Professor Florian Breuer
Professor
School of Mathematical and Physical Sciences (Mathematics)
 Email:florian.breuer@newcastle.edu.au
 Phone:0240339609
The lure of pure mathematics
Professor Florian Breuer’s research is guided by a profound passion and curiosity for pure mathematics. His specialty area is number theory, a fascinating field of mathematics that seeks to better understand whole numbers and their properties.
Professor Florian Breuer values pure mathematics for its beauty, ingenuity and creativity. While many of his peers are driven to apply mathematics to solve immediate problems—like how to build better phones—as a pure mathematician, Florian is more interested in developing new mathematics for its own sake.
“A lot of mathematics that underlies our technological civilisation was discovered decades or centuries ago by mathematicians in the course of purely curiositydriven research,” explains Florian.
“They couldn’t possibly have known that their discoveries would eventually find applications in the distant future. Likewise, I have no idea how much of my work may one day be applied. Instead, I am guided by my curiosity and by a sense for what’s beautiful, interesting and deep.”
Florian believes in honouring the research process and allowing it to lead him in novel directions, regardless of whether the realworld application is immediately obvious. Much of what he studies has the tantalising potential to drive yetunknown future technological innovation—just like the discoveries of mathematicians before him.
“To most people, mathematics is invisible, running quietly in the background, but it underlies our modern society. For example, the electronics in your phone rely on quantum mechanics and other physical processes, which we can only understand with the help of advanced mathematics.”
Number theory and cryptography
Florian’s work in number theory, one of the oldest branches of mathematics, studies sets of whole numbers. Most people would recognise basic sets of numbers such as even, odd and prime numbers. Florian’s work carries this same idea into much more sophisticated and complex territory, exploring new and unexpected relationships between natural numbers. It’s a branch of mathematics sometimes referred to as “higher arithmetic”.
“Mostly my research goals are quite abstract and difficult to convey to a lay audience, but here is one theme. Number theory is divided into two ‘parallel worlds’.
“The first classical world deals with ordinary numbers and is called the world of number fields (a ‘field’ is a number system within which one can consistently add, subtract, multiply and divide).
“Parallel to this is the world of function fields. Here the ordinary numbers are replaced by other objects, which nevertheless behave just like numbers (they also form fields). You can imagine these objects to be like numbers, except that when one adds or multiplies two of them you can discard any ‘carries’ that occur in the digitwise additions, e.g. 7+8 = 5 instead of ^{1}5.
“These two worlds are remarkably similar, and most phenomena in number fields have their analogues in function fields, far more so than one should reasonably expect. Why is this? What are the similarities and what are the differences? This has been a recurrent theme in my own research.”
Most research in number theory is not solely motivated by realworld applications. However, one important application is cryptography: the study of how messages can be safely encrypted and authenticated.
“A lot of widelyused cryptography relies on relatively simple number theory that was discovered centuries ago, but only applied to cryptography since the 1970s.”
Cryptography helps protect users from having their messages read or their online activity impersonated. As technology advances rapidly around the world, number theory is increasingly being applied in new, more complex ways to facilitate greater technological security.
“The advent of quantum computers, which will be able to crack many widely used cryptographic systems, necessitates the development of postquantum cryptography, which relies on much more abstract and sophisticated topics in number theory.
“I am very fortunate to be already familiar with these more advanced topics from my own work, and I am very excited to become more involved in postquantum cryptography.”
Challenges and collaborations
Florian’s studies and research work have seen him travel widely, pursuing learning and collaboration opportunities with some of the brightest minds in his field.
He completed his D.E.A. (Masters) at the Université Pierre et Marie Curie, France, his PhD at the Université Denis Diderot, France, and two postdoctoral fellowships in Taiwan and Germany. He also taught at Stellenbosch University, South Africa, and undertook research semesters in Germany and Switzerland.
“There is a near infinite amount of mathematics to learn, and mathematics really builds up on other mathematics to an extent not seen in other disciplines. To understand one topic, you must first master three others”
