The UNSW Number Theory Seminar has been running in the School of Mathematics and Statistics since 2015. The seminar usually takes place every two weeks on Wednesdays, 2 - 4 pm, with two talks scheduled each seminar, followed by refreshments and discussions.
The talks take place in the room H13, Lawrence East, 4082, which is better known
under its previous name Red Centre, RC-4082.
The refreshments and discussions will take place in the common room of the school on the 3rd floor.
Next talks:
22 July, 2026:
14:00 - 15:00: Daniel Altman (Stanford University)
Title: Transference in the quantitative polynomial Szemerédi theorem
Abstract: We will discuss a framework which uniformly recovers and extends all known cases of quantitative bounds in the finite field polynomial Szemerédi theorem. We will also discuss its usage in the integer setting, showing in particular quantitative bounds for Szemerédi's theorem with (fixed) polynomial common difference. No prior knowledge of the area will be assumed. Based on joint work with Mehtaab Sawhney.
15:00 - 16:00: Florian Breuer (University of Newcastle)
Title: Can AI do Number Theory? Adventures with Fable and Sol
Abstract: There has recently been a lot of hype regarding mathematical breakthroughs achieved by AI, but the models in question are not available to ordinary mathematicians. In this talk, I will report on my attempts to do Number Theory research with the latest publicly accessible AI models.
Upcoming talks:
5 August: Yixiu Xiao (UNSW Sydney)
19 August:
23 September:
14:00 - 15:00: Hong Ziwei (Renmin University of China)
Title: The first moment of primitive cubic L-functions with fixed genus
Abstract: We investigate the mean value of the first moment of primitive cubic $L$-functions over $\mathbb{F}_q(T)$ in the non-Kummer setting. Employing a double Dirichlet series approach, we establish an error term of size $q^{(7/8+\epsilon)g}$. Our method significantly reduces computational complexity by avoiding the computation of the residue.
Title: On the distribution of analytic ranks of elliptic curves over number fields
Abstract: I will discuss recent work establishing upper bounds for moments of analytic ranks of elliptic curves over number fields. The approach is to study moments of the 1-level density of elliptic curve L-functions. The proof combines tools from arithmetic geometry and analytic number theory, including counting of points of bounded height on weighted projective stacks, the explicit formula for L-functions, and estimates for sums of class numbers. Part of this work is joint with Saahil Sharma.
15:00 - 16:00: John Voight (University of Sydney)
Title: Experiments and expectations in computational algebra
Abstract: Experimental results have long played a key role in number theory, suggesting conjectures, testing heuristics, and guiding the development of algorithms. At the same time, the scale and complexity of modern computations raise new horizons for these experiments and new questions for how we use computational algebra software. In this talk, I will discuss several recent examples. The first is joint work with Blair Butler and Edgar Costa on a statistical approach to understanding ranks of elliptic curves. The second is joint work with Andreas-Stephan Elsenhans on computing class groups of number fields. I will conclude with some general questions about our expectations for computer algebra systems and how these expectations may evolve as formalization and AI/machine learning tools become increasingly integrated into mathematical practice.
10 June, 2026:
14:00 - 15:00: Raymond van Bommel (University of Bristol),
Title: Reduction of plane quartics and Cayley octads
Abstract: For a long time, number theorists have been interested in studying the reduction modulo $p$ of algebraic varieties defined over number fields. For example, in the case of an elliptic curve $E$, where we distinguish between good, multiplicative, and additive reduction, the Birch and Swinnerton-Dyer conjecture predicts that the reduction plays a crucial role in understanding the rank of $E(\mathbb{Q})$. For hyperelliptic curves $y^2 = f(x)$, the reduction has been studied extensively through the Weierstrass points, i.e. the roots of $f(x)$. In this talk, I will tell about recent work joint with Jordan Docking, Vladimir Dokchitser, Reynald Lercier, Elisa Lorenzo Garcia, and Andreas Pieper, in which we study the situation for the first case of non-hyperelliptic curves: plane quartics. As a result of numerous computations, we made a prediction how the reduction type of a plane quartic can be determined from the Cayley octad, a set of eight points in $\mathbb{P}^3$ associated to the curve.
15:00 - 16:00: Youming Qiao (UTS)
Title: Algebra isomorphism, random matrices, and character sums
Abstract: In this talk, we show how the algorithmic problem of algebra isomorphism naturally leads to questions about random matrices over finite fields. To settle these questions, we need to extend both the generating function approach as in Neumann and Praeger (J. London Math. Soc., 1998) and the characteristic sum method of Gorodetsky and Rodgers (Trans. Amer. Math. Soc., 2021). Several open questions remain for further exploration.
