Multiple \(t\) values are an analogue of multiple zeta values formed by summing over odd denominators. They can directly be expressed via alternating multiple zeta values, but using motivic methods Murakami proved the surprising fact that any multiple \( t \) value of maximal height (i.e. all entries \(\geq 2\)) is a rational-linear combination of multiple zeta values, no alternation required. In this talk we give an explicit formula (in the manner of Zhao's generalised 2-1 formula), which directly expresses such a multiple \( t \) value of maximal height via multiple zeta values. We will also mention some applications to evaluations of multiple zeta values. This is joint work with Michael Hoffman and Nobuo Sato.
A brief introduction to multiple zeta values, the connections to modular forms, and \(\zeta(2,\ldots,2,4,2,\ldots,2)\).
Multiple zeta (star) values are a multivariable generalisation of Riemann zeta values like \(\zeta(2)=\pi^2/6\), \(\zeta(3)\); their arithmetic nature (irrational or transcendental) is still largely conjectural. They play a rather important role in high-energy physics calculations, acting as a bridge between number theory and particle physics. One of the main goals is to understand their algebraic structure, and all of the relations and identities they satisfy.
I'll discuss some on-going work with Danylo Radchenko, on the limit behaviour of multiple zeta star values, and the applications to understanding some special evaluations, new and old. The motivation comes from a strange conjectural identity for a weight \(8n + 4\) multiple zeta star value as a multiple of \( \pi^{8n+4} \), and its possible generalisations.
Dirichlet's famous Class Number Formula expresses the residue of the Dedekind zeta function \(\zeta_F(s)\) of a number field \(F\) in terms of important arithmetic information about the number field, including, in particular, the regulator which is a determinant involving logarithms of units of \(\mathcal{O}_F\). Zagier's Polylogarithm Conjecture extends this to express values of \(\zeta_F(n)\), \(n = 2, 3, 4, \ldots \), in terms of suitable combinations of \(n\)-logarithms, giving `higher' units.
Goncharov outlined a strategy to tackle Zagier's Polylogarithm Conjecture, by understanding first the maximal depth multiple polylogarithms and the geometrically defined Grassmannian polylogarithm, before attempting to successively reduce the depth, to reach the classical $n$-logarithm. I will outline some of the history of Zagier's Polylogarithm Conjecture, and of Goncharov's Programme. I will then briefly discuss some recent developments, including work by Goncharov-Rudenko proving \(\zeta_F(4)\); by C-Gangl-Radchenko on formulas for Grassmannian polylogarithms making this explicit; and some separate work by Matveiakin-Rudenko and by myself which establishes new reductions in weight 6.
Multiple polylogarithms \( \operatorname{Li}_{k_1,\ldots,k_d}(x_1,\ldots,x_d) \) are a class of multi-variable special functions generalising the natural logarithm \( \Li_1(x) = -\log(1-x) \). These functions appear in connection with \( K \)-theory, hyperbolic geometry, values of $L$-functions, mixed Tate motives, high-energy physic, and many other areas.
One of the main challenges in the study of MPL's revolves around understanding on how many variables a MPL (or `interesting' combinations thereof) actually depend (``the depth''). It is well known, for example, that \( \operatorname{Li}_{1,1} \) can already be expressed via \( \Li_2 \), likewise \( \Li_{1,1,1} \) can be expressed via \( \Li_3 \). Goncharov gave a conjectural criterion (``the Depth Conjecture'') to determine this, using the motivic coproduct, as part of his programme to investigate Zagier's Polylogarithm Conjecture on the special values of the Dedekind zeta function \( \zeta_F(m) \).
I will give an overview of Goncharov's Depth Conjecture, and its implications. I will discuss what is currently known, including recent progress in weight 6. In particular, the conjecture predicts that a certain weight 6 function (essentially a small modification of \( \Li_{4,1,1}(x,y,z) \)) should satisfy the 6-fold dilogarithm symmetries \( \lambda \mapsto \lambda^{-1}, 1-\lambda \) in each variable independently, modulo depth \( \leq 2 \) terms.
