methods for Maxwell's Equations. New York: Oxford University Press Inc., 2003.
S. C. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods. Springer, 2008.

Karel Van Bockstal

## 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

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

Promotor: Michael Ruzhansky

Supervisor: Zhirayr Avetisyan

Description: 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.

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:

 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.  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.  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.  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

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

Promotor: Michael Ruzhansky

Supervisor: Zhirayr Avetisyan

Description: 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.

Michael Ruzhansky

## Non-commutative analysis performed on the basic example of the Heisenberg group

Promotor: Michael Ruzhansky

Supervisor:Marianna Chatzakou

Description: The idea of this project is to understand the notion and methodology, in terms of the associated Fourier analysis and the development of pseudo-differential operators, in the first basic example of the non-commutative setting of the Heisenberg group. After becoming acquainted with the necessary tools, one can aim to study sub-elliptic equations in this setting.

Michael Ruzhansky