Joint work with Tiange Li, Yinan Li, Dacheng Tao, Yingjie Wang. Paper available at https://arxiv.org/pdf/2604.00591.
29 April, 2026:
14:00 - 15:00: David Harvey (UNSW Sydney)
Title: Deterministic methods for finding elements of large multiplicative order
Abstract: I will discuss my recent paper with Markus Hittmeir
(https://arxiv.org/abs/2601.11131) on deterministic algorithms for
finding elements of large order in the multiplicative group of integers
modulo $N$.
15:00 - 16:00: Jack J. Garzella (University of California San Diego)
Title: Zeta functions on projective hypersurfaces via controlled reduction
Abstract: We describe a cohomological approach to this problem of computing the zeta function of a variety in characteristic $p$, which involves a $p$-adic formula for the Frobenius action on cohomology. Costa and Harvey came up with a fast algorithm called *controlled reduction* which uses this method and is the state of the art for varieties of dimension greater than one. We describe various ways one can improve the algorithms of Costa and Harvey. Using our algorithms, we find many examples of varieties with interesting arithmetic invariants (like Newton polygons and domino numbers). All work is joint with Batubara, Huang, and Mellberg.
29 April, 2026:
14:00 - 15:00: Chandler Corrigan (UNSW Sydney)
Title: Transformation formulæ for exponential sums twisted by Hecke eigenvalues
Abstract: Estimates for exponential sums twisted by a fixed arithmetic function play a central rôle in analytic number theory. In particular, such estimates can be used to study certain properties of the object associated to the chosen arithmetic function. In this talk, we discuss an analog of the van der Corput B-process in the case of Hecke eigenvalues for congruence subgroups.
15:00 - 16:00: Muhammad Afifurrahman (UNSW Sydney)
Title: On a restricted divisor function over function fields
Abstract: There have been recent interests on studying restricted divisor functions over integers, which naturally appear in various problems in number theory. In this (hopefully my last) talk (as a student in the campus), we discuss an analogue of these functions over $\mathbb{F}_q[T]$ and provide some related asymptotic formulas.
15 April, 2026:
14:00 - 15:00: Kéva Djambaé (University of French Polynesia)
Title: Explicit constructions of Galois Groups
Abstract: In this talk, I will introduce the concepts and questions arising in the explicit description of Galois groups over finite fields, and present a geometric construction based on isogeny cycles that allows for an explicit computation of Galois automorphisms. This is based on the preprint arXiv:2603.19428 (https://arxiv.org/abs/2603.19428).
15:00 - 16:00: Siddharth Iyer (UNSW Sydney)
Title: Cubic Polynomials and Sums of Two squares
Abstract: We establish a lower bound for the frequency with which an irreducible monic cubic polynomial can be expressed as a sum of two squares ($\square_{2}$). This provides a quantitative answer to a question posed by Grechuk (2021) concerning the infinitude of such values. Our proof relies on a two-dimensional unit argument and the arithmetic of degree six number fields. For example, we show that if $h \equiv 2 \pmod{4}$, then
\begin{align*}
\# \{n : n^3+h \in \square_{2}, \ 1 \leq n \leq x \} \gg x^{1/3-o(1)}.
\end{align*}
These arguments may be generalised to study the representation of irreducible monic cubic polynomials by the quadratic form $x^2+ny^2$, where $n \in \mathbb{N}$.
18 March, 2026:
14:00 - 15:00: Jesse Jääsaari (University of Turku)
Title: On the real zeros of half-integral weight Hecke cusp forms
Abstract: In this talk I will discuss recent work concerning the distribution of zeros of half-integral weight Hecke cusp forms of large weight on the surface $\Gamma_0(4)\backslash \mathbb{H}$. In particular, I will focus on the so-called real zeros, that is zeros on certain geodesic segments on which the cusp form takes real values, and give lower bounds for the number of these zeros.
15:00 - 16:00: Timothy Trudgian (UNSW Canberra)
Title: I've got Euclid's, I've got Al Gore, I've got Rhythm, who could ask for anything more?
