Talks
New Polylogarithm Depth Reductions in Weight 5 and 6
13 Sept 2023, at 15:50
Polylogarithms, Cluster Algebras, and Scattering Amplitudes, Brin Center, UMD
Abstract: Goncharov sketched a programme to tackle Zagier’s Polylogarithm Conjecture on \(\zeta_F(m)\) by understanding the structure of multiple polylogarithms in weight m, in particular how the motivic framework should provide a characterisation of the depth of a multiple polylogarithm by a filtration arising from iterating the coproduct/cobracket. In weights 2 and 3, this is essentially equivalent to the result that one can write every multiple polylogarithm in terms of \(\Li_2\) and \(\Li_3\) respectively. In weight 4 however, the function \(\Li_{3,1}\) (or \(I_{3,1}\) as an integral) is genuinely of depth 2 and cannot be reduced, but the framework predicts that \(I_{3,1}(\text{dilogarithm 5-term relation}, z)\) should reduce. In 2011, Gangl gave this reduction explicitly, and provided 122 \(\Li_4\) terms (whose arguments typically involved structured products of up to 4 cross-ratios) found with perspicacious experimentation and computer assistance; a conceptual derivation was given later, in 2018, by Goncharov and Rudenko as a consequence of a beautiful and simple weight 4 identity, with a cluster-geometric flavour. Since then various subsets of Matveiakin, Rudenko, Gangl, Radchenko, and myself, have worked to extend these cluster-geometric identities, and in particular the consequent depth reduction identities, to higher weight. I will report on the progress, so far, of the known depth reductions in weight 5 and weight 6, what is still left for us to do, and what this means for trying to tackle \(\zeta_F(5)\) and \(\zeta_F(6)\).
Multiple polylogarithms, depth reductions and Zagier's conjecture
10 Aug 2023, at 15:00
MPI-Oberseminar, MPIM Bonn
Abstract: Polylogarithms (and the many variable generalisation, the multiple polylogarithms) are an important class of special functions which appear in many areas of pure mathematics (K-theory, number theory, hyperbolic geometry, differential geometry, ...) and high-energy physics (computation of Feynman integrals and of scattering amplitudes, ...).
I will give an introduction to the prominent results and conjectures on the structure of multiple polylogarithms, primarily originating with Goncharov, motivated by his programme to tackle Zagier's conjecture on special values \(\zeta_F(n)\) of the Dedekind zeta function. I will then explain some of the/our recent results (involving collaborations of various subsets of myself, Andrei Matveiakin, Danylo Radchenko, Daniil Rudenko, and Herbert Gangl), wherein they/we establish identities which reduce the depth (number of arguments) of important combinations of multiple polylogarithms. These results should be relevant for tackling Zagier's conjecture on \(\zeta_F(5)\) and \(\zeta_F(6)\).
Depth reductions of multiple polylogarithms: expectations, techniques, approaches
20 Jul 2023
Talk for REU students, UMD
Abstract: An overview of the expectations, and techniques for polylogarithm depth reduction, with the reduction of \( \Li_{2,1} \) to \( \Li_3 \), and its 22-term relation corollary, as the main example.
Multiple zeta values in (differential) geometry and number theory
31 May 2023, at 09:30 CEST / 15:20 CST
BIMSA-BIT Differential Geometry Seminar, BIMSA, Beijing
Abstract: Multiple zeta values are a mysterious class of real numbers that appear in many branches of pure mathematics and in theoretical physics. I will explain some of the basic theory and problems surrounding multiple zeta values (from a more algebraic or number theoretic viewpoint). I will then discuss where multiple zeta values (or slight generalisations thereof) appear some more geometric or analytic contexts, such as the area expansion of families of constant mean curvature surfaces (as studied by Heller, Heller and Traizet), or in the Dirichlet eigenvalues of regular polygons (as studied by Berghaus, Georgiev, Monien and Radchenko).
Generators of multiple t values, and alternating multiple zeta values
3 May 2023, at 14:30
Number Theory Lunch Seminar, MPIM Bonn
5 June 2023, at 14:00
Oberseminar Zahlentheorie, Universität zu Köln
Abstract: Multiple zeta values, and their relatives including the multiple \(t\) values, are a prominent but mysterious class of real numbers, which appear in various areas from high energy physics and knot theory, to number theory and the periods of mixed Tate motives. I will review some work by Francis Brown, and some recent work by Takuya Murakami, on how to prove certain elements \(\zeta(\text{2's and 3's})\), and \(t(\text{2's and 3's})\), generate the space of multiple zeta values. I will then extend Murakami's work to show \(t(\text{1's and 2's})\) generate the space of multiple \(t\) values and alternating multiple zeta values, and make some progress towards Saha's conjecture that \(t(\text{1's and 2's, 2 or 3})\) are a basis for convergent MtV’s.