Through challenges and successes, Florian’s love for pure mathematics continues to propel his research forward. His passion for collaboration has also led to firstoftheirkind activities such as the online 2020 Number Theory conference which, on behalf of the Priority Research Centre for ComputerAssisted Research Mathematics and its Applications (CARMA), brought together researchers from all areas of number theory worldwide.
“Facilitating collaborations between mathematicians is critically important for enabling as yet undreamedof technologies. Fundamental research is essentially an investment in the future, providing a solid basis for new possibilities and applications.
“Mathematical research can be psychologically quite tough. Progress only occurs in occasional bursts—sometimes months apart—and most of the time I’m stuck on a problem. When I do have a good idea and make progress on something, it feels great!”
The lure of pure mathematics
Professor Florian Breuer’s research is guided by a profound passion and curiosity for pure mathematics. His specialty area is number theory, a fascinating field of mathematics that seeks to better understand whole numbers and their properties.
Career Summary
Biography
Research Interests: Florian works in Number Theory, especially in elliptic curves, Drinfeld modules, Drinfeld modular forms and also elementary topics such as Ducci sequences.
Biography: Florian grew up in Stellenbosch, South Africa, where he completed school as well as his undergraduate studies in Mathematics and Theoretical Physics. He then obtained a bursary from the French government to complete his graduate studies in Paris, where he completed his D.E.A. (Masters) at the Université Pierre et Marie Curie (Paris 6) in 1999 and his PhD at the Université Denis Diderot (Paris 7) in 2002, both under supervision of Marc Hindry.
After two postdoctoral fellowships in Taiwan and Germany, he returned to Stellenbosch University in July 2004 as a senior lecturer. He was promoted to associate professor in 2007 and spent a semester in Germany and Switzerland in 2009 on an AlexandervonHumboldt Fellowship for Experienced Researchers. Florian served as head of the Mathematics Division at Stellenbosch University from 2012 to 2015, and was promoted to full professor in 2013.
In April 2018 Florian and his family moved to the University of Newcastle. He still holds an appointment as extraordinary professor at Stellenbosch University.
Florian is the director of the Priority Research Centre for ComputerAssisted Research Mathematics and its Applications (CARMA) and serves on the editorial boards of the Journal of Number Theory and Quaestiones Mathematicae.
Research: In modern Number Theory, there exist the parallel worlds of number fields and function fields. Most of the wellknown results and phenomena of Number Theory, such as multiplicative Number Theory, the Riemann Hypothesis, Class Field Theory and modular forms have parallels in the function field world. In particular, Drinfeld modules are function field objects whose behaviour closely parallels that of elliptic curves over number fields.
Florian's main research area is the arithmetic of function fields. This started with his PhD thesis, in which he proved an analogue of the AndréOort Conjecture for products of Drinfeld modular curves. He later extended this conjecture to subvarieties of Drinfeld modular varieties, and proved it in certain special cases. From here his research voyage has lead him in a natural way to consider Drinfeld modular polynomials in higher rank, Galois representations associated to Drinfeld modules (which allowed him to prove the Generalised Iteration Conjecture of Abhyankar) and to Drinfeld modular forms. Most recently he was involved in joint work with former PhD student Dirk Basson (Stellenbosch), and friend and mentor Richard Pink (ETHZurich), which laid the foundations for the analytic theory of Drinfeld modular forms in arbitrary rank. This research has opened the door to a variety of exciting new directions.
In other work, Florian is interested in elliptic curves, and has contributed results on the growth of torsion subgroups of elliptic curves over number fields and has studied Heegner points on elliptic curves.
A more elementary, but rich and interesting, topic is Ducci sequences. Florian's first research experience as an undergraduate student concerned periods of Ducci sequences, and he keeps returning to this topic over the years. Recently, Florian uncovered links between Ducci sequences, multiplicative orders of elements in finite fields, and the arithmetic of real quadratic fields.
Keywords
 Drinfeld modular forms
 Drinfeld modules
 Elliptic curves
 Number Theory
Languages
 German (Mother)
 English (Fluent)
 French (Fluent)
 Afrikaans (Fluent)
Fields of Research
Code  Description  Percentage 