Abstract: Inspired by Gershwin, the title refers to the oldest (?) mathematical algorithm: the Euclid's algorithm for division. This allows us to divide two numbers, keep track of remainders, rinse 'n' repeat, and recover GCDs. I will discuss other algebraic settings: some rings are known to be Euclidean (meaning they have this algorithm), some are known not to be; many are unknown. I will end with a summary of recent work done by Bagger, Booker, Kerr, McGown, Starichkova, and me that resolves completely the case of cyclic cubic fields.
5 November, 2025:
14:00 - 15:00: Benjamin Ward (University of York)
Title: Sets of Exact(er) approximation order
Abstract: In this talk, which is joint work with Simon Baker (Loughborough, UK), I will introduce a quantitative notion of exactness within Diophantine approximation. Given functions $\Psi:(0,\infty)\to (0, \infty)$ and $\omega : (0,\infty)\to (0, 1)$, we study the set of points that are $\Psi$-well approximable but not $\Psi(1-\omega)$-well approximable, denoted $E(\Psi,\omega)$. This generalises the set of $\Psi$-exact approximation order as studied by Bugeaud (Math. Ann. 2003). We prove results on the cardinality and Hausdorff dimension of $E(\Psi,\omega)$. In particular, for certain functions $\Psi$ we find a critical threshold on ω whereby the set $E(\Psi,\omega)$ drops from positive Hausdorff dimension to empty when $\omega$ is multiplied by a constant. The results discussed can be found in [2510.18451] A quantitative framework for sets of exact approximation order by rational numbers.
15:00 - 16:00: Subham Bhakta (UNSW Sydney)
Title: Character sums with division polynomials of elliptic curves
Abstract: In this talk, I will take you on a journey through the character sums of division polynomials evaluated at rational points on elliptic curves over prime fields; a topic that first caught my attention near the end of my PhD, inspired by a 2009 paper of I. E. Shparlinski and K. E. Stange. These character values exhibit an “almost multiplicative” behaviour. Motivated by Chowla’s conjectures on correlations of multiplicative functions, I will first present a recent joint work with I. E. Shparlinski (2025) on the correlations of these character sums under shifts. I will then discuss some bounds for these sums when twisted by various multiplicative functions.
29 October, 2025:
14:00 - 15:00: Andreas-Stephan Elsenhans (University of Sydney and University of Würzburg)
Title: Numerical verification of the Collatz Conjecture
Abstract: The Collatz conjecture (also known as the 3n+1 problem) is one of the most
popular open problems in number theory. In this talk I will give an introduction to
a theoretical analysis of the problem and explain which strategies are used for
a numerical verification.
15:00 - 16:00: Youming Qiao (UTS)
Title: A quantum algorithm for $2\times 2\times 2$ tensor isomorphism over $\mathbb{Z}$
Abstract: We present a quantum polynomial-time algorithm that decides whether two tensors in $\mathbb{Z}^2\otimes\mathbb{Z}^2\otimes\mathbb{Z}^2$ are in the same orbit under the natural action of $\mathrm{GL}(2, \mathbb{Z})\times\mathrm{GL}(2, \mathbb{Z})\times\mathrm{GL}(2, \mathbb{Z})$. This algorithm is a natural consequence of the works of Gauss (on composition laws), Bhargava (on higher composition laws), and Hallgren (on quantum algorithms for the principal ideal problem). An intriguing question is the case of $\mathbb{Z}^3\otimes\mathbb{Z}^3\otimes\mathbb{Z}^3$.
15 October, 2025:
14:00 - 15:00: Lee Zhao (UNSW Sydney)
Title: When the eggs are fried
Abstract: Grey, dear friends, is all unproven theory. Thus I mar the immortal words of a very witty and most unjustly abused immortal. At the most recent meeting of Number Theory Down Under, it was suggested that the work of Kerr-Shparlinski-Wu-Xi on Kloosterman sums might be applied to improve an asymptotic formula of Gao-Zhao for the twisted fourth moment of Dirichlet $L$-functions to certain prime power moduli, as this latter result was presented. Can this idea work? We heeded Sancho Panza's counsel that "you'll see when the eggs are fried" and greened the untested theory. More specifically, taking the above recommendation, as well as doing other things, we extended the aforesaid moment result to general moduli and significantly improved the error term. I shall report on this recent work (arXiv:2509.24690), joint with P. Gao and X. Wu, during this talk.
15:00 - 16:00: Chiara Bellotti (UNSW Canberra)
Title: A New Zero-Density Estimate for the Riemann Zeta Function close to the 1-line
Abstract: In this talk we present a new type of zero-density estimate for the Riemann zeta function close to the one-line. In particular, we show that the number of zeros in this region remains bounded by an absolute constant when approaching the left edge of the Korobov-Vinogradov zero-free region. As a consequence, we obtain an essentially optimal refinement of a result due to Pintz concerning the error term in the prime number theorem.