I will then describe the computational background and tools involved in investigating and proving these symmetries. In particular, one has to consider many possible degenerations (to boundary components of \( {\frak M}_{0,n} \)) of the Matveiakin-Rudenko quadrangular polylogarithm functional equations, to iteratively find weaker symmetries of \( \Li_{4,1,1} \) and useful short identities. To investigate higher weight analogues will require a more structure approach and understanding of this degeneration process.
%In his programme to investigate Zagier's Polylogarithm Conjecture on the values of \( \zeta_F(m) \), Goncharov gave a conjectural criterion -- the Depth Conjecture -- to determine the depth (number of variables) of a linear combination of multiple polylogarithms using the motivic coproduct. I will give an overview of this conjecture and its implications; in particular this Conjecture explains why all multiple polylogarithms of weight 2, and weight 3 can be expressed via \( \Li_2 \) and \( \Li_3 \) respectively.
In weight 4, this Conjecture predicts that \( \Li_{3,1}(x,y) \) (rather, some small modification thereof) should satisfy the dilogarithm 5-term relation independently in each variable, modulo depth 1 terms \( \Li_4 \). This was established by Gangl, via an explicit 122-term reduction to \( \Li_4 \)'s, and conceptually understood by Goncharov and Rudenko in their proof of Zagier's Conjecture for \( \zeta_F(4) \).
In weight 6, Matveiakin and Rudenko were likewise able to show that \( \Li_{4,1,1}(x,y,z) \) (or small modification thereof) satisfies the dilogarithm 5-term in each variable, but only modulo depth 2 and modulo the 6-fold symmetries \( \lambda \mapsto \lambda^{-1}, 1-\lambda \). Goncharov's Depth Conjecture predicts that \( \Li_{4,1,1} \) should also satisfy these symmetries modulo depth 2, but unlike in weight 4, these symmetries proved much harder to establish. I will explain how to explicitly derive these symmetries from the degenerations of the Matveiakin-Rudenko quadrangular polylogarithms to boundary components of \( \overline{\mathfrak{M}}_{0,9}\). This establishes Matveiakin-Rudenko's 5-term reduction unconditionally, and hence Goncharov's Depth Conjecture in weight 6, depth 3.
Multiple zeta values, connections to modular forms, and \( \zeta(2, \ldots, 2, 4, 2, \ldots, 2) \) in 3 minutes.
Multiple zeta values, connections to modular forms, and \( \zeta(2, \ldots, 2, 4, 2, \ldots, 2) \) in 3 minutes.
An introductiont to my research in polylogarithms, and its connections elsewhere.
An overview of MZV's, periods, and strange evaluations. How to prove some new ones conceptually, without WZ methods?
Multiple polylogarithms \( \Li_{k_1,\ldots,k_d}(x_1,\ldots,x_d) \) are a class of multi-variable special functions appearing in connection with K-theory, hyperbolic geometry, values of zeta functions/L-functions/Mahler measures, mixed Tate motives, and in high-energy physics.
One of the main challenges in the study of multiple polylogarithms revolves around understanding how on many variables a multiple polylogarithm function (or 'interesting' combinations thereof) actually depend (''the depth''), as for example \( \Li_{1,1} \) can already be expressed via \( \Li_2 \). Goncharov gave a conjectural criterion (''the Depth Conjecture'') for determining this, using the motivic coproduct, as part of his programme to investigate Zagier's Polylogarithm Conjecture on values of the Dedekind zeta function \( \zeta_F(m) \).
I will give an overview of multiple polylogarithms, Goncharov's Depth Conjecture, and its implications. I will try to discuss what is currently known, including recent results in weight 6, and what we are still trying to investigate.
I will discuss Goncharov's approach to MPL parity and how it can be extended to a symmetry theorem of MtV's (conjectured by Hoffman) and to MMV's in the setting of Xu-Yan-Zhao.
One of the main challenges in the study of multiple polylogarithms revolves around understanding how on many variables a multiple polylogarithm function (or `interesting' combinations thereof) actually depend (``the depth''), as for example Li_{1,1} can already be expressed via Li_2. Goncharov gave a conjectural criterion (``the Depth Conjecture'') for determining this, using the motivic coproduct, as part of his programme to investigate Zagier's Polylogarithm Conjecture on values of the Dedekind zeta function zeta_F(m).