010102  Algebraic and Differential Geometry  10 
010101  Algebra and Number Theory  90 
Professional Experience
UON Appointment
Title  Organisation / Department 

Professor  University of Newcastle School of Mathematical and Physical Sciences Australia 
Academic appointment
Dates  Title  Organisation / Department 

1/10/2007  31/12/2012  Associate Professor  Stellenbosch University Mathematical Sciences South Africa 
1/7/2004  30/9/2007  Senior Lecturer  Stellenbosch University Mathematical Sciences South Africa 
1/2/2003  31/10/2003  Postdoctoral Fellow  National Tsing Hua University National Center for Theoretical Sciences Taiwan, Province of China 
1/1/2004  30/4/2004  Postdoctoral Fellow  Max Planck Institute MaxPlanckInstitute for Mathematics Germany 
1/4/2018  31/3/2021  Extraordinary Professor  Stellenbosch University Mathematical Sciences South Africa 
1/1/2013  31/3/2018  Professor  Stellenbosch University Mathematical Sciences South Africa 
Awards
Prize
Year  Award 

2008 
MeiringNaude Medal The Royal Society of South Africa 
Scholarship
Year  Award 

2009 
AlexandervonHumboldt Fellowship for Experienced Researchers Alexander Von Humboldt Foundation 
Teaching
Code  Course  Role  Duration 

3120 
Algebra The University of Newcastle 
Lectuer, course coordinator  30/7/2018  30/11/2018 
1210 
Mathematical Discovery 1 The University of Newcastle 
Professor  1/4/2018  30/6/2018 
1120 
Mathematics for Engineering, Science and Technology, 2 The University of Newcastle 
Lecturer, course coordinator  25/2/2019  29/6/2019 
2340 
Linearity and Continuity The University of Newcastle 
Lecturer  25/2/2020  17/7/2020 
4104 
Number Theory The University of Newcastle 
Lecturer, course coordinator  30/7/2018  30/11/2018 
1120 
Mathematics for Engineering, Science and Technology 2 The University of Newcastle 
Lecturer / Course Coordinator  24/2/2020  17/7/2020 
1210 
Mathematical Discovery 1 The University of Newcastle 
Lecturer, course coordinator  25/2/2019  29/6/2019 
4101 
Introduction to Valued Fields (ACE Course) The University of Newcastle On the field of rational numbers, besides the usual notion of absolute value, one can also define a padic absolute value for each prime p which measures divisibility by p. Essentially, a rational number in lowest terms has small padic absolute value if its numerator is highly divisible by p. Just like the usual absolute value, the padic absolute value is multiplicative, but satisfies a stronger form of the triangle inequality called the ultrametric inequality. The usual absolute value defines a metric on the field of rational numbers, whose completion is the field of real numbers. Similarly, completion with respect to a padic absolute value gives the field of padic numbers. The analytic theory of padic numbers is rich and interesting, and in many cases one obtains stronger results than in real or complex analysis, due to the stronger ultrametric inequality. This course will study the general theory of absolute values on fields and the resulting topologies and analysis. These foundations lead to important concepts in number theory (such as the behaviour of primes in field extensions) and the theory of topological groups, such as totally disconnected locally compact (t.d.l.c.) groups. Besides the intrinsic interest of this basic area of mathematics, this course serves as an introduction to research topics of current interest at the University of Newcastle and elsewhere. Taught jointly with Prof. George Willis 
Lecturer  25/2/2019  29/6/2019 
Publications
For publications that are currently unpublished or inpress, details are shown in italics.
Chapter (1 outputs)
Year  Citation  Altmetrics  Link  

2005 
Breuer F, Pink R, 'Monodromy groups associated to nonisotrivial Drinfeld modules in generic characteristic', , BIRKHAUSER BOSTON 6169 (2005)

Journal article (29 outputs)
Year  Citation  Altmetrics  Link  

2020 
Breuer F, Shparlinski IE, 'Lower bounds for periods of Ducci sequences', Bulletin of The Australian Mathematical Society, 102 3138 (2020) [C1]


2019 
Breuer F, 'Periods Of Ducci Sequences And Odd Solutions To A Pellian Equation', Bulletin of the Australian Mathematical Society, 100 201205 (2019) [C1]


2017 
Basson D, Breuer F, 'On certain Drinfeld modular forms of higher rank', Journal de Théorie des Nombres de Bordeaux, 29 827843 (2017)


2016 
Breuer F, 'A note on Gekeler's hfunction', ARCHIV DER MATHEMATIK, 107 305313 (2016)


2016 
Breuer F, 'Explicit Drinfeld moduli schemes and Abhyankar's Generalized Iteration Conjecture', JOURNAL OF NUMBER THEORY, 160 432450 (2016)


