Alexander Kasprzyk

Rahul Sarkar (Stanford)27 September 2023"A framework for generating inequality conjectures"In this talk, I'll present some recent and ongoing work, where we ... Rahul Sarkar (Stanford)

Rahul Sarkar (Stanford) speaking on "A framework for generating inequality conjectures" at our #MachineLearning #Math seminar back in September 2023. #MathSky

Growth of Mahler Measure and Algebraic Entropy of Dynamics with the Laurent Property We consider the growth rate of the Mahler measure in discrete dynamical systems with the Laurent property, and in cluster algebras, and compare this with other measures of growth. In particular, we formulate the conjecture that the growth rate of the logarithmic Mahler measure coincides with the algebraic entropy, which is defined in terms of degree growth. Evidence for this conjecture is provided by exact and numerical calculations of the Mahler measure for a family of Laurent polynomials generated by rank 2 cluster algebras, for a recurrence of third order related to the Markoff numbers, and for the Somos-4 recurrence. Also, for the sequence of Laurent polynomials associated with the Kronecker quiver (the cluster algebra of affine type \tilde{A}_1 we prove a precise formula for the leading order asymptotics of the logarithmic Mahler measure, which grows linearly.

"Growth of Mahler Measure and Algebraic Entropy of Dynamics with the Laurent Property" by Andrew N. W. Hone. #ExperimentalMath #ClusterAlgebra #MathSky

Combinatorial mutations and block diagonal polytopes Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton polytopes, and more recently have been shown to give rise to toric degenerations of various families of varieties. Whenever a matching field gives rise to a toric degeneration, the associated polytope of the toric variety coincides with the matching field polytope. We study combinatorial mutations, which are analogues of cluster mutations for polytopes, of matching field polytopes and show that the property of giving rise to a toric degeneration of the Grassmannians, is preserved by mutation. Moreover, the polytopes arising through mutations are Newton-Okounkov bodies for the Grassmannians with respect to certain full-rank valuations. We produce a large family of such polytopes, extending the family of so-called block diagonal matching fields.

The work described by Ollie Clarke - joint with Akihiro Higashitani and Fatemeh Mohammadi - was published in Collectanea Mathematica.
link.springer.com/article/10.1...

Ollie Clarke (Bristol and Ghent)12 August 2021"Combinatorial mutations and block diagonal polytopes"Matching fields were introduced by Sturmfels and Zelevins... Ollie Clarke (Bristol and Ghent)

Ollie Clarke (Bristol and Ghent) speaking on "Combinatorial mutations and block diagonal polytopes" at our #AlgebraicGeometry #Math seminar back in August 2021. #MathSky

BBC Scotland - The Secret Letters of Mary Queen of Scots Deciphering a mysterious cache of coded letters from Mary, Queen of Scots.

A 1-hour documentary on Mary Stuart's decoded letters - first revealed in a 2023 paper published in Cryptologia - is being aired on BBC Scotland this Sat, 8pm!📺

Follow the experts' journey to the 'most important discovery on the Queen for 100 years'👑

www.bbc.co.uk/programmes/m...

Fundamental Components of Deep Learning: A category-theoretic approach Deep learning, despite its remarkable achievements, is still a young field. Like the early stages of many scientific disciplines, it is marked by the discovery of new phenomena, ad-hoc design…

See Bruno Gavranović's arXiv preprint from last year.
arxiv.org/abs/2403.13001

Bruno Gavranović (Strathclyde)20 September 2023"Fundamental Components of Deep Learning: A category-theoretic approach"Deep learning, despite its remarkable ... Bruno Gavranović (Strathclyde)

Bruno Gavranović (Strathclyde) speaking on "Fundamental Components of Deep Learning: A category-theoretic approach" at our #MachineLearning #Math seminar back in September 2023. #MathSky

📢 Calling all Early Career Researchers!

Join our expert panel for a free online workshop series that will cover the basics of applying for academic jobs.

Session 1 starts Thursday 16 October at midday (GMT).

For further details and to register ➡️ lms.ac.uk/events/CDP-1

Dhruv Ranganathan (Cambridge)5 August 2021"Toric contact cycles in the moduli space of curves"The toric contact cycles are loci in the moduli space of curves... Dhruv Ranganathan (Cambridge)

Dhruv Ranganathan (Cambridge) speaking on "Toric contact cycles in the moduli space of curves" at our #AlgebraicGeometry #Math seminar back in August 2021. #MathSky

Clasper Presentations of Habegger-Lin’s Action on String Links Habegger and Lin gave a classification of the link-homotopy classes of links as the link-homotopy classes of string links modulo the actions of conjugations and partial conjugations for string links. In this paper, we calculate the actions of the partial conjugations and the conjugations explicitly for 4- and 5-component string links which gave classifications (presentations) of the link-homotopy classes of 4- and 5-component links. As an application, we can run Habegger and Lin’s algorithm which determines whether given two links are link-homotopic or not for 4- and 5-component links.

