Professor Florian Breuer
Professor
School of Information 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 curiosity-driven 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 real-world application is immediately obvious. Much of what he studies has the tantalising potential to drive yet-unknown 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 digit-wise additions, e.g. 7+8 = 5 instead of 15.
“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 real-world applications. However, one important application is cryptography: the study of how messages can be safely encrypted and authenticated.
“A lot of widely-used 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 post-quantum 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 post-quantum 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 first-of-their-kind activities such as the online 2020 Number Theory conference which, on behalf of the Priority Research Centre for Computer-Assisted 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 undreamed-of 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 Alexander-von-Humboldt 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 Computer-Assisted 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 well-known 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 André-Oort 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 (ETH-Zurich), 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.
Qualifications
- Doctoral Degree in Mathematics (Equiv PhD), University of Paris - France
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 |
---|---|---|
490402 | Algebraic and differential geometry | 30 |
490401 | Algebra and number theory | 60 |
490404 | Combinatorics and discrete mathematics (excl. physical combinatorics) | 10 |
Professional Experience
UON Appointment
Title | Organisation / Department |
---|---|
Professor | University of Newcastle School of Mathematical and Physical Sciences Australia |
Academic appointment
Dates | Title | Organisation / Department |
---|---|---|
1/4/2018 - 31/3/2021 | Extraordinary Professor | Stellenbosch University Mathematical Sciences South Africa |
1/1/2013 - 31/3/2018 | Professor | Stellenbosch University Mathematical Sciences South Africa |
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 |
1/1/2004 - 30/4/2004 | Postdoctoral Fellow | Max Planck Institute Max-Planck-Institute for Mathematics Germany |
1/2/2003 - 31/10/2003 | Postdoctoral Fellow | National Tsing Hua University National Center for Theoretical Sciences Taiwan, Province of China |
Awards
Prize
Year | Award |
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2008 |
Meiring-Naude Medal The Royal Society of South Africa |
Scholarship
Year | Award |
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2009 |
Alexander-von-Humboldt 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 p-adic absolute value for each prime p which measures divisibility by p. Essentially, a rational number in lowest terms has small p-adic absolute value if its numerator is highly divisible by p. Just like the usual absolute value, the p-adic 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 p-adic absolute value gives the field of p-adic numbers. The analytic theory of p-adic 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 in-press, details are shown in italics.
Chapter (2 outputs)
Year | Citation | Altmetrics | Link | ||
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2021 |
Böckle G, Breuer F, 'The generic monodromy of drinfeld modular varieties in special characteristic', Abelian Varieties and Number Theory, American Mathematical Society, Providence, RI 147-159 (2021) [B1]
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Nova | |||
2005 |
Breuer F, Pink R, 'Monodromy groups associated to non-isotrivial Drinfeld modules in generic characteristic', , BIRKHAUSER BOSTON 61-69 (2005)
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Journal article (32 outputs)
Year | Citation | Altmetrics | Link | |||||
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2021 |
Breuer F, 'Multiplicative orders of Gauss periods and the arithmetic of real quadratic fields', Finite Fields and their Applications, 73 (2021) [C1] We obtain divisibility conditions on the multiplicative orders of elements of the form ¿+¿-1 in a finite field by exploiting a link to the arithmetic of real quadratic fields.... [more] We obtain divisibility conditions on the multiplicative orders of elements of the form ¿+¿-1 in a finite field by exploiting a link to the arithmetic of real quadratic fields.
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Nova | ||||||
2021 |
Breuer F, Pazuki F, Razafinjatovo MH, 'Heights and isogenies of Drinfeld modules', ACTA ARITHMETICA, 197 111-128 (2021) [C1]
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Nova | ||||||
2020 |
Osborn JA, Larkins JA, McBain B, Ellerton P, Black J, Borwein N, et al., 'Foundations of the DEFT Project: tertiary educators Developing Expertise Fostering Thinking', International Journal of Innovation in Science and Mathematics Education, 28 2-15 (2020) [C1] We describe the rationale, creation, and activity of a long-term co-constructed voluntary professional development initiative for tertiary educators. This is a Community of Practi... [more] We describe the rationale, creation, and activity of a long-term co-constructed voluntary professional development initiative for tertiary educators. This is a Community of Practice (CoP) formed to investigate ¿thinking¿ as a topic which may be explicitly taught. The aim of this paper is to share the value of this CoP in one context and insights into how similar approaches may be useful to other tertiary educators. The project has run for a year to date, involving a small butgrowing collective of tertiary educators, withmembers from one Canadian and several Australian Universities. Our methodology is participatory: we regularly meet,reflect,and record our reflections. Our records contain data relatingto our motivation, our insights, and the impact of these upon our choices in our teaching practices. In particular,our rationaleincludes the mutual desire to invest in developing understanding of our teaching challenges, to enable us to create thoughtful teaching approaches fit for our purposes and contexts. Hence, the central focus of our CoP isthe Development of our Expertise in Fostering Thinking (DEFT). This focus hasilluminatedgaps in existing scholarly literaturepertainingto communal development of theory, personal development of schemata, capacity for reflexivity, and instantiation in our disciplines. Opportunities and risks associated with our other sources of professional learningare identified and discussed. We elaborate on adouble-layered approach, in which we explore the constructionofour ownschemataas a precursor to helpingstudents buildtheirschemataas a foundation for their own understanding, and the role offlexible, critical,and creativethinking on our part. We utilise the scholarship of expertise, frequently returning to such questionsas ¿How do we know what our students are thinking?¿Insights gleaned from our reflections are shared, and recommendations are presented on the formation of similar projects.