2016 
Breuer F, Rueck HG, 'Drinfeld modular polynomials in higher rank II: Kronecker congruences', JOURNAL OF NUMBER THEORY, 165 114 (2016)


2012 
Breuer F, 'Newton Identities for Weierstrass Products', AMERICAN MATHEMATICAL MONTHLY, 119 796799 (2012)


2012 
Breuer F, 'Special subvarieties of Drinfeld modular varieties', JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, 668 3557 (2012)


2010 
Breuer F, 'Torsion bounds for elliptic curves and Drinfeld modules', JOURNAL OF NUMBER THEORY, 130 12411250 (2010)


2010 
Breuer F, 'Ducci sequences and cyclotomic fields', JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS, 16 847862 (2010)


2009 
Breuer F, Rueck HG, 'Drinfeld modular polynomials in higher rank', JOURNAL OF NUMBER THEORY, 129 5983 (2009)


2008 
Breuer F, Im BH, 'Heegner points and the rank of elliptic curves over large extensions of global fields', CANADIAN JOURNAL OF MATHEMATICSJOURNAL CANADIEN DE MATHEMATIQUES, 60 481490 (2008)


2007 
Breuer F, 'CM points on products of Drinfeld modular curves', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 359 13511374 (2007)


2007 
Breuer F, Lotter E, van der Merwe B, 'Duccisequences and cyclotomic polynomials', FINITE FIELDS AND THEIR APPLICATIONS, 13 293304 (2007)


2007 
Breuer F, 'The AndreOort conjecture for Drinfeld modular varieties', COMPTES RENDUS MATHEMATIQUE, 344 733736 (2007)


2005 
Breuer F, 'The AndreOort conjecture for products of Drinfeld modular curves', JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, 579 115144 (2005)


2004 
Breuer F, 'Images of isogeny classes on modular elliptic curves', MATHEMATICAL RESEARCH LETTERS, 11 649651 (2004)


2004 
Breuer F, 'Higher Heegner points on elliptic curves over function fields', JOURNAL OF NUMBER THEORY, 104 315326 (2004)


2002 
Breuer F, 'Distinguished liftings and the andréoort conjecture', Quaestiones Mathematicae, 25 363380 (2002) In this paper we study liftings of affine varieties from finite fields to number fields, such that the lifted varieties contain specified ¿canonical¿ lifts of points. If this cano... [more] In this paper we study liftings of affine varieties from finite fields to number fields, such that the lifted varieties contain specified ¿canonical¿ lifts of points. If this canonical lifting of points corresponds to the DeuringSerreTate lift of jinvariants of ordinary elliptic curves, then the resulting lifting problem is closely related to the AndréOort conjecture. We explore this connection, prove some results related to the AndréOort conjecture, and then apply these results together with other known special cases of the conjecture to our lifting problems. © 2002, Taylor & Francis Group, LLC. All rights reserved.


2002 
Breuer F, 'The AndréOort conjecture for the product of two Drinfeld modular curves', Comptes Rendus Mathematique, 335 867870 (2002) We prove an analogue of the AndréOort conjecture for the product of two Drinfeld modular curves, following S.J. Edixhoven's approach. © 2002 Académie des sciences/Éditions s... [more] We prove an analogue of the AndréOort conjecture for the product of two Drinfeld modular curves, following S.J. Edixhoven's approach. © 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS.


2001 
Breuer F, 'Heights of CM points on complex affine curves', Ramanujan Journal, 5 311317 (2001) In this note we show that, assuming the generalized Riemann hypothesis for quadratic imaginary fields, an irreducible algebraic curve in C'' is modular if and only if it... [more] In this note we show that, assuming the generalized Riemann hypothesis for quadratic imaginary fields, an irreducible algebraic curve in C'' is modular if and only if it contains a CM point of sufficiently large height. This is an effective version of a theorem of Edixhoven.


