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Papers by Christian Haase

Research paper thumbnail of Fine Polyhedral Adjunction Theory

arXiv (Cornell University), Feb 8, 2023

Research paper thumbnail of The reflexive dimension of a lattice polytope

arXiv (Cornell University), Jun 23, 2004

Research paper thumbnail of Classifier Construction in Boolean Networks Using Algebraic Methods

We investigate how classifiers for Boolean networks (BNs) can be constructed and modified under c... more We investigate how classifiers for Boolean networks (BNs) can be constructed and modified under constraints. A typical constraint is to observe only states in attractors or even more specifically steady states of BNs. Steady states of BNs are one of the most interesting features for application. Large models can possess many steady states. In the typical scenario motivating this paper we start from a Boolean model with a given classification of the state space into phenotypes defined by high-level readout components. In order to link molecular biomarkers with experimental design, we search for alternative components suitable for the given classification task. This is useful for modelers of regulatory networks for suggesting experiments and measurements based on their models. It can also help to explain causal relations between components and phenotypes. To tackle this problem we need to use the structure of the BN and the constraints. This calls for an algebraic approach. Indeed we ...

Research paper thumbnail of Maximum number of modes of Gaussian mixtures

Information and Inference: A Journal of the IMA, 2019

Gaussian mixture models are widely used in Statistics. A fundamental aspect of these distribution... more Gaussian mixture models are widely used in Statistics. A fundamental aspect of these distributions is the study of the local maxima of the density or modes. In particular, it is not known how many modes a mixture of kkk Gaussians in ddd dimensions can have. We give a brief account of this problem’s history. Then, we give improved lower bounds and the first upper bound on the maximum number of modes, provided it is finite.

Research paper thumbnail of Competitive equilibrium always exists for combinatorial auctions with graphical pricing schemes

ArXiv, 2021

We show that a competitive equilibrium always exists in combinatorial auctions with anonymous gra... more We show that a competitive equilibrium always exists in combinatorial auctions with anonymous graphical valuations and pricing, using discrete geometry. This is an intuitive and easy-to-construct class of valuations that can model both complementarity and substitutes, and to our knowledge, it is the first class besides gross substitutes that have guaranteed competitive equilibrium. We prove through counter-examples that our result is tight, and we give explicit algorithms for constructive competitive pricing vectors. We also give extensions to multi-unit combinatorial auctions (also known as product-mix auctions). Combined with theorems on graphical valuations and pricing equilibrium of Candogan, Ozdagar and Parillo, our results indicate that quadratic pricing is a highly practical method to run combinatorial auctions.

Research paper thumbnail of Theorems of Brion, Lawrence, and Varchenko on rational generating functions for cones

We discuss and give elementary proofs of results of Brion and of Lawrence-Varchenko on the lattic... more We discuss and give elementary proofs of results of Brion and of Lawrence-Varchenko on the lattice-point enumerator generating functions for polytopes and cones. This largely expository note contains a new proof of Brion's Formula using irrational decompositions, and a generalization of the Lawrence-Varchenko formula.

Research paper thumbnail of Face-subgroups of permutation polytopes

In [Baumeister, H., Nill, Paffenholz, On permutation polytopes, Adv. Math. 222 (2009), 431-452 / ... more In [Baumeister, H., Nill, Paffenholz, On permutation polytopes, Adv. Math. 222 (2009), 431-452 / arXiv:0709.1615] we conjectured a characterization of subgroups H of a permutation group G so that, on the level of permutation polytopes, P(H) is a face of P(G). Here we present the embarrassingly simple proof of this conjecture.

Research paper thumbnail of Examples and counterexamples for Perles' conjecture

The combinatorial structure of a d-dimensional simple convex polytope can be reconstructed from i... more The combinatorial structure of a d-dimensional simple convex polytope can be reconstructed from its abstract graph [Blind & Mani 1987, Kalai 1988]. However, no polynomial/efficient algorithm is known for this task, although a polynomially checkable certificate for the correct reconstruction was found by [Joswig, Kaibel & Koerner 2000]. A much stronger certificate would be given by the following characterization of the facet subgraphs, conjectured by M. Perles: ``The facet subgraphs of the graph of a simple d-polytope are exactly all the (d-1)-regular, connected, induced, non-separating subgraphs'' [Perles 1970]. We give examples for the validity of Perles conjecture: In particular, it holds for the duals of cyclic polytopes, and for the duals of stacked polytopes. On the other hand, we identify a topological obstruction that must be present in any counterexample to Perles' conjecture; thus, starting with a modification of ``Bing's house'', we construct explic...