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Nova | ||||||
2020 |
Breuer F, Shparlinski IE, 'Lower bounds for periods of Ducci sequences', Bulletin of The Australian Mathematical Society, 102 31-38 (2020) [C1]
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Nova | ||||||
2019 |
Breuer F, 'Periods Of Ducci Sequences And Odd Solutions To A Pellian Equation', Bulletin of the Australian Mathematical Society, 100 201-205 (2019) [C1]
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Nova | ||||||
2016 |
Breuer F, 'A note on Gekeler's h-function', ARCHIV DER MATHEMATIK, 107 305-313 (2016) [C1]
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2016 |
Breuer F, 'Explicit Drinfeld moduli schemes and Abhyankar's Generalized Iteration Conjecture', JOURNAL OF NUMBER THEORY, 160 432-450 (2016) [C1]
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Nova | ||||||
2016 |
Breuer F, Rueck H-G, 'Drinfeld modular polynomials in higher rank II: Kronecker congruences', JOURNAL OF NUMBER THEORY, 165 1-14 (2016) [C1]
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Nova | ||||||
2012 |
Breuer F, 'Newton Identities for Weierstrass Products', AMERICAN MATHEMATICAL MONTHLY, 119 796-799 (2012)
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2012 |
Breuer F, 'Special subvarieties of Drinfeld modular varieties', JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, 668 35-57 (2012)
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2010 |
Breuer F, 'Torsion bounds for elliptic curves and Drinfeld modules', JOURNAL OF NUMBER THEORY, 130 1241-1250 (2010)
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2010 |
Breuer F, 'Ducci sequences and cyclotomic fields', JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS, 16 847-862 (2010)
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2009 |
Breuer F, Rueck H-G, 'Drinfeld modular polynomials in higher rank', JOURNAL OF NUMBER THEORY, 129 59-83 (2009)
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2008 |
Breuer F, Im B-H, 'Heegner points and the rank of elliptic curves over large extensions of global fields', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 60 481-490 (2008)
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2007 |
Breuer F, 'CM points on products of Drinfeld modular curves', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 359 1351-1374 (2007)
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2007 |
Breuer F, Lotter E, van der Merwe B, 'Ducci-sequences and cyclotomic polynomials', FINITE FIELDS AND THEIR APPLICATIONS, 13 293-304 (2007)
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2007 |
Breuer F, 'The Andre-Oort conjecture for Drinfeld modular varieties', COMPTES RENDUS MATHEMATIQUE, 344 733-736 (2007)
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2005 |
Breuer F, 'The Andre-Oort conjecture for products of Drinfeld modular curves', JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, 579 115-144 (2005)
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2004 |
Breuer F, 'Images of isogeny classes on modular elliptic curves', MATHEMATICAL RESEARCH LETTERS, 11 649-651 (2004)
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2004 |
Breuer F, 'Higher Heegner points on elliptic curves over function fields', JOURNAL OF NUMBER THEORY, 104 315-326 (2004)
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2002 |
Breuer F, 'Distinguished liftings and the andré-oort conjecture', Quaestiones Mathematicae, 25 363-380 (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 Deuring-Serre-Tate lift of j-invariants of ordinary elliptic curves, then the resulting lifting problem is closely related to the André-Oort conjecture. We explore this connection, prove some results related to the André-Oort 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.
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2002 |
Breuer F, 'The André-Oort conjecture for the product of two Drinfeld modular curves', Comptes Rendus Mathematique, 335 867-870 (2002) We prove an analogue of the André-Oort 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 André-Oort 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.
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2001 |
Breuer F, 'Heights of CM points on complex affine curves', Ramanujan Journal, 5 311-317 (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.