The usefulness of two-one formulas
22 March 2023, at 13:30
Geometries and Special Functions for Physics and Mathematics
Multiple zeta values in block degree 2, and the period polynomial relations
5 December 2022, at 14:00
Séminaire de théorie des nombres de l'IMJ-PRG, Paris
20 December 2022, at 11:30
- MZVs in block degree 2, and the period polynomial relations notes
- An introduction to MZV's (for Introductory part of Groningen seminar)
Abstract: I introduced the block decomposition on multiple zeta values in order to understand and generalise some (conjectural) families of relations. It was extended to a filtration on motivic multiple zeta values by Francis Brown and further extended by Adam Keilthy, who showed it gives a route to understanding the structure of the motivic Lie algebra. I will discuss a recent project with Keilthy where we are able to understand the structure in block degree 2 by evaluating \(\zeta(2,\ldots,2,4,2,\ldots,2)\) in terms of double zeta values, and where we showed how the famous period polynomial relations for double zeta values arise in an explicit way from the so-called block relations introduced in Keilthy’s thesis.
Multiple zeta values and modular forms
30 June 2022, at 18:10
Point Configurations: Deformations and Rigidity
Supplement to Modular forms, universal optimality and Fourier interpolation, minicourse by D. Radchenko.
Abstract: Multiple zeta values (MZV's) are a prominent but mysterious class of real numbers, generalising the values of the Riemann zeta function to several arguments. They appear surprisingly often in many branches of mathematics and in high energy physics. I will give a brief introduction and overview of MZV's, and then explain some work by Gangl, Kaneko and Zagier which connected modular forms with double zeta value identities.
Zagier's polylogarithm conjecture and an explicit 4-ratio
23 June 2022, at 10:55
[KA2W02] Arithmetic geometry, cycles, Hodge theory, regulators, periods and heights
[KAH2] \( K \)-theory, algebraic cycles and motivic homotopy theory
Abstract: In his celebrated proof of Zagier's polylogarithm conjecture for weight 3 Goncharov introduced a "triple ratio", a projective invariant akin to the classical cross-ratio. He has also conjectured the existence of "higher ratios" that should play an important role for Zagier's conjecture in higher weights. Recently, Goncharov and Rudenko proved the weight 4 case of Zagier's conjecture with a somewhat indirect method where they avoided the need to define a corresponding "quadruple ratio". We propose an explicit candidate for such a "quadruple ratio" and as a by-product we get an explicit formula for the Borel regulator of \( K_7(F) \) in terms of the tetralogarithm function (joint work with H. Gangl and D. Radchenko).
Functional equations for Nielsen polylogarithms
16 June 2022, at 16:30
Abstract: Nielsen polylogarithms \( S_{p,q} \) are perhaps the simplest examples of higher depth multiple polylogarithms, but beyond some simple symmetries, relatively little seems to be known about their identities and functional relations. I will report on some joint work with Herbert Gangl, and Danylo Radchenko, wherein we establish that \( S_{3,2} \) satisfies the dilogarithm 5-term relation, modulo explicit \( \Li_5 \) terms. From this we can always extract corresponding results for \( S_{3,2} \) whenever a dilogarithm identity is accessible through the 5-term relation. I will also try to give a flavour of some of our results and evaluations in higher weight, and how this 5-term relation for \( S_{3,2} \) could be useful in trying to prove Zagier’s conjecture on \( \zeta_F(5) \).
Computing \( \zeta(n_1,\ldots,n_r) \) numerically -- explaination of Zagier's approach and some extensions
12 May 2022, at 17:00 JST
Computing Multiple Zeta Seminar
Abstract: First, I will explain how to compute the values of truncated MZV's \( \zeta_M(n_1,\ldots,n_r) \), where we sum the terms up to some finite bound. I will point out some problems and pitfalls with the naive implementation(s) of this, and show how to do this more efficiently. Then I will discuss how to find recursively the asymptotic series which can be used to approximate the tail of \( \zeta(n_1,\ldots,n_r) \), and how to obtain a numerical value for \( \zeta(n_1,\ldots,n_r) \) from this. I will give implementations in both gp/pari and Mathematica. I will also indicate how one can extend this approach to evaluate alternating MZV's or multiple \(t\) values. (This is in some sense a continuation of the previous seminar talk.)
