Dr Adrian Dudek
Dr

Adrian Dudek

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Background

Adrian grew up in Perth and double majored in Pure Mathematics and Applied Mathematics at the University of Western Australia. Soonafter, he ventured to Canberra to undertake a PhD, focussing on analytic number theory: an enchanting area where one perplexingly uses calculus and analysis to study discrete structures such as the set of prime numbers.

After this, he worked as a derivatives trader at Optiver APAC for five years and stayed on there as Head of Academic Partnerships. He currently straddles both industry and academia and believes they both have much to offer mathematicians.

Adrian is available (and invariably keen) to supervise honours, masters and PhD projects in analytic number theory.

Availability

Dr Adrian Dudek is:
Available for supervision

Fields of research

Algebra and number theory Applied mathematics Combinatorics and discrete mathematics (excl. physical combinatorics) Financial mathematics Mathematical Sciences Pure mathematics

Research interests

  • Analytic Number Theory

    Analytic Number Theory is the application of tools from analysis to study the structure of discrete sets such as the prime numbers. This is a beautiful field with much interplay between the continuous and discrete.

  • The Distribution of Prime Numbers

    The prime numbers are simple to define, yet try to study them in depth and they are notoriously evasive. I enjoy all manner of problems involving prime numbers, from results that establish the existence of prime numbers in short intervals to additive problems similiar to Goldbach's conjecture.

  • Construction of Expander Graphs

    An expander graph is a graph that is a strong candidate to be used as some sort of network. Mathematicians have a straightfoward way of characterising the expansion of a graph, that is, how well information travels through it. It then becomes an interesting problem to construct arbitrarily large graphs with high expansion. Solutions to this problem call on various seemingly disconnected areas of maths.

Search Professor Adrian Dudek’s works on UQ eSpace

16 works between 2015 and 2023
All (16) Journal Article (16)

1 - 16 of 16 works

2023

Journal Article

An explicit mean-value estimate for the prime number theorem in interval

Cully-Hugill, Michaela and Dudek, Adrian W. (2023). An explicit mean-value estimate for the prime number theorem in interval. Journal of the Australian Mathematical Society, 1-15. doi: 10.1017/S1446788723000113

An explicit mean-value estimate for the prime number theorem in interval

2022

Journal Article

A conditional explicit result for the prime number theorem in short intervals

Cully-Hugill, Michaela and Dudek, Adrian W. (2022). A conditional explicit result for the prime number theorem in short intervals. Research in Number Theory, 8 (3) 61. doi: 10.1007/s40993-022-00358-1

A conditional explicit result for the prime number theorem in short intervals

2019

Journal Article

Explicit short intervals for primes in arithmetic progressions on GRH

Dudek, Adrian W., Grenie, Loic and Molteni, Giuseppe (2019). Explicit short intervals for primes in arithmetic progressions on GRH. International Journal of Number Theory, 15 (4), 825-862. doi: 10.1142/s1793042119500441

Explicit short intervals for primes in arithmetic progressions on GRH

2019

Journal Article

Note on the number of divisors of reducible quadratic polynomials

Dudek, Adrian W., Pankowski, Lukasz and Scharaschkin, Victor (2019). Note on the number of divisors of reducible quadratic polynomials. Bulletin of the Australian Mathematical Society, 99 (1), 1-9. doi: 10.1017/S0004972718000734

Note on the number of divisors of reducible quadratic polynomials

2017

Journal Article

On the sum of a prime and a square-free number

Dudek, Adrian W. (2017). On the sum of a prime and a square-free number. Ramanujan Journal, 42 (1), 233-240. doi: 10.1007/s11139-015-9736-2

On the sum of a prime and a square-free number

2016

Journal Article

An explicit result for primes between cubes

Dudek, Adrian W. (2016). An explicit result for primes between cubes. Functiones Et Approximatio Commentarii Mathematici, 55 (2), 177-197. doi: 10.7169/facm/2016.55.2.3

An explicit result for primes between cubes

2016

Journal Article

On the Success of Mishandling Euclid's Lemma

Dudek, Adrian W. (2016). On the Success of Mishandling Euclid's Lemma. American Mathematical Monthly, 123 (9), 924-927. doi: 10.4169/amer.math.monthly.123.9.924

On the Success of Mishandling Euclid's Lemma

2016

Journal Article

On the Spectrum of the Generalised Petersen Graphs

Dudek, Adrian W. (2016). On the Spectrum of the Generalised Petersen Graphs. Graphs and Combinatorics, 32 (5), 1843-1850. doi: 10.1007/s00373-016-1676-0

On the Spectrum of the Generalised Petersen Graphs

2016

Journal Article

Primes in explicit short intervals on RH

Dudek, Adrian W., Grenie, Loic and Molteni, Giuseppe (2016). Primes in explicit short intervals on RH. International Journal of Number Theory, 12 (5), 1391-1407. doi: 10.1142/s1793042116500858