1999 
Breuer F, 'Ducci sequences over abelian groups', COMMUNICATIONS IN ALGEBRA, 27 59996013 (1999)


1998 
Breuer F, 'A note on a paper by Glaser and Schoffl', FIBONACCI QUARTERLY, 36 463466 (1998)


1998 
Breuer F, Robson JM, 'Strategy and complexity of the game of squares', BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, 30 274282 (1998)


Show 26 more journal articles 
Conference (2 outputs)
Year  Citation  Altmetrics  Link 

2012  Breuer F, 'On Abhyankar s Generalized Iteration Conjecture', DMVJahrestagung 2012 (2012)  
1998  Poretti E, Mantegazza L, Koen C, Martinez P, Breuer F, De Alwis D, Haupt H, 'Line Profile Variations in the Spectra of the Dor Star HR 2740', SymposiumInternational Astronomical Union, Cambridge University Press (1998) 
Other (1 outputs)
Year  Citation  Altmetrics  Link 

2013  Breuer F, 'The parallel worlds of number theory', (2013) [O1] 
Thesis / Dissertation (1 outputs)
Year  Citation  Altmetrics  Link 

2002  Breuer F, Sur la conjecture d AndréOort et courbes modulaires de Drinfeld, (2002) 
Grants and Funding
Summary
Number of grants  16 

Total funding  $278,720 
Click on a grant title below to expand the full details for that specific grant.
20201 grants / $5,000
Number Theory Online Conference$5,000
Funding body: AMSI Australian Mathematical Sciences Institute
Funding body  AMSI Australian Mathematical Sciences Institute 

Project Team  Florian Breuer, Michael Coons, Thomas Morrill, Alina Ostafe 
Scheme  Small Event Funding 
Role  Lead 
Funding Start  2020 
Funding Finish  2020 
GNo  
Type Of Funding  C1700  Aust Competitive  Other 
Category  1700 
UON  N 
20191 grants / $15,320
AlexandervonHumboldt Renewed Research Stay$15,320
Funding body: Alexander Von Humboldt Foundation
Funding body  Alexander Von Humboldt Foundation 

Project Team  Florian Breuer, Gebhard Böckle 
Scheme  AlexandervonHumboldt Fellowship for Experienced Researchers 
Role  Lead 
Funding Start  2019 
Funding Finish  2019 
GNo  
Type Of Funding  International  Competitive 
Category  3IFA 
UON  N 
20182 grants / $55,000
Startup Grant$50,000
Funding body: The University of Newcastle
Funding body  The University of Newcastle 

Project Team  Florian Breuer 
Scheme  School of Mathematical and Physical Sciences 
Role  Lead 
Funding Start  2018 
Funding Finish  2019 
GNo  
Type Of Funding  Internal 
Category  INTE 
UON  N 
AMSI/AustMS Workshop on Mathematical Thinking$5,000
Funding body: AMSI Intern Australian Mathematical and Physical Sciences
Funding body  AMSI Intern Australian Mathematical and Physical Sciences 

Project Team  Florian Breuer, Ljiljana Brankovic, Judyanne Osborn, Tomothy Trudgian 
Scheme  Small Event Funding 
Role  Lead 
Funding Start  2018 
Funding Finish  2018 
GNo  
Type Of Funding  Aust Competitive  Commonwealth 
Category  1CS 
UON  N 
20172 grants / $25,300
StellenboschAIMS Number Theory Conference$21,300
Funding body: CoEMASS
Funding body  CoEMASS 

Project Team  Florian Breuer, Barry Green, Patrick Rabarison 
Scheme  Number Theory 
Role  Lead 
Funding Start  2017 
Funding Finish  2017 
GNo  
Type Of Funding  External 
Category  EXTE 
UON  N 
Africa Collaboration Grant  conference funding$4,000
Funding body: Stellenbosch University
Funding body  Stellenbosch University 

Project Team  Florian Breuer, Barry Green, Patrick Rabarison 
Scheme  Africa Collaboration Grant 
Role  Lead 
Funding Start  2017 
Funding Finish  2017 
GNo  
Type Of Funding  Internal 
Category  INTE 
UON  N 
20161 grants / $39,000
Drinfeld modular forms and LFunctions$39,000
Funding body: University of Stellenbosch
Funding body  University of Stellenbosch 

Project Team  Florian Breuer, Luca Demangos 
Scheme  SubcommitteeB Postdoc Grant 
Role  Lead 
Funding Start  2016 
Funding Finish  2018 
GNo  
Type Of Funding  Internal 
Category  INTE 
UON  N 
20111 grants / $2,000
Africa Collaboration Grant  Exchange with Madagascar$2,000
Funding body: Stellenbosch University
Funding body  Stellenbosch University 