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1999 |
Breuer F, 'Ducci sequences over abelian groups', COMMUNICATIONS IN ALGEBRA, 27 5999-6013 (1999)
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1998 |
Breuer F, 'A note on a paper by Glaser and Schoffl', FIBONACCI QUARTERLY, 36 463-466 (1998)
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1998 |
Breuer F, Robson JM, 'Strategy and complexity of the game of squares', BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, 30 274-282 (1998)
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Show 29 more journal articles |
Conference (2 outputs)
Year | Citation | Altmetrics | Link |
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2012 | Breuer F, 'On Abhyankar s Generalized Iteration Conjecture', DMV-Jahrestagung 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', Symposium-International Astronomical Union, Cambridge University Press (1998) |
Creative Work (2 outputs)
Year | Citation | Altmetrics | Link |
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2021 | Breuer F, 2021 Annual Conference of the Australian Mathematical Society, NuSpace + Online, Newcastle (2021) | ||
2021 | Breuer F, 2021 Annual Conference of the Australian Mathematical Society, NuSpace + Online, Newcastle (2021) |
Presentation (1 outputs)
Year | Citation | Altmetrics | Link |
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2013 | Breuer F, 'The parallel worlds of number theory', (2013) |
Thesis / Dissertation (1 outputs)
Year | Citation | Altmetrics | Link |
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2002 | Breuer F, Sur la conjecture d André-Oort et courbes modulaires de Drinfeld, (2002) |
Grants and Funding
Summary
Number of grants | 19 |
---|---|
Total funding | $324,245 |
Click on a grant title below to expand the full details for that specific grant.
20231 grants / $9,595
Harnessing Artificial Intelligence to Support Psychological Well-Being$9,595
This research will be underpinned by two research questions:
1) Can AI generate content and responses for online discussions groups/forums regarding psychological well-being that are comparable to that generated by two clinicians (psychologist and social worker)?
2) How do e-health users assess AI generated content for online discussions groups/forums regarding psychological well-being compared to content generated by clinicians?
The research aims are as follows:
a) To develop a purpose-built AI chatbot to facilitate generation of content psychological well-being content and responses-.
b) To compare the empathy, quality, trust, and applicability of responses generated by the AI bot with written by clinicians.
c) To provide recommendations and guidelines for further research, development, and implementation of similar AI in health settings.
Funding body: Healthy Minds Research Program HMRI
Funding body | Healthy Minds Research Program HMRI |
---|---|
Project Team | Danielle Simmonette, Gumeher Gulhati, Dr Dara Sampson, Dr Louise Thornton, Dr Erica Breuer, Prof. Florian Breuer, Dr Jane Rich, Gillian Mason, Jess Wilcox, Dr Nerida Paterson, Professor Frances Kay-Lambkin |
Scheme | Healthy Minds Seed Funding Grant |
Role | Investigator |
Funding Start | 2023 |
Funding Finish | 2023 |
GNo | |
Type Of Funding | Internal |
Category | INTE |
UON | N |
20221 grants / $5,810
Develop the Squid assessment creation and processing system to a marketable state$5,810
Squid is a system for creating and processing Mathematics assessments and consists of three main components.
The first component helps academics create pools of randomised Mathematics questions. The second component allows us to quickly and easily combine these question pools into complete assessments, which then can be either uploaded to the learning management system (Canvas) to be deployed online, or printed directly as test papers to be deployed in person.
In the case of test papers written in person, the resulting papers are scanned to PDF, and the third component of the system then processes the results, grading the multiple-choice questions, recording the marks awarded to any written-answer questions and uploading the final results to Canvas, along with feedback for each student.
This greatly decreases the workload associated to setting assessments and processing the results, as only the written-answer questions are still marked by hand while everything else is automated.
The system is currently being used for all assessments in MATH1120, and we aim to expand its use also to other courses.