Symmetries of multiple \( t \) values
13 April 2022, at 12:15 (part 1)
20 April 2022, at 12:15 (part 2)
Seminar: Arithmetische Geometrie und Zahlentheorie, Hamburg
Abstract: Joint work with Michael Hoffman. We establish a symmetry theorem for multiple \( t \) values, and give some applications. arXiv:2204:14183.
The Goncharov coproduct and motivic multiple zeta values
12 January 2022, at 12:15
Seminar: Arithmetische Geometrie und Zahlentheorie, Hamburg
Abstract: An introduction to the Goncharov coproduct on iterated integrals and motivic MZV's, with some examples of applications to transcendence questions.
Generators of multiple t values, and alternating multiple zeta values
15 December 2021, at 12:15
Seminar: Arithmetische Geometrie und Zahlentheorie, Hamburg
17 December 2021, at 14:15
Abstract: Multiple zeta values, and their relatives including the multiple \(t\) values, are a prominent but mysterious class of real numbers, which appear in various areas from high energy physics and knot theory, to number theory and the periods of mixed Tate motives. I will review some work by Francis Brown, and some recent work by Takuya Murakami, on how to prove certain elements \(\zeta(\text{2's and 3's})\), and \(t(\text{2's and 3's})\), generate the space of multiple zeta values. I will then extend Murakami's work to show \(t(\text{1's and 2's})\) generate the space of multiple \(t\) values and alternating multiple zeta values, and make some progress towards Saha's conjecture that \(t(\text{1's and 2's, 2 or 3})\) are a basis for convergent MtV’s.
Functional equations for Nielsen polylogarithms
6 July 2021, at 9:00 CEST / 17:00 JST
JENTE seminar, zoom
Abstract: Nielsen polylogarithms \( S_{p,q} \) are perhaps the simplest examples of higher depth multiple polylogarithms, but beyond some simple symmetries, relatively little seems to be known about their identities and functional relations. I will report on some joint work with Herbert Gangl, and Danylo Radchenko, wherein we establish that \( S_{3,2} \) satisfies the dilogarithm 5-term relation, modulo explicit \(\operatorname{Li}_5\) terms. From this we can always extract corresponding results for \(S_{3,2}\) whenever a dilogarithm identity is accessible through the 5-term relation. I will also try to give a flavour of some of our results and evaluations in higher weight.
Multiple polylogarithms in weight 4 and weight 5
21, 28 April 2021, at 12:15
Seminar: Arithmetische Geometrie und Zahlentheorie, Hamburg
Abstract: An introduction to multiple polylogarithms, and an in depth look at questions in weight 4 and weight 5 connected to Zagier's polylogarithm conjecture, and Goncharov's freeness conjecture.
Zagier's polylogarithm conjecture and an explicit 4-ratio
22 June 2020, at 9:00 CEST = 16:00 JST
MZV Seminar, Kyushu University
15 July 2020, at 14:30
Number Theory Lunch Seminar, MPIM
Abstract: In his celebrated proof of Zagier's polylogarithm conjecture for weight 3 Goncharov introduced a "triple ratio", a projective invariant akin to the classical cross-ratio. He has also conjectured the existence of "higher ratios" that should play an important role for Zagier's conjecture in higher weights. Recently, Goncharov and Rudenko proved the weight 4 case of Zagier's conjecture with a somewhat indirect method where they avoided the need to define a corresponding "quadruple ratio". We propose an explicit candidate for such a "quadruple ratio" and as a by-product we get an explicit formula for the Borel regulator of \(K_7\) in terms of the tetralogarithm function (joint work with H. Gangl and D. Radchenko).