Project Team  Florian Breuer, David Holgate, Stephan Wagner 
Scheme  Africa Collaboration Grant 
Role  Lead 
Funding Start  2011 
Funding Finish  2011 
GNo  
Type Of Funding  Internal 
Category  INTE 
UON  N 
20094 grants / $106,500
Incentive Funding for Rated Researchers$38,500
Funding body: National Research Foundation
Funding body  National Research Foundation 

Project Team  Florian Breuer 
Scheme  Incentive funding for rated researchers 
Role  Lead 
Funding Start  2009 
Funding Finish  2018 
GNo  
Type Of Funding  External 
Category  EXTE 
UON  N 
Drinfeld modular forms of higher rank$30,300
Funding body: Alexander Von Humboldt Foundation
Funding body  Alexander Von Humboldt Foundation 

Project Team  Florian Breuer, HansGeorg Rueck 
Scheme  AlexandervonHumboldt Fellowship for Experienced Researchers 
Role  Lead 
Funding Start  2009 
Funding Finish  2009 
GNo  
Type Of Funding  International  Competitive 
Category  3IFA 
UON  N 
Drinfeld modular forms in higher rank$29,700
Funding body: National Research Foundation
Funding body  National Research Foundation 

Project Team  Florian Breuer 
Scheme  Blue Skies Research Grant 
Role  Lead 
Funding Start  2009 
Funding Finish  2011 
GNo  
Type Of Funding  External 
Category  EXTE 
UON  N 
ALGANT Mobility Grant$8,000
Funding body: ALGANT
Funding body  ALGANT 

Project Team  Florian Breuer 
Scheme  ALGANT mobility grant 
Role  Lead 
Funding Start  2009 
Funding Finish  2009 
GNo  
Type Of Funding  External 
Category  EXTE 
UON  N 
20061 grants / $1,500
IMU Travel Grant$1,500
Funding body: International Mathematical Union
Funding body  International Mathematical Union 

Project Team  Florian Breuer 
Scheme  IMU Travel Grant 
Role  Lead 
Funding Start  2006 
Funding Finish  2006 
GNo  
Type Of Funding  International  Competitive 
Category  3IFA 
UON  N 
20052 grants / $24,000
Arithmetic Geometry$14,000
Funding body: Stellenbosch University
Funding body  Stellenbosch University 

Project Team  Florian Breuer 
Scheme  SubcommitteeB Fund for Promising Young Researchers 
Role  Lead 
Funding Start  2005 
Funding Finish  2007 
GNo  
Type Of Funding  Internal 
Category  INTE 
UON  N 
Arithmetic Geometry$10,000
Funding body: Stellenbosch University
Funding body  Stellenbosch University 

Project Team  Florian Breuer 
Scheme  SubcommitteeB Research Grant 
Role  Lead 
Funding Start  2005 
Funding Finish  2006 
GNo  
Type Of Funding  Internal 
Category  INTE 
UON  N 
20041 grants / $5,100
Conference travel grants$5,100
Funding body: Stellenbosch University
Funding body  Stellenbosch University 

Project Team  Florian Breuer 
Scheme  Faculty of Science 
Role  Lead 
Funding Start  2004 
Funding Finish  2008 
GNo  
Type Of Funding  Internal 
Category  INTE 
UON  N 
Research Supervision
Number of supervisions
Current Supervision
Commenced  Level of Study  Research Title  Program  Supervisor Type 

2011  PhD  Bounds on coefficients of Drinfeld modular polynomials  Mathematics, University of Antananarivo  Principal Supervisor 
Past Supervision
Year  Level of Study  Research Title  Program  Supervisor Type 

