All Outputs (11)

Polytopes and machine learning (2023)
Journal Article
Bao, J., He, Y., Hirst, E., Hofscheier, J., Kasprzyk, A., & Majumder, S. (2023). Polytopes and machine learning. International Journal of Data Science in the Mathematical Sciences, 1(2), 181-211. https://doi.org/10.1142/S281093922350003X

We introduce machine learning methodology to the study of lattice polytopes. With supervised learning techniques, we predict standard properties such as volume, dual volume, reflexivity, etc, with accuracies up to 100%. We focus on 2d polygons and 3d... Read More about Polytopes and machine learning.

Machine Learning: The Dimension of a Polytope (2023)
Book Chapter
Coates, T., Hofscheier, J., & Kasprzyk, A. M. (2023). Machine Learning: The Dimension of a Polytope. In Machine Learning in Pure Mathematics and Theoretical Physics (85-104). World Scientific. https://doi.org/10.1142/9781800613706_0003

We use machine learning to predict the dimension of a lattice polytope directly from its Ehrhart series. This is highly effective, achieving almost 100% accuracy. We also use machine learning to recover the volume of a lattice polytope from its Ehrha... Read More about Machine Learning: The Dimension of a Polytope.

Hilbert series, machine learning, and applications to physics (2022)
Journal Article
Bao, J., He, Y., Hirst, E., Hofscheier, J., Kasprzyk, A., & Majumder, S. (2022). Hilbert series, machine learning, and applications to physics. Physics Letters B, 827, Article 136966. https://doi.org/10.1016/j.physletb.2022.136966

We describe how simple machine learning methods successfully predict geometric properties from Hilbert series (HS). Regressors predict embedding weights in projective space to ∼1 mean absolute error, whilst classifiers predict dimension and Gorenstei... Read More about Hilbert series, machine learning, and applications to physics.

A generalization of a theorem of White (2021)
Journal Article
Batyrev, V., & Hofscheier, J. (2022). A generalization of a theorem of White. Moscow Journal of Combinatorics and Number Theory, 10(4), 281-296. https://doi.org/10.2140/moscow.2021.10.281

An m-dimensional simplex ∆ in Rm is called empty lattice simplex if ∆ ∩ Zm is exactly the set of vertices of ∆. A theorem of White states that if m = 3 then, up to an affine unimodular transformation of the lattice Zm, any empty lattice simplex ∆ ⊂ R... Read More about A generalization of a theorem of White.

Splittings of Toric Ideals (2021)
Journal Article
Favacchio, G., Hofscheier, J., Keiper, G., & Van Tuyl, A. (2021). Splittings of Toric Ideals. Journal of Algebra, 574, 409-433. https://doi.org/10.1016/j.jalgebra.2021.01.012

Let I ⊆ R = K[x1, . . . , xn] be a toric ideal, i.e., a binomial prime ideal. We investigate when the ideal I can be “split” into the sum of two smaller toric ideals. For a general toric ideal I, we give a sufficient condition for this splitting in t... Read More about Splittings of Toric Ideals.

Smooth Centrally Symmetric Polytopes in Dimension 3 are IDP (2019)
Journal Article
Beck, M., Haase, C., Higashitani, A., Hofscheier, J., Jochemko, K., Katthän, L., & Michałek, M. (2019). Smooth Centrally Symmetric Polytopes in Dimension 3 are IDP. Annals of Combinatorics, 23(2), 255-262. https://doi.org/10.1007/s00026-019-00418-x

In 1997 Oda conjectured that every smooth lattice polytope has the integer decomposition property. We prove Oda’s conjecture for centrally symmetric 3-dimensional polytopes, by showing they are covered by lattice parallelepipeds and unimodular simpli... Read More about Smooth Centrally Symmetric Polytopes in Dimension 3 are IDP.

On Ehrhart Polynomials of Lattice Triangles (2018)
Journal Article
Hofscheier, J., Nill, B., & Öberg, D. (2018). On Ehrhart Polynomials of Lattice Triangles. Electronic Journal of Combinatorics, 25(1), https://doi.org/10.37236/6624

The Ehrhart polynomial of a lattice polygon PP is completely determined by the pair (b(P),i(P))(b(P),i(P)) where b(P)b(P) equals the number of lattice points on the boundary and i(P)i(P) equals the number of interior lattice points. All possible pair... Read More about On Ehrhart Polynomials of Lattice Triangles.

Ehrhart Theory of Spanning Lattice Polytopes (2017)
Journal Article
Hofscheier, J., Katthän, L., & Nill, B. (2018). Ehrhart Theory of Spanning Lattice Polytopes. International Mathematics Research Notices, 2018(19), 5947-5973. https://doi.org/10.1093/imrn/rnx065

© 2018 Oxford University Press. All rights reserved. The key object in the Ehrhart theory of lattice polytopes is the numerator polynomial of the rational generating series of the Ehrhart polynomial, called h-polynomial. In this article, we prove a n... Read More about Ehrhart Theory of Spanning Lattice Polytopes.

The generalized Mukai conjecture for symmetric varieties (2016)
Journal Article
Gagliardi, G., & Hofscheier, J. (2017). The generalized Mukai conjecture for symmetric varieties. Transactions of the American Mathematical Society, 369(4), 2615-2649. https://doi.org/10.1090/tran/6738

©2016 American Mathematical Society. We associate to any complete spherical variety X a certain nonnegative rational number ℘(X), which we conjecture to satisfy the inequality ℘(X) ≤ dimX − rankX with equality holding if and only if X is isomorphic t... Read More about The generalized Mukai conjecture for symmetric varieties.

Gorenstein spherical Fano varieties (2015)
Journal Article
Gagliardi, G., & Hofscheier, J. (2015). Gorenstein spherical Fano varieties. Geometriae Dedicata, 178(1), 111-133. https://doi.org/10.1007/s10711-015-0047-y

We obtain a combinatorial description of Gorenstein spherical Fano varieties in terms of certain polytopes, generalizing the combinatorial description of Gorenstein toric Fano varieties by reflexive polytopes and its extension to Gorenstein horospher... Read More about Gorenstein spherical Fano varieties.

Homogeneous spherical data of orbits in spherical embeddings (2015)
Journal Article
Gagliardi, G., & Hofscheier, J. (2015). Homogeneous spherical data of orbits in spherical embeddings. Transformation Groups, 20(1), 83-98. https://doi.org/10.1007/s00031-014-9297-2

Let G be a connected reductive complex algebraic group. Luna assigned to any spherical homogeneous space G/H a combinatorial object called a homogeneous spherical datum. By a theorem of Losev, this object uniquely determines G/H up to G-equivariant i... Read More about Homogeneous spherical data of orbits in spherical embeddings.