Cluster polylogarithms and identities
5 March 2020, at 13:30
Cluster Algebras and the Geometry of Scattering Amplitudes, Higgs Centre, Edinburgh
Zagier's polylogarithm conjecture and an explicit 4-ratio
29 January 2020, at 12:15
Arithmetische Geometrie und Zahlentheorie Seminar, Universität Hamburg
4 February 2020, at 13:00
Arithmetic Study Group, Durham University
5 February 2020, at 16:00
Heilbronn Number Theory Seminar, Bristol University
Abstract: In his celebrated proof of Zagier's polylogarithm conjecture for weight 3 Goncharov introduced a "triple ratio", a projective invariant akin to the classical cross-ratio. He has also conjectured the existence of "higher ratios" that should play an important role for Zagier's conjecture in higher weights. Recently, Goncharov and Rudenko proved the weight 4 case of Zagier's conjecture with a somewhat indirect method where they avoided the need to define a corresponding "quadruple ratio". We propose an explicit candidate for such a "quadruple ratio" and as a by-product we get an explicit formula for the Borel regulator of \(K_7\) in terms of the tetralogarithm function (joint work with H. Gangl and D. Radchenko).
Clean single-valued multiple polylogarithms
9 April 2019, at 9:00
Workshop on Modular forms, periods and scattering amplitudes, ETH Zürich
Abstract: Based on joint work with Duhr, Dulat and Gangl, we define a new class of so-called clean single-valued multiple polylogarithms \(C(a_1,\ldots,a_n;z)\). We show that these functions satisfy the same functional relations as the usual multiple polylogarithms, but with all product terms eliminated, leaving only clean functional relations. In particular, identities on the level of the symbol modulo products always lift to numerically verifiable identities between these clean functions.
MZV's speed talk
5 September 2018, Speed Talks Session
Elementare und Analytische Zahlentheorie, MPIM Bonn
Abstract: MZV identities in 60 seconds.
Cyclic insertion on MZV's and the alternating block decomposition
24 April 2018, at 14:00
Zahlentheorie Seminare, Köln
- Cyclic insertion on MZV's and the alternating block decomposition slides
- Cyclic insertion on MZV's and the alternating block decomposition with notes
Abstract: A generalisation of the cyclic insertion conjecture on MZV's, and progress towards a proof using the motivic MZV framework.
Various aspects of (multiple) polylogs
15 March 2018, at 15:00
Oberseminar (New Guests), MPIM Bonn
Abstract: An introduction/overview of my research in multiple polylogarithms
Bowman-Bradley type identities for symmetrised MZV's
Tuesday 30 January 2018, at 14:00
Periods and regulators workship, HIM Bonn
Abstract: Motivated by the corresponding result for finite MZV's, I will discuss a Bowman-Bradley type identity for symmetrised MZV's.
Motivic MZV's and the cyclic insertion conjecture
Wednesday 17 January 2018, at 15:00
Periods and regulators workship, HIM Bonn
- Motivic MZV's and the cyclic insertion conjecture slides
- Motivic MZV's and the cyclic insertion conjecture with notes
Abstract: I will start by recalling two conjectural families of MZV identities proposed by Borwein-Bradley-Broadhurst-Lisonek, and by Hoffman. I will show how both of these conjectures can be unified into a larger conjectural family of identities by using the so-called block decomposition of iterated integrals introduced here.
Using the motivic MZV framework of Brown I will show that a symmetrised version of this conjecture holds up to \( \Q \). This will give a proof of Hoffman's identity, up to \( \Q \) and an improvement of the Bowman-Bradley theorem giving some progress towards the BBBL conjecture.
Relating MPL's in weight \( \geq 5 \)
Saturday 11 November 2017, at 11:20
Polylogs, multiple zetas and related topics, Tohoku (東北)
Abstract: Multiple polylogarithms, a multi-variable variant of the classical polylogarithms, are important functions both in number theory, and in theoretical physics. Understanding their identities and functional equations is of considerable interest. Here we investigate some of the symmetries and relations between multiple polylogarithms at weight 5. Using an observation due to Goncharov, on the co-boundary of \( I^+_{4,1}(x,y) = \frac{1}{2} (I_{4,1}(x,y) + I_{4,1}(x,1/y)) \), we are able to obtain identities reducing certain combinations \( I^+_{4,1}(\text{ \( \Li_2 \) functional equation}, y) \) or \( I^+_{4,1}(x, \text{\( \Li_3 \) functional equation}) \) to \( \Li_5 \)'s and so obtain new functional equations for \( \Li_5 \). We can generalise this approach to weight 6 using \( I_{5,1}^+(\text{\( \Li_3 \) functional equation}, \text{\( \Li_3 \) functional equation}) = \Li_6's \) to obtain new \( \Li_6 \) functional equations. We indicate some potential approaches and partial results for higher weight \( \geq 7 \).