2020  PhD  Torsion bounds for Drinfeld modules with complex multiplication  Mathematics, Stellenbosch University  Sole Supervisor 
2019  PhD  Drinfeld modular forms of higher rank from a latticeoriented point of view  Mathematics, Stellenbosch University  CoSupervisor 
2018  PostDoctoral Fellowship  Quantum jinvariants and class fields of function fields  Mathematics, Stellenbosch University  Sole Supervisor 
2016  Masters 
Geometry of Complex Polynomials: On Sendov’s Conjecture Sendov’s conjecture states that if all the zeroes of a complex polynomial<br />P(z) of degree at least two lie in the unit disk, then within a unit distance<br />of each zero lies a critical point of P(z). In a paper that appeared in 2014,<br />Dégot proved that, for each a in (0, 1), there is an integer N such that for any<br />polynomial P(z) with degree greater than N, P(a) = 0 and all zeroes inside<br />the unit disk, the disk z  a <= 1 contains a critical point of P(z). Basing<br />on this result, we derive an explicit formula N(a) for each a in (0, 1) and,<br />furthermore, obtain a uniform bound N for all a in [alpha, beta] where 0 < alpha < beta <<br />1. This addresses the questions posed in Dégot’s paper.<br /> 
Mathematics, Stellenbosch University  Principal Supervisor 
2016  Masters 
Elliptic Curve Cryptography In this thesis we present a selection of DiffieHellman cryptosystems, which<br />were classically formulated using the multiplicative group of a finite field, but<br />which may be generalised to use other group varieties such as elliptic curves.<br />We also describe known attacks on special cases of such cryptosystems, which<br />manifest as solutions to the discrete logarithm problem for group varieties,<br />and the elliptic curve discrete logarithm problem in particular. We pursue<br />a computational approach throughout, with a focus on the development of<br />practical algorithms. 
Mathematics, Stellenbosch University  Sole Supervisor 
2015  PostDoctoral Fellowship  Drinfeld modular forms in higher rank  Mathematics, Stellenbosch University  Sole Supervisor 
2013  PhD 
On the coefficients of Drinfeld modular forms of higher rank <p><span lang="ENGB">While defined on a different number system than the usual real numbers, Drinfeld modular forms are functions which exhibit remarkable symmetry properties. The 1dimensional Drinfeld modular forms are well understood and correspond closely to classical modular forms which have a central position in the solutions of many important problems in modern mathematics. Recently, higher dimensional Drinfeld modular forms have been defined, but not much is known about them at present. The candidate has made important progress toward the understanding of these functions.</span></p> 
Mathematics, Stellenbosch University  Sole Supervisor 
2013  Masters 
Riemann Hypothesis for the zeta function of a function field over a finite field. Let K be a function field over a finite field. Fix a place (\infty) of K, which<br />we shall call the prime at infinity. We consider the ring A of elements of K regular away from infinity, which we call the ring of integers<br />of K with respect to (\infty). There is a bijection between the set of proper ideals<br />of A and the places of K different from (\infty). We define the zeta function Z_A(s)<br />for the ring A in a way analogous to the Dedekind zeta function of the ring of<br />integers of a number field. The analogue of the Riemann Hypothesis for Z_A(s)<br />was first proved by André Weil in 1948, and our goal is to give an exposition<br />of a simpler proof of this theorem due to Enrico Bombieri. 
Mathematics, Stellenbosch University  Sole Supervisor 
2012  Masters 
Drinfeld modules and their application to polynomial factorization <p><span style="fontfamily:Arial, Helvetica, sansserif;">Major works done in Function Field Arithmetic show strong analogy between </span></p><p><span style="fontfamily:Arial, Helvetica, sansserif;">the ring of integers Z and ring of polynomials over a finite field F [T]. While</span></p><p><span style="fontfamily:Arial, Helvetica, sansserif;">an algorithm has been discovered to factor integers using elliptic curves, the</span><br /><span style="fontfamily:Arial, Helvetica, sansserif;">discovery of Drinfeld modules, which are analogous to elliptic curves made it</span><br /><span style="fontfamily:Arial, Helvetica, sansserif;">possible to exhibit an algorithm for factorising polynomials in the ring F [T].</span><br /><span style="fontfamily:Arial, Helvetica, sansserif;">In this thesis, we will introduce the notion of Drinfeld modules by studying</span><br /><span style="fontfamily:Arial, Helvetica, sansserif;">some notion within it. Then we will show an evidence of the analogy between</span><br /><span style="fontfamily:Arial, Helvetica, sansserif;">Drinfeld modules and Elliptic curves. Finally, we will confirm the analogy by</span><br /><span style="fontfamily:Arial, Helvetica, sansserif;">giving the algorithm for factoring polynomials over finite field using Drinfeld</span><br /><span style="fontfamily:Arial, Helvetica, sansserif;">modules.</span></p> 
