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...

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…

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

Malik Amir (SolutionAI and Montréal)25 August 2023"Data-Driven Insights into the Rank of Elliptic Curves of Prime Conductors"In this presentation, we explore...

We study the distribution of the length of longest monotone subsequences in random (fixed-point free) involutions of n integers as n grows large, establishing asymptotic expansions in powers of n^{−1/6} in the general case and in powers of n^{−1/3} in the fixed-point free cases. Whilst the limit laws were shown by Baik and Rains to be one of the Tracy–Widom distributions F_β for β = 1 or β = 4, we find explicit analytic expressions of the first few expansions terms as linear combinations of higher order derivatives of F_β with rational polynomial coefficients. Our derivation is based on a concept of generalized analytic de-Poissonization and is subject to the validity of certain hypotheses for which we provide compelling (computational) evidence. In a preparatory step expansions of the hard-to-soft edge transition laws of LβE are studied, which are lifted into expansions of the generalized Poissonized length distributions for large intensities.

The combinatorial mutation of polygons, which transforms a given lattice polygon into another one, is an important operation to understand mirror partners for two-dimensional Fano manifolds, and the mutation-equivalent polygons give ℚ-Gorenstein deformation-equivalent toric varieties. On the other hand, for a dimer model, which is a bipartite graph described on the real two-torus, one can assign a lattice polygon called the perfect matching polygon. It is known that for each lattice polygon P there exists a dimer model having P as the perfect matching polygon and satisfying certain consistency conditions. Moreover, a dimer model has rich information regarding toric geometry associated with the perfect matching polygon. In this paper, we introduce a set of operations which we call deformations of consistent dimer models, and show that the deformations of consistent dimer models realize the combinatorial mutations of the associated perfect matching polygons.

Yusuke Nakajima (Kyoto)24 June 2021"Combinatorial mutations and deformations of dimer models"The combinatorial mutation of a polytope was introduced in the c...

We describe recent progress in a program to understand the classification of three-dimensional Fano varieties with Q-factorial terminal singularities using mirror symmetry. As part of this we give an improved and more conceptual understanding of Laurent inversion, a technique that sometimes allows one to construct a Fano variety X directly from a Laurent polynomial f that corresponds to it under mirror symmetry.

Conjectures have historically played an important role in the development of pure mathematics. We propose a systematic approach to finding abstract patterns in mathematical data, in order to generate…

Challenger Mishra (Cambridge)25 August 2023"Mathematical Conjecture Generation and Machine Intelligence"Conjectures hold a special status in mathematics. Goo...

Study of the level curve Re(\eta(s))=0 for \eta(s)=\pi^{s/2}\Gamma(s/2)\zeta'(s) gives a new classification of the zeros of \zeta(s) and of \zeta'(s). We conjecture that for type 2 zeros, liminf(\beta'-1/2)log \gamma'=0 if and only if liminf(\gamma^+ - \gamma^-)log \gamma'=0, and reduce the conjecture to a lower bound on the curvature of the level curve. We compute and classify 10^6 zeros of \zeta'(s) near T=10^10. The Riemann Hypothesis is assumed throughout. An appendix develops the analogous classification for characteristic polynomials of unitary matrices.

Tropical geometry and the theory of Newton-Okounkov bodies are two methods which produce toric degenerations of an irreducible complex projective variety. Kaveh-Manon showed that the two are related.…

Laura Escobar (Washington)17 June 2021"Wall-crossing phenomenon for Newton-Okounkov bodies"A Newton-Okounkov body is a convex set associated to a projective ...