The block decomposition of iterated integrals, and cyclic insertion on MZV's
Monday 17 October 2017, at 16:50
MZV Seminar, Kyushu (九州)
Abstract: As some background, I will first discuss two (conjectural) families of MZV identities -- the cyclic insertion conjecture of Borwein et al, and an identity of a similar flavour, presented by Hoffman. Using the motivic framework due to Goncharov and Brown, I will explain how one can gain some insight into the structure of these identities. I will then present a (conjectural) unification of these identities described using the so-called alternating block decomposition of iterated integrals, and prove a certain symmetrised version always holds for motivic MZV's.
Cuspidal types and characters: tame parametrisation theorem
Part 1: Tuesday 27 June 2017, at 15:15
Part 2: Tuesday 4 July 2017, at 14:15
OSAZ, Tübingen
Motives and multiple zeta values
Wednesday 5 April at 14:00
British Mathematical Colloquium, Durham
Abstract: In this talk I will introduce multiple zeta values (MZV's), a rather mysterious class of real numbers about which many things are conjectured, but relatively little is known.
Their analytic definition frequently causes transcendentality problems and makes understanding the structure of MZV's difficult. To circumvent these problems, we can introduce a purely algebraic lifting -- the so-called `motivic' MZV's of Goncharov, and of Brown. Motivic MZV's form a graded Hopf algebra, giving them a much more rigid structure, which we can exploit.
I will aim to discuss some conjectural families of relations on MZV's that I have been able to better understand, and to generalise, with this motivic point of view.
Computation of arithmetic cohomology
Part 1: Tuesday 17 January 2017, at 14:15
Part 2: Tuesday 24 January 2017, at 14:15
OSAZ, Tübingen
(Motivic) multiple zeta values, cyclic insertion and the block decopmosition
Part 1: Tuesday 18 October 2016, at 14:15
Part 2: Tuesday 25 October 2016, at 14:15
OSAZ, Tübingen
Mathematics of the Rubik's cube
Tuesday 14 June 2016, at 11:00
Post exam activity, Durham
- Mathematics of the Rubik's cube handout
- Mathematics of the Rubik's cube notes
- Mathematics of the Rubik's cube slides
Abstract: Everyone has probably played with a Rubik's cube at some point. Some people might even have learned how to solve it. But wouldn't it be much more satisfying to figure out a solution on your own? Group theory gives you all the tools you need to do this!
Bring along a Rubik's cube and see how to create your own solution. We'll see how commutators and conjugation let you build your own algorithms to move specific pieces. And how invariants prove certain configurations are impossible. I'll also bring along plenty of other twisty puzzles to play with.
Twisty puzzles and group theory
Wednesday 15 December 2015, at 16:00
Gandalf seminar, Durham
- Mathematics of the Rubik's cube handout
- Mathematics of the Rubik's cube notes
- Mathematics of the Rubik's cube slides
Abstract: Everyone has probably played with a Rubik's cube at some point. Some people might have even learned how to solve it. But wouldn't it be much more satisfying if you could figure out your own solution? Using the ideas of commutators and conjugation from group theory I will explain how you can do this, not only for the Rubik's cube but for various other twisty puzzle you might encounter.
I will also bring along plenty of different puzzles for people to play with!
Primes of the form \( x^2 + ny^2 \)
Wednesday 21 October 2015, at 16:00
Gandalf seminar, Durham
- Primes of the form \( x^2 + ny^2 \) notes
- Primes of the form \( x^2 + ny^2 \) slides
- Resources from last time
Abstract: Fermat observed that (except for \( p = 2 \)) a prime \( p \) can be written as the sum of two squares if and only if \( p \equiv 1 \pmod{4} \). This result motivates our basic question: which primes does a given quadratic form represent?
To begin to answer this, we will relate the question of primes represented by a quadratic form to questions about ideal classes in quadratic number fields. And we will then be able to study these questions using the powerful tools of class field theory.
The main goal of this talk will to give a complete answer to this question for a specific class of quadratic forms, the so-called principal forms \( x^2 + ny^2 \). In this case the answer has the following form: there exists a polynomial \( f_n(t) \) such that \( p = x^2 + ny^2 \) if and only if \( f_n(t) \) has a root modulo \( p \). And for squarefree \( n \), this polynomial \( f_n(t) \) has an explicit interpretation as the polynomial describing the `Hilbert class field' of \( \mathbb{Q}(\sqrt{n}) \).