I will give an overview of Goncharov's Depth Conjecture, and its implications. I will discuss what is currently known, including recent results in weight 6, in particular: my proof of the depth reduction of a weight 6 depth 3 function under the dilogarithm symmetries x \mapsto 1-x, 1/x, and Matveiakin-Rudenko's proof of depth reduction of this function under the 5-term relation (modulo the symmetries).
With Gangl, Radchenko, and Rudenko we showed that every depth 2 MPL can be expressed via Li_{a+b-1,1}, with sufficiently high degree roots of unity. What does this say about the structure of high level coloured MZV's?
Multiple polylogarithms \( \Li_{k_1,\ldots,k_d}(x_1,\ldots,x_d) \) are a class of multi-variable special functions appearing in connection with K-theory, hyperbolic geometry, values of zeta functions/L-functions/Mahler measures, mixed Tate motives, and in high-energy physics.
One of the main challenges in the study of multiple polylogarithms revolves around understanding how on many variables a multiple polylogarithm function (or 'interesting' combinations thereof) actually depend (`the depth'), as for example \( \Li_{1,1} \) can already be expressed via \( \Li_2 \). Goncharov gave a conjectural criterion (`the Depth Conjecture') for determining this, using the motivic coproduct, as part of his programme to investigate Zagier's Polylogarithm Conjecture on values of the Dedekind zeta function \( \zeta_F(m) \).
I will give an overview of multiple polylogarithms, Goncharov's Depth Conjecture, and its implications. I will try to discuss what is currently known, including recent results in weight 6, and what we are still trying to investigate.
Multiple polylogarithms \( \Li_{k_1,\ldots,k_d}(x_1,\ldots,x_d) \) are a class of multi-variable special functions appearing in connection with K-theory, hyperbolic geometry, values of zeta functions/L-functions/Mahler measures, mixed Tate motives, and in high-energy physics.
One of the main challenges in the study of multiple polylogarithms revolves around understanding how on many variables a multiple polylogarithm function (or 'interesting' combinations thereof) actually depend (`the depth'), as for example \( \Li_{1,1} \) can already be expressed via \( \Li_2 \). Goncharov gave a conjectural criterion (`the Depth Conjecture') for determining this, using the motivic coproduct, as part of his programme to investigate Zagier's Polylogarithm Conjecture on values of the Dedekind zeta function \( \zeta_F(m) \).
I will give an overview of multiple polylogarithms, Goncharov's Depth Conjecture, and its implications. I will try to discuss what is currently known, including recent results in weight 6, and what we are still trying to investigate.
Multiple polylogarithms \( \Li_{k_1,\ldots,k_d}(x_1,\ldots,x_d) \) are a class of multi-variable special functions appearing in connection with K-theory, hyperbolic geometry, values of zeta functions/L-functions/Mahler measures, mixed Tate motives, and in high-energy physics.
One of the main challenges in the study of multiple polylogarithms revolves around understanding how on many variables a multiple polylogarithm function (or 'interesting' combinations thereof) actually depend (`the depth'), as for example \( \Li_{1,1} \) can already be expressed via \( \Li_2 \). Goncharov gave a conjectural criterion (`the Depth Conjecture') for determining this, using the motivic coproduct, as part of his programme to investigate Zagier's Polylogarithm Conjecture on values of the Dedekind zeta function \( \zeta_F(m) \).
I will give an overview of multiple polylogarithms, Goncharov's Depth Conjecture, and its implications. I will try to discuss what is currently known, including recent results in weight 6, and what we are still trying to investigate.
Multiple polylogarithms \( \Li_{k_1,\ldots,k_d}(x_1,\ldots,x_d) \) are a class of multi-variable special functions appearing in connection with K-theory, hyperbolic geometry, values of zeta functions/L-functions/Mahler measures, mixed Tate motives, and in high-energy physics.
One of the main challenges in the study of multiple polylogarithms revolves around understanding how on many variables a multiple polylogarithm function (or 'interesting' combinations thereof) actually depend (`the depth'), as for example \( \Li_{1,1} \) can already be expressed via \( \Li_2 \). Goncharov gave a conjectural criterion (`the Depth Conjecture') for determining this, using the motivic coproduct, as part of his programme to investigate Zagier's Polylogarithm Conjecture on values of the Dedekind zeta function \( \zeta_F(m) \).