1 October, 2025:
14:00 - 15:00: Bittu Chahal (IIIT Delhi)
Title: Chebyshev's bias for irrational factor function
Abstract: Chebyshev's bias is the phenomenon that the number of prime quadratic nonresidues of a given modulus predominate over the prime quadratic residues, in other words, primes are biased toward quadratic nonresidues. We study this bias question in the context of the irrational factor function $I_k(n)$, defined by $I_k(n)=\prod_{i=1}^lp_i^{\beta_i}$, where $n=\prod_{i=1}^lp_i^{\alpha_i}$ and $$\beta_i=
\left\{\begin{array}{cc}
\alpha_i, & \mbox{if } \alpha_i < k,\\
\frac{1}{\alpha_i}, & \mbox{if } \alpha_i\geq k.\end{array}\right.$$
In particular, we introduce the irrational factor function in both number field and function field settings, derive asymptotic formulas for their average value, and establish $\Omega$-results for the error term in the asymptotic formulas. Furthermore, we study the Chebyshev's bias phenomenon for number field and function field analogues of sum of the irrational factor function, under assumptions on the real zeros of Hecke $L$-functions associated with Hecke characters in the number field case.
15:00 - 16:00: Muhammad Afifurrahman (UNSW Sydney)
Title: Counting multiplicatively dependent integer vectors on a hyperplane
Abstract: We give several asymptotic formulas and bounds for the number of multiplicativly dependent integer vectors of bounded height that lies on a hyperplane, extending the work of Pappalardi, Sha, Shparlinski and Stewart. Joint work with Valentio Iverson and Gian Cordana Sanjaya (University of Waterloo).
17 September, 2025:
14:00 - 15:00: Lewis Combes (University of Sydney)
Title: Selmer groups for mod p Galois representations
Abstract: Selmer groups are an important construction in modern number theory, with their ranks expected to encode arithmetic information associated to their underlying objects. This is most obvious in conjectures like that of Bloch-Kato, relating an $L$-value to the rank of a Selmer group of a $p$-adic Galois representation. In recent years, mod p Galois representations have started to receive similar attention, partly due to Scholze's proof that many torsion classes have their own associated representations. In this talk we will cover some basics of Selmer groups, how to compute them for mod $p$ Galois representations, and how to formulate and test interesting conjectures regarding their ranks.
15:00 - 16:00: Alexander Fish (University of Sydney)
Title: Ehrhart spectra of large subsets in $\mathbb{Z}^n$
Abstract: The Ehrhart spectrum of a set $E$ in $\mathbb{Z}^n$, defined as the set of all Ehrhart polynomials of simplices with vertices in $E$, generalizing the notion of volume spectrum. We show that for any $E$ in $\mathbb{Z}^n$ with positive upper Banach density, there is some integer $k$ such that the Ehrhart spectrum of $k\mathbb{Z}^n$ is contained in the Erhard spectrum of $E$. This is a joint work with Michael Bjorkludn and Rickard Cullman both from Chalmers.
14:00 - 15:00: Dmitry Badziahin (University of Sydney)
Title: Can we generate "RSA-safe" values of polynomials
Abstract: A crucial part of various cryptosystems such as RSA is to generate composite numbers n=pq that are almost impossible to factorise. Among other restrictions, that means that n needs to be huge (e.g. 2048 bits) and p and q need to be primes of a similar size. Such numbers are not difficult to generate. But what if, on top of that, we require n to be a value P(m) of a given polynomial P with integer coefficients at an integer point m? Then the problem becomes much less trivial. In this talk I will discuss how one can randomly generate such triples (p,q,m) for quadratic and cubic polynomials P. We will also see that p and q can be generated in such a way that p/q is close to any given positive real number.
15:00 - 16:00: Shanta Laishram (Indian Statistical Institute, New Delhi)
Title: On a class of Monogenic polynomials
Abstract: Let $f(x) \in \mathbb{Z}[x]$ be an irreducible polynomial of degree
$n$ and $\theta$ be a root of $f(x)$. Let $K=\mathbb{Q}(\theta)$ be
the number field and $\mathbb{Z}_K$ be the ring of algebraic integers
of $K$. We say $f(x)$ is monogenic if $\{1, \theta, \ldots,
\theta^{n-1} \}$ is a $\mathbb{Z}$-basis of $\mathbb{Z}_K$.