The coproduct on multiple zeta values, and `almost' identities
Tuesday 20 October 2015, at 14:00
Arithmetic Study Group, Durham
Abstract: Multiple zeta values are a mysterious and intriguing set of real numbers, about which many results are conjectured, but relatively little is proven. One is typically interested in finding all relations between MZVs, and completely understanding them, but transcendentality problems make this difficult to approach directly. One way to make progress with these questions is by lifting MZVs to purely algebraic objects which have additional, more rigid, structure.
I will start by giving an introduction to MZVs and some of the various standard results about them. From here we will lift to Brown's motivic MZVs, and look at the coproduct structure they acquire. Then using this coproduct, I will show how one can sometimes get easy combinatorial proofs of `almost' identities (identities up to a non-explicit rational), even in cases where the explicit identity remains conjectural.
The coproduct on multiple zeta values, and `almost' identities
Friday 7 November 2014, at 11:00
Algebra + Combinatorics Seminar, ICMat, Madrid
Abstract: Multiple zeta values are a mysterious and intriguing set of real numbers, about which many results are conjectured, but relatively little is known. One is typically interested in finding all relations between MZVs, and completely understanding them. One way to make progress with these questions is by lifting the MZVs to purely algebraic objects which have additional, and more rigid, structure.
In this talk I'll discuss MZVs, and the coproduct structure they acquire when lifted to motivic MZVs. Using this, I'll show how one can sometimes get easy combinatorial proofs of `almost' identities, identities up to a non-explicit factor, even in cases where the explicit identity remains conjectural.
Surreal numbers
Thursday 16 October 2014, at 15:00
Gandalf seminar, Durham
Abstract: Surreal numbers were invented by Conway, and used in his study of game theory. While the definition of a surreal number is surprisingly simple, it rapidly leads to a rich and deep structure encompassing not only the usual real numbers, but infinities, infinitesimals and more. In this talk I'll give an introduction to how surreal numbers work and an overview of the some of the weirdness that ensues.
Polylogarithms and double scissors congruence groups
Tuesday 18 February 2014, at 14:00
Arithmetic Study Group, Durham
Abstract: Polylogarithms are a class of special functions which have applications throughout the mathematics and physics worlds. I will begin by introducing the basic properties of polylogarithms and some reasons for interest in them, such as their functional equations and the role they play in Zagier's polylogarithm conjecture. From here I will turn to a Aomoto polylogarithms, a more general class of functions and explain how they motivate a geometric view of polylogarithms as configurations of hyperplanes in \(\mathbb{P}^n\). This approach has been used by Goncharov to establish Zagier's conjecture for \(n = 3\).
Polylogarithms and double scissors congruence groups (practice run)
Thursday 13 February 2014, at 15:00
Gandalf seminar, Durham
Multiple zeta values
Friday 1 November 2013, at 14:00
St Mary's Postgraduate Talks Day, St Mary's College, Durham
Abstract: An introduction to Multiple Zeta Values, and some discussion about my research. The talk is part of a day of talks aimed at final year undergraduate students who might be considering a PhD.
Multiple zeta values (practice run)
Thursday 24 October 2013, at 16:30
Gandalf seminar, Durham
Local \( \zeta \)-functions - Tate's Thesis section 2.4
20 July 2013, at 14:00
Student seminar on Tate's Thesis, Durham
Primes of the form \( x^2 + ny^2 \)
Wednesday 28 November 2012, at 16:00
Gandalf seminar, Durham
- Primes of the form \( x^2 + ny^2 \) slides
- Primes of the form \( x^2 + ny^2 \) notes
- Maple worksheets implementing some of the criteria discussed for: \( x^2 + y^2 \), \( x^2 + 2y^2 \), \( x^2 + 3y^2 \), \( x^2 + 5y^2 \), \( x^2 + 17y^2 \), \( x^2 - 2y^2 \), \( x^2 - 142y^2 \)
- Maple worksheet implementing the criteria for the cubic form \( a^3 + 11b^3 + 121c^3 - 33abc \)
- Maple worksheet to count the number of solutions to \( N = x^2 + 5y^2 \) and \( N = 2x^2 + 2xy + 3y^2 \)
Abstract: Fermat's observation about which primes can be written as the sum of two squares motivates the question: which primes does a given quadratic form represent? After relating quadratic forms with ideals in quadratic fields, we show how Class Field Theory can be applied to construct general criteria describing these primes. (This talk is a very much expanded version of my fourth year project presentation.)