Funding body: College of Engineering, Science and Environment, University of Newcastle
Funding body | College of Engineering, Science and Environment, University of Newcastle |
---|---|
Project Team | Florian Breuer |
Scheme | Impact Translator Scheme |
Role | Lead |
Funding Start | 2022 |
Funding Finish | 2022 |
GNo | |
Type Of Funding | Internal |
Category | INTE |
UON | N |
20202 grants / $35,120
FSCI Course Optimisation Grant$30,120
Funding body: Faculty of Science | University of Newcastle
Funding body | Faculty of Science | University of Newcastle |
---|---|
Project Team | Florian Breuer, Björn Rüffer |
Scheme | N/A |
Role | Investigator |
Funding Start | 2020 |
Funding Finish | 2020 |
GNo | |
Type Of Funding | Internal |
Category | INTE |
UON | N |
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
Alexander-von-Humboldt Renewed Research Stay$15,320
Funding body: Alexander Von Humboldt Foundation
Funding body | Alexander Von Humboldt Foundation |
---|---|
Project Team | Florian Breuer, Gebhard Böckle |
Scheme | Alexander-von-Humboldt 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, Judy-anne 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
Stellenbosch-AIMS Number Theory Conference$21,300
Funding body: CoE-MASS
Funding body | CoE-MASS |
---|---|
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 L-Functions$39,000
Funding body: University of Stellenbosch
Funding body | University of Stellenbosch |
---|---|
Project Team | Florian Breuer, Luca Demangos |
Scheme | Subcommittee-B 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, Hans-Georg Rueck |
Scheme | Alexander-von-Humboldt 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 | Subcommittee-B 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 | Subcommittee-B 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 |
---|---|---|---|---|
2023 | PhD | Number Theory Results With A Focus On Elliptic Curves | PhD (Mathematics), College of Engineering, Science and Environment, The University of Newcastle | Principal Supervisor |
2021 | PhD | Topology and Deep Learning | PhD (Computer Science), College of Engineering, Science and Environment, The University of Newcastle | Co-Supervisor |
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 |
---|---|---|---|---|
2023 | PhD | Heights and Drinfeld A-modules | PhD (Mathematics), College of Engineering, Science and Environment, The University of Newcastle | Principal Supervisor |
2022 | PhD | The Spectral Theory of Regular Sequences | PhD (Mathematics), College of Engineering, Science and Environment, The University of Newcastle | Co-Supervisor |
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 lattice-oriented point of view | Mathematics, Stellenbosch University | Co-Supervisor |
2018 | Post-Doctoral Fellowship | Quantum j-invariants 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 Diffie-Hellman 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 | Post-Doctoral 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="EN-GB">While defined on a different number system than the usual real numbers, Drinfeld modular forms are functions which exhibit remarkable symmetry properties. The 1-dimensional 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="font-family:Arial, Helvetica, sans-serif;">Major works done in Function Field Arithmetic show strong analogy between </span></p><p><span style="font-family:Arial, Helvetica, sans-serif;">the ring of integers Z and ring of polynomials over a finite field F [T]. While</span></p><p><span style="font-family:Arial, Helvetica, sans-serif;">an algorithm has been discovered to factor integers using elliptic curves, the</span><br /><span style="font-family:Arial, Helvetica, sans-serif;">discovery of Drinfeld modules, which are analogous to elliptic curves made it</span><br /><span style="font-family:Arial, Helvetica, sans-serif;">possible to exhibit an algorithm for factorising polynomials in the ring F [T].</span><br /><span style="font-family:Arial, Helvetica, sans-serif;">In this thesis, we will introduce the notion of Drinfeld modules by studying</span><br /><span style="font-family:Arial, Helvetica, sans-serif;">some notion within it. Then we will show an evidence of the analogy between</span><br /><span style="font-family:Arial, Helvetica, sans-serif;">Drinfeld modules and Elliptic curves. Finally, we will confirm the analogy by</span><br /><span style="font-family:Arial, Helvetica, sans-serif;">giving the algorithm for factoring polynomials over finite field using Drinfeld</span><br /><span style="font-family:Arial, Helvetica, sans-serif;">modules.</span></p> |
Mathematics, Stellenbosch University | Sole Supervisor |
2012 | PhD |
An analogue of the André-Oort conjecture for products of Drinfeld modular surfaces <span lang="EN-GB" style="font-size:10.0pt;font-family:'Arial',sans-serif;">The André-Oort 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 Latimer-MacDuffee Theorem for polynomials over finite fields. Latimer & MacDuffee showed that there is a one-to-one 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 Latimer-MacDuffee 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 sign-normalised 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 Grothendieck-Teichmuller group. |
Mathematics, Stellenbosch University | Sole Supervisor |
News
News • 28 May 2020
Mathematicians forge on despite pandemic
While disruptions caused by COVID-19 are affecting conferences worldwide, mathematicians in the Faculty of Science are embracing cutting-edge technology to ensure important progress is not stifled.
Professor Florian Breuer
Position
Professor
School of Information and Physical Sciences
College of Engineering, Science and Environment
Focus area
Mathematics
Contact Details
florian.breuer@newcastle.edu.au | |
Phone | 0240339609 |
Link | Personal webpage |
Office
Room | SR-220 |
---|---|
Building | Mathematics Building |
Location | Callaghan University Drive Callaghan, NSW 2308 Australia |