"Clasper Presentations of Habegger-Lin’s Action on String Links" by Yuka Kotorii and Atsuhiko Mizusawa. #ExperimentalMath #LinkHomotopy #MathSky

Discrete neural nets and polymorphic learning Theorems from universal algebra such as that of Murskiĭ from the 1970s have a striking similarity to universal approximation results for neural nets along the lines of Cybenko's from the 1980s. We…

For further details see the arXiv preprint by Charlotte Aten.
arxiv.org/abs/2308.00677

Charlotte Aten (Denver)6 September 2023"Discrete neural nets and polymorphic learning"Classical neural network learning techniques have primarily been focuse... Charlotte Aten (Denver)

Charlotte Aten (Denver) speaking on "Discrete neural nets and polymorphic learning" at our #MachineLearning #Math seminar back in September 2023. #MathSky

Local positivity and effective Diophantine approximation In this paper we present a new approach to prove effective results in Diophantine approximation. This approach involves measures of local positivity of divisors combined with Faltings’s version of…

For further details see Matthias Nickel's paper in Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg.
link.springer.com/article/10.1...

Matthias Nickel (Frankfurt)29 July 2021"Local positivity and effective Diophantine approximation"In this talk I will discuss a new approach to prove effectiv... Matthias Nickel (Frankfurt)

Matthias Nickel (Frankfurt) speaking on "Local positivity and effective Diophantine approximation" at our #AlgebraicGeometry #Math seminar back in July 2021. #MathSky

Some Singular Curves in Mukai’s Model of \bar{M}_7 Mukai showed that the GIT quotient Gr(7,16)//Spin(10) is a birational model of the moduli space of Deligne-Mumford stable genus 7 curves \bar{M}_7. The key observation is that a general smooth genus 7 curve can be realized as the intersection of the orthogonal Grassmannian OG(5,10) in P^15 with a six-dimensional projective linear subspace. What objects appear on the boundary of Mukai’s model? As a first step in this study, computer calculations in Macaulay2, Magma, and Sage are used to find and analyze linear spaces yielding three examples of singular curves: a 7-cuspidal curve, the balanced ribbon of genus 7, and a family of genus 7 reducible nodal curves. Spin(10)-semistability is established by constructing and evaluating an invariant polynomial.

"Some Singular Curves in Mukai’s Model of \bar{M}_7" by David Swinarski. #ExperimentalMath #AlgebraicGeometry #MathSky

Thousands more university jobs cut as financial crisis deepens - BBC News University workers will vote on national strike action this month over a 1.4% pay offer made in the summer.

We pick up strike action again next week and this recent article from the BBC shows just how much we are fighting for.

First step: take compulsory redundancies off the table.

#StopTheCuts #SaveHE

www.bbc.co.uk/news/article...

Dundee, UWS and Nottingham latest to face further strikes UCU Scotland members back taking action over job cuts plans, while staff at Nottingham reject deal that would have postponed any compulsory redundancies

If the university is actually "committed to minimising any disruption to our students and our research," they should show it.

No compulsory redundancies or no labour from the people that make this university what it is. Your choice

#StoptheCuts #SaveHE

www.timeshighereducation.com/news/dundee-...

Kristin DeVleming (UCSD)22 July 2021"K moduli of quartic K3 surfaces"We will discuss a family of compactifications of moduli spaces of log Fano pairs coming ... Kristin DeVleming (UCSD)

Kristin DeVleming (UCSD) speaking on "K moduli of quartic K3 surfaces" at our #AlgebraicGeometry #Math seminar back in July 2021. #MathSky

Teachouts Striking lecturers do care about your learning experience and, in fact enjoy teaching. Teachouts are an opportunity for students and staff to engage with each other, but no material relevant to exams…

We're gearing up for another two weeks of strike action starting on Monday and part of that involves organising teach outs!

More details will be announced as they come in, but you can find the growing list of learning opportunities here: uonucu.org/teachouts/

Deep learning symmetries and their Lie groups, algebras, and subalgebras from first principles We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset. We use fully connected neural networks to model the symmetry transformations and the corresponding generators. The constructed loss functions ensure that the applied transformations are symmetries and the corresponding set of generators forms a closed (sub)algebra. Our procedure is validated with several examples illustrating different types of conserved quantities preserved by symmetry. In the process of deriving the full set of symmetries, we analyze the complete subgroup structure of the rotation groups SO(2), SO(3), and SO(4), and of the Lorentz group SO(1,3). Other examples include squeeze mapping, piecewise discontinuous labels, and SO(10), demonstrating that our method is completely general, with many possible applications in physics and data science. Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.

See the article "Deep learning symmetries and their Lie groups, algebras, and subalgebras from first principles" by Roy Forestano, Konstantin Matchev, Katia Matcheva, Alexander Roman, Eyup Unlu, and Sarunas Verner, in Machine Learning: Science and Technology.
iopscience.iop.org/article/10.1...