In this talk, we consider the family of polynomials $f(x)=x^{n-km}
(x^k+a)^m+b \in \mathbb{Z}[x]$, $1\leq km< n$. We provide a necessary
and sufficient conditions for $f(x)$ to be monogenic. As an
application, we get a class of monogenic polynomials having non
square-free discriminant and Galois group $S_n$, the symmetric group
on $n$ symbols. This is a joint work with A. Jakhar and P. Yadav.
Title: Tackling the $\varepsilon$ for primes in short arithmetic progressions
Abstract: Given a zero-free region and an average zero-density estimate for all Dirichlet $L$-functions modulo $q$, we refine the error terms of the prime number theorem in all and almost all short arithmetic progressions. If we e.g. assume the Generalized Density Hypothesis, then as $x\rightarrow \infty$ the prime number theorem holds for any arithmetic progression modulo $q\leq \log^\ell x$ for any $\ell>0$ and in the interval $(x,x+\sqrt{x}\exp(\log^{2/3+\varepsilon} x)]$ for any $\varepsilon>0$. This refines the classic interval $(x,x+x^{1/2+\varepsilon}]$.
25 June, 2025:
14:00 - 15:00: Florian Breuer (University of Newcastle)
Title: Coefficients of modular polynomials
Abstract: Modular polynomials encode isogenies between pairs of elliptic curves and have applications to cryptography. Famously, these polynomials have very large coefficients. In this talk I will outline some recent results on the sizes and divisibility properties of these coefficients. Time permitting, I will also touch on the analogous situation for Drinfeld modular polynomials.
15:00 - 16:00: Bryce Kerr (UNSW Canberra)
Title: Poissonian pair correlation for real sequences
Abstract: The Poissonian pair correlation is a local statistic that captures strong pseudo-randomness in deterministic sequences. In a forthcoming paper with Lianf, we provide new sufficient conditions under which a real sequence exhibits the metric Poissonian property. This will be a continuation of Liang's talk a few weeks ago.
18 June, 2025:
15:00 - 16:00: John Voight (University of Sydney)
Title: An overview of Magma
Abstract: Magma is a computer algebra system that provides a rigorous environment for computations in algebra, number theory, geometry, combinatorics, and more! Our goal is to support and accelerate maths research. In this introductory talk, we'll provide an interactive overview of some of Magma's features, focused on concrete examples.
11 June, 2025:
14:00 - 15:00: Aaron Manning (UNSW Sydney)
Title: Counting Matrices Over Finite Rank Multiplicative Groups
Abstract: There have been many recent works regarding arithmetic statistics questions related to matrices with entries from sets of number theoretic interest. This includes, in particular, providing upper bounds on the number of matrices with a prescribed rank, determinant, or characteristic polynomial, over such a set. Motivated by some recent work by Alon and Solymosi (2023), we consider matrices with entries from finitely generated subgroups of the group of units of a field of characteristic zero. Such sets require a considerably different approach to many that have been studied previously. The primary tools we require follow from the Subspace Theorem of Schmidt (1972) on the simultaneous approximation of algebraic numbers by rational numbers.
15:00 - 16:00: Timothy Trudgian (UNSW Canberra)
Title: A convex hull, a boundary drawn
Envelops points from dusk till dawn
Abstract: Many results in number theory rely on bounding exponential sums. The title (written, like so many student assignments, by ChatGPT) mentions a convex hull. The more we know about this set of points, the better our knowledge of exponential sums. Applications abound! I will mention these and an online database in which everyone can contribute
Abstract: Following a brief survey on the recent advances in the problem of counting rational points near planar curves, we will take a particular interest in the case of the brownian motion. Recent results in collaboration with Evgenyi Zorin (York) and Volodymyr Pavlenkov (York) indeed enable us to solve a problem asked by Sprindzuk (1979).
15:00 - 16:00: Liang Wang (UNSW Canberra)
Title: On the metric Poissonian pair correlation of real sequences
Abstract: The Poissonian pair correlation is a local statistic that captures strong pseudo-randomness in deterministic sequences. In a forthcoming paper (potentially on arXiv by the talk), joint work with Bryce Kerr, we provide new sufficient conditions under which a real sequence exhibits the metric Poissonian property. Moreover, our criterion refines and improves existing quantitative bounds on the energy estimates that govern such behaviour. As an application of our results, we show that both convex and polynomial sequences are metric Poissionian.