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.

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

Deciphering a mysterious cache of coded letters from Mary, Queen of Scots.

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…

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

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...

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.

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…

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

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…

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

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.

University workers will vote on national strike action this month over a 1.4% pay offer made in the summer.

UCU Scotland members back taking action over job cuts plans, while staff at Nottingham reject deal that would have postponed any compulsory redundancies

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 ...

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 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.

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

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}.

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...

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.

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...

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.

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...