Katia Matcheva (Florida)25 August 2023"Deep Learning Symmetries in Physics from First Principles"Symmetries are the cornerstones of modern theoretical physic... Katia Matcheva (Florida)

Katia Matcheva (Florida) speaking on "Deep Learning Symmetries in Physics from First Principles" at our #MachineLearning #Math #DANGER workshop back in August 2023. #MathSky

Conjectural Criteria for the Most Singular Points of the Hilbert Schemes of Points We provide conjectural necessary and (separately) sufficient conditions for the Hilbert scheme of points of a given length to have the maximum dimension tangent space at a point. The sufficient condition is claimed for 3D and reduces the original problem to a problem in convex geometry. Proving either of the two conjectural statements will in particular resolve a long-standing conjecture by Briançon and Iarrobino back in the ’70s for the case of the powers of the maximal ideal. Furthermore, for specific classes of lengths, we conjecturally classify points satisfying the conjectural sufficient conditions. This in particular (conjecturally) provides many new explicit families of examples of maximum dimension tangent space at a point of the Hilbert schemes of points of lengths strictly between two consecutive tetrahedral numbers {3+k \choose 3}.

"Conjectural Criteria for the Most Singular Points of the Hilbert Schemes of Points" by Fatemeh Rezaee. #ExperimentalMath #AlgebraicGeometry #MathSky

Sergey Galkin (PUC-Rio and HSE)15 July 2021"Graph potentials and combinatorial non-abelian Torelli"I will introduce graph potentials and discuss some of thei... Sergey Galkin (PUC-Rio and HSE)

Sergey Galkin (PUC-Rio and HSE) speaking on "Graph potentials and combinatorial non-abelian Torelli" at our #AlgebraicGeometry #Math seminar back in July 2021. #MathSky

New Calabi–Yau manifolds from genetic algorithms Calabi–Yau manifolds can be obtained as hypersurfaces in toric varieties built from reflexive polytopes. We generate reflexive polytopes in various dimensions using a genetic algorithm. As a proof of principle, we demonstrate that our algorithm reproduces the full set of reflexive polytopes in two and three dimensions, and in four dimensions with a small number of vertices and points. Motivated by this result, we construct five-dimensional reflexive polytopes with the lowest number of vertices and points. By calculating the normal form of the polytopes, we establish that many of these are not in existing datasets and therefore give rise to new Calabi–Yau four-folds. In some instances, the Hodge numbers we compute are new as well.

The research described by Elli Heyes - joint with Per Berglund, Yang-Hui He, Edward Hirst, Vishnu Jejjala, and Andre Lukas - was published in Physics Letters B last year.
www.sciencedirect.com/science/arti...

Elli Heyes (LIMS)25 August 2023"New Calabi-Yau Manifolds from Genetic Algorithms"Calabi-Yau manifolds can be obtained as hypersurfaces in toric varieties bui... Elli Heyes (LIMS)

Elli Heyes (LIMS) speaking on "New Calabi-Yau Manifolds from Genetic Algorithms" at our #MachineLearning #Math #DANGER workshop back in August 2023. #MathSky

New Bounds and Progress Towards a Conjecture on the Summatory Function (-2)^{\Omega(n)} In this article, we study the summatory function W(x) = \sum_{n <= x}(-2)^{\Omega(n)} where \Omega(n) counts the number of prime factors of n, with multiplicity. We prove W(x) = O(x), and in particular, that |W(x)| < 2260x for all x >= 1. This provides new progress toward a conjecture of Sun, which asks if |W(x)| < x for all x >= 3078. To obtain our results, we also compute new explicit bounds on the Mertens function M(x). These bounds may be of independent interest. Moreover, we obtain similar results and make further conjectures that pertain to the more general function W_a(x) = \sum_{n <= x}(-a)^{\Omega(n)} for any real a > 0.

"New Bounds and Progress Towards a Conjecture on the Summatory Function (-2)^{\Omega(n)}" by Daniel R. Johnston, Nicol Leong, and Sebastian Tudzi. #ExperimentalMath #NumberTheory #MathSky

Chengxi Wang (UCLA)8 July 2021"Varieties of general type with small volume"By Hacon-McKernan, Takayama, and Tsuji, there is a constant r_n such that for ever... Chengxi Wang (UCLA)

Chengxi Wang (UCLA) speaking on "Varieties of general type with small volume" at our #AlgebraicGeometry #Math seminar back in July 2021. #MathSky

On the liftability of the automorphism group of smooth hypersurfaces of the projective space - Israel Journal of Mathematics Let X be a smooth hypersurface of dimension n ≥ 1 and degree d ≥ 3 in the projective space given as the zero set of a homogeneous form F. If (n, d) ≠ (1, 3), (2, 4) it is well known that every…

See the article "On the liftability of the automorphism group of smooth hypersurfaces of the projective space" by Víctor González-Aguilera, Alvaro Liendo, and Pedro Montero, published in the Israel Journal of Mathematics.
link.springer.com/article/10.1...

Pedro Montero (Valparaíso)1 July 2021"On the liftability of the automorphism group of smooth hypersurfaces of the projective space"Smooth hypersurfaces are c... Pedro Montero (Valparaíso)

Pedro Montero (Valparaíso) speaking on "On the liftability of the automorphism group of smooth hypersurfaces of the projective space" at our #AlgebraicGeometry #Math seminar back in July 2021. #MathSky