Primes in explicit short intervals on RH

2016

Journal Article

On the number of divisors of n2−1

Dudek, Adrian W. (2016). On the number of divisors of n2−1. Bulletin of the Australian Mathematical Society, 93 (2), 194-198. doi: 10.1017/s0004972715001136

On the number of divisors of n2−1

2016

Journal Article

On the sum of the square of a prime and a square-free number

Dudek, Adrian W. and Platt, David J. (2016). On the sum of the square of a prime and a square-free number. LMS Journal of Computation and Mathematics, 19 (1), 16-24. doi: 10.1112/s1461157015000297

On the sum of the square of a prime and a square-free number

2016

Journal Article

Almost-Ramanujan graphs and prime gaps (vol 43, pg 204, 2015)

Dudek, Adrian W. (2016). Almost-Ramanujan graphs and prime gaps (vol 43, pg 204, 2015). European Journal of Combinatorics, 51, 533-534. doi: 10.1016/j.ejc.2015.06.002

Almost-Ramanujan graphs and prime gaps (vol 43, pg 204, 2015)

2015

Journal Article

An explicit result for |L(1+it,χ)|

Dudek, Adrian W. (2015). An explicit result for |L(1+it,χ)|. Functiones Et Approximatio Commentarii Mathematici, 53 (1), 23-29. doi: 10.7169/facm/2015.53.1.2

An explicit result for |L(1+it,χ)|

2015

Journal Article

On the Riemann hypothesis and the difference between primes

Dudek, Adrian W. (2015). On the Riemann hypothesis and the difference between primes. International Journal of Number Theory, 11 (3), 771-778. doi: 10.1142/s1793042115500426

On the Riemann hypothesis and the difference between primes

2015

Journal Article

Almost-Ramanujan graphs and prime gaps

Dudek, Adrian W. (2015). Almost-Ramanujan graphs and prime gaps. European Journal of Combinatorics, 43, 204-209. doi: 10.1016/j.ejc.2014.09.001

Almost-Ramanujan graphs and prime gaps

2015

Journal Article

On Solving a Curious Inequality of Ramanujan

Dudek, Adrian W. and Platt, David J. (2015). On Solving a Curious Inequality of Ramanujan. Experimental Mathematics, 24 (3), 289-294. doi: 10.1080/10586458.2014.990118

On Solving a Curious Inequality of Ramanujan

Availability

Dr Adrian Dudek is:
Available for supervision

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Available projects

  • Prime Numbers in Short Intervals (PhD, Masters, Honours)

    The Riemann Hypothesis is possibly the most well known unsolved problem in all of mathematics. Incredibly, the more we know about the zeroes of the Riemann zeta-function, the more we know about the set of prime numbers. There are entire books full of unsolved problems on prime numbers; recent advances in studying zeta-function zeroes can be called upon to tackle these.

    One such problem involves proving the existence of a function h(x) such that the interval (x, x+h(x)) always contains at least one prime. The goal of this area of research is to find slow-growing functions that do the trick; this would ensure the existence of primes in quite small intervals. Much of this research is motivated by Legendre's conjecture: the unproved assertion that there is always a prime number between any two square numbers.

  • Problems in Additive Number Theory (PhD, Masters, Honours)

    Goldbach's conjecture is the assertion that every even integer greater than 2 can be written as the sum of two prime numbers. Despite enormous efforts, this has still not been proven and remains out of reach.

    There are plenty of problems of the above form that one can prove using a variety of techniques. In this project, the candidate will work on tangible problems in additive number theory using classic analytic approaches such as the distribution of prime numbers and sieve methods.

  • Applications of Number Theory to Graph Theory and Group Theory (PhD, Masters, Honours)

    Many problems in algebra reduce to establishing results in number theory. For example, the construction of expander graphs or estimating the number of non-isomorphic groups of a given order can be tackled using number theoretic tools.

    In this project, one will work at the intersection, tackling problems of an algebraic birthplace with tools and results from (analytic) number theory.

  • Bounded Gaps Between Prime Numbers (Honours)

    The Twin Prime Conjecture states that there are infinitely many pairs of primes that are two apart. This remains unproven but astounding progress was made in 2013 when Yitang Zhang proved that there exists some finite number such that there are infinitely many primes with this number as their difference.

    In this project, students will learn some analytic number theory towards understanding how such a result can be proven. They will also be able to highlight where progress is expected to be made and also work on some related but more tractable problems.

  • Fourier Analysis in Number Theory (Honours)

    It is well known (see, for example, the Riemann Hypothesis) that there is an intimate connection between the zeroes of the Riemann zeta-function and the distribution of prime numbers. This can be seen more directly in the setting of Fourier transforms; indeed, many modern proofs of the Prime Number Theorem opt for this approach. In this project, we will work to understand this connection more deeply as well as apply it towards new results in analytic number theory.

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