I will give an overview of multiple polylogarithms, Goncharov's Depth Conjecture, and its implications. I will try to discuss what is currently known, including recent results in weight 6, and what we are still trying to investigate.
As part of a programme to tackle Zagier's Polylogarithm Conjecture and understand the structure of multiple polylogarithms, Goncharov proposed an ambitious Depth Conjecture giving an exact criterion, in terms of the motivic cobracket, to determine when a linear combination of MPL's has a certain depth. In particular, this explains why all weight 2 and 3 multiple polylogarithms can be expressed via depth 1; it was also one of the main catalysts for simplifying the 2-loop 6-point remainder function \(R_6^{(2)}\), and expressing it purely via classical polylogarithms.
In weight 6 depth 3, Goncharov's Depth Conjecture predicts that \(\Li_{3 ; 1,1,1}(x,y,z)\) (closely related to \(\Li_{4,1,1}(x y z, 1/x, 1/y) \)) should satisfy dilogarithm functional equations in argument, modulo terms of depth 2. Using the quadrangular polylogarithm relation, Matveiakin and Rudenko showed the 5-term part of this holds, but only by working modulo the 6-fold dilogarithm symmetries \(\Li_{3 ;1,1,1}(x,y,z) + Li_{3 ; 1,1,1}(1-x,y,z)\), and \(Li_{3;1,1,1}(x,y,z) + \Li_{3;1,1,1}(1/x,y,z)\) which they assumed would reduce to depth 2.
I will explain how to show that \(\Li_{3 ; 1,1,1}(x,y,z)\) satisfies these 6-fold symmetries, by systematically understanding how the quadrangular polylogarithm relation degenerates to boundary components of (the compactification of) \( \mathfrak{M}_{0,9} \). Together with Matveiakin and Rudenko's proof of the 5-term part, this means Goncharov's Depth Conjecture holds in weight 6 depth 3. Finally, I can try to indicate some expectations and future directions for investigating the Depth Conjecture.
Informal talk about about Lemma of Lindström, Gressel and Viennot, how it is useful for determinant formulae for Schur MZV's, and whether of not it might give an approach to or generalisation of the Basso-Dixon integral formulae.
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 \(\zeta_F(6)\).
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 \(\zeta_F(6)\).
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 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).
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.
A short talk about the two-one formula, and its generalisations (also connected to Hirose-Sato's iterate beta integrals). These formulas trivialise many complicated evaluations, like Zagier's \(\zeta(2, \ldots, 2, 3, 2, \ldots, 2) \) evaluation. What else can they do?
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 (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.
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).
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 \( \zeta_F(5) \).
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.)
Joint work with Michael Hoffman. We establish a symmetry theorem for multiple \( t \) values, and give some applications. arXiv:2204:14183.
An introduction to the Goncharov coproduct on iterated integrals and motivic MZV's, with some examples of applications to transcendence questions.
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.
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.
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.
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).
Overview of some recent work with Gangl and Radchenko, where we found new identities in the spirit of the \( \mathbf{Q}_3 \) and \( \mathbf{Q}_4 \) identities used by Goncahrov and Rudenko for \( \zeta_F(4) \). We used these identities to establish new depth reductions in weight 5, 6 and 7.
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).
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.
A generalisation of the cyclic insertion conjecture on MZV's, and progress towards a proof using the motivic MZV framework.
An introduction/overview of my research in multiple polylogarithms
Motivated by the corresponding result for finite MZV's, I will discuss a Bowman-Bradley type identity for symmetrised MZV's.
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.
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 \( \geq 7 \).
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.
Section 5 of "The Local Langlands Conjecture of GL(2)", Bushnell, Henniart.
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.
Comology of arithmetic groups seminar , on the topic of computation of arithmetic cohomology, following [Gunnels] Gunnells, Paul: Lectures on computing cohomology of arithmetic groups. Computations with modular forms, 3–45, Contrib. Math. Comput. Sci., 6, Springer, Cham, 2014.
Introduction to motivic MZV's.
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!
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.)
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.
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 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 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 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 \(n = 3\).
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.
Section 2.4 of Tate's thesis.
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.)