Mathematics, Stellenbosch University  Sole Supervisor 
2012  PhD 
An analogue of the AndréOort conjecture for products of Drinfeld modular surfaces <span lang="ENGB" style="fontsize:10.0pt;fontfamily:'Arial',sansserif;">The AndréOort Conjecture states that the Zariski closure of a set of special points on a complex Shimura variety is a union of special subvarieties. Under the classical analogy between number fields and function fields, one may ask for analogous statements over function fields. In this thesis, an analogue is considered in which the Shimura variety is replaced by a product of Drinfeld modular varieties, and the corresponding conjecture is proved in the case of a product of two Drinfeld modular surfaces. In this case, the special subvarieties consist of fibres and graphs of Hecke correspondences.</span> 
Mathematics, Stellenbosch University  Principal Supervisor 
2011  PhD 
On the LatimerMacDuffee Theorem for polynomials over finite fields. Latimer & MacDuffee showed that there is a onetoone correspondence between equivalence<br />classes of matrices with a given minimum polynomial and equivalence classes of ideals of a<br />certain ring. In this dissertation, we develop an algorithm to produce a representative in each<br />equivalence class of matrices taken over the ring of polynomials over a finite field, we prove a<br />modified version of the LatimerMacDuffee theorem which holds for proper equivalence<br />classes of matrices and we define an operation that makes the set of equivalence classes of<br />matrices into an Abelian group, the class group. 
Mathematics, Stellenbosch University  Sole Supervisor 
2011  Masters 
Cyclotomic Polynomials (in the parallel worlds of number theory) It is well known that the ring of integers Z and the ring of polynomials A = Fr[T] over a<br />finite field Fr have many properties in common. It is due to these properties that almost all<br />the famous (multiplicative) number theoretic results over Z have analogues over A. In this<br />thesis, we are devoted to utilising this analogy together with the theory of Carlitz modules.<br />We do this to survey and compare the analogues of cyclotomic polynomials, the size of their<br />coefficients and cyclotomic extensions over the rational function field k = Fr(T). 
Mathematics, Stellenbosch University  Sole Supervisor 
2007  Masters 
Explicit class field theory for rational function fields Class field theory describes the abelian extensions of a given field K in terms of various<br />class groups of K, and can be viewed as one of the great successes of 20th century<br />number theory. However, the main results in class field theory are pure existence<br />results, and do not give explicit constructions of these abelian extensions. Such<br />explicit constructions are possible for a variety of special cases, such as for the field Q<br />of rational numbers, or for quadratic imaginary fields. When K is a global function<br />field, however, there is a completely explicit description of the abelian extensions of<br />K, utilising the theory of signnormalised Drinfeld modules of rank one. In this thesis<br />we give detailed survey of explicit class field theory for rational function fields over<br />finite fields, and of the fundamental results needed to master this topic. 
Mathematics, Stellenbosch University  Principal Supervisor 
2006  Masters 
Geometric actions of the absolute Galois group This thesis gives an introduction to some of the ideas originating from A. Grothendieck's<br />1984 manuscript Esquisse d'un programme. Most of these ideas are related to a new<br />geometric approach to studying the absolute Galois group over the rationals by considering<br />its action on certain geometric objects such as dessins d'enfants (called stick figures in<br />this thesis) and the fundamental groups of certain moduli spaces of curves.<br />I start by defining stick figures and explaining the connection between these innocent<br />combinatorial objects and the absolute Galois group. I then proceed to give some background<br />on moduli spaces. This involves describing how Teichmuller spaces and mapping<br />class groups can be used to address the problem of counting the possible complex structures<br />on a compact surface. In the last chapter I show how this relates to the absolute<br />Galois group by giving an explicit description of the action of the absolute Galois group<br />on the fundamental group of a particularly simple moduli space. I end by showing how<br />this description was used by Y. Ihara to prove that the absolute Galois group is contained<br />in the GrothendieckTeichmuller group. 
Mathematics, Stellenbosch University  Sole Supervisor 
News
Mathematicians forge on despite pandemic
May 28, 2020
Professor Florian Breuer
Position
Professor
School of Mathematical and Physical Sciences
Faculty of Science
Focus area
Mathematics
Contact Details
florian.breuer@newcastle.edu.au  
Phone  0240339609 
Link  Personal webpage 
Office
Room  SR220 

Building  Mathematics Building 
Location  Callaghan University Drive Callaghan, NSW 2308 Australia 