Research paper thumbnail of Polyhedral adjunction theory

In this paper we give a combinatorial view on the adjunction theory of toric varieties. Inspired ... more In this paper we give a combinatorial view on the adjunction theory of toric varieties. Inspired by classical adjunction theory of polarized algebraic varieties we define two convex-geometric notions: the Q-codegree and the nef value of a rational polytope P. We define the adjoint polytope P^(s) as the set of those points in P, whose lattice distance to every facet of P is at least s. We prove a structure theorem for lattice polytopes P with high Q-codegree. If P^(s) is empty for some s < 2/(dim(P)+2), then the lattice polytope P has lattice width one. This has consequences in Ehrhart theory and on polarized toric varieties with dual defect. Moreover, we illustrate how classification results in adjunction theory can be translated into new classification results for lattice polytopes.

Research paper thumbnail of Quasi-period collapse and GL_n(Z)-scissors congruence in rational polytopes

Quasi-period collapse occurs when the Ehrhart quasi-polynomial of a rational polytope has a quasi... more Quasi-period collapse occurs when the Ehrhart quasi-polynomial of a rational polytope has a quasi-period less than the denominator of that polytope. This phenomenon is poorly understood, and all known cases in which it occurs have been proven with ad hoc methods. In this note, we present a conjectural explanation for quasi-period collapse in rational polytopes. We show that this explanation applies to some previous cases appearing in the literature. We also exhibit examples of Ehrhart polynomials of rational polytopes that are not the Ehrhart polynomials of any integral polytope. Our approach depends on the invariance of the Ehrhart quasi-polynomial under the action of affine unimodular transformations. Motivated by the similarity of this idea to the scissors congruence problem, we explore the development of a Dehn-like invariant for rational polytopes in the lattice setting.

Research paper thumbnail of 3 an Inequality for Adjoint Rational Surfaces

Research paper thumbnail of Integral Affine Structures on Spheres Complete Intersections

Abstract. We extend our model for affine structures on toric Calabi-Yau hypersurfaces [HZ02] to t... more Abstract. We extend our model for affine structures on toric Calabi-Yau hypersurfaces [HZ02] to the case of complete intersections. 1.

Research paper thumbnail of Lattice polytopes in algebra, physics

Research paper thumbnail of Cayley decompositions of lattice polytopes and upper bounds for h*-polynomials

Journal für die reine und angewandte Mathematik (Crelles Journal), 2009

Research paper thumbnail of Algebraic hyperbolicity for surfaces in toric threefolds

arXiv: Algebraic Geometry, 2019

Adapting focal loci techniques used by Chiantini and Lopez, we provide lower bounds on the genera... more Adapting focal loci techniques used by Chiantini and Lopez, we provide lower bounds on the genera of curves contained in very general surfaces in Gorenstein toric threefolds. We illustrate the utility of these bounds by obtaining results on algebraic hyperbolicity of very general surfaces in toric threefolds.

Research paper thumbnail of Permutation Polytopes of Cyclic Groups

arXiv: Combinatorics, 2011

We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permu... more We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permutation groups, i.e., the convex hulls of cyclic groups of permutation matrices. We give formulas for their dimension and vertex degree. In the situation that the generator of the group consists of at most two orbits, we can give a complete combinatorial description of the associated permutation polytope. In the case of three orbits the facet structure is already quite complex. For a large class of examples we show that there exist exponentially many facets.

Research paper thumbnail of Integral affine structures on spheres III: complete intersections

arXiv: Algebraic Geometry, 2005

We extend our model for affine structures on toric Calabi-Yau hypersurfaces math.AG/0205321 to th... more We extend our model for affine structures on toric Calabi-Yau hypersurfaces math.AG/0205321 to the case of complete intersections.

Research paper thumbnail of Contemporary Mathematics Problems from the Cottonwood Room

This collection was compiled by Christian Haase and Bruce Reznick from problems presented at the ... more This collection was compiled by Christian Haase and Bruce Reznick from problems presented at the problem sessions, and submissions solicited from the participants of the AMS/IMS/SIAM summer Research Conference on Integer points in polyhedra.

Research paper thumbnail of An inequality for adjoint rational surfaces

arXiv: Algebraic Geometry, 2013

We generalize an inequality for convex lattice polygons -- aka toric surfaces -- to general ratio... more We generalize an inequality for convex lattice polygons -- aka toric surfaces -- to general rational surfaces.

Research paper thumbnail of Toric Newton-Okounkov functions with an application to the rationality of certain Seshadri constants on surfaces

We initiate a combinatorial study of Newton-Okounkov functions on toric varieties with an eye on ... more We initiate a combinatorial study of Newton-Okounkov functions on toric varieties with an eye on the rationality of asymptotic invariants of line bundles. In the course of our efforts we identify a combinatorial condition which ensures a controlled behavior of the appropriate Newton-Okounkov function on a toric surface. Our approach yields the rationality of many Seshadri constants that have not been settled before.

Research paper thumbnail of Fine Polyhedral Adjunction Theory

arXiv (Cornell University), Feb 8, 2023

Research paper thumbnail of The reflexive dimension of a lattice polytope

arXiv (Cornell University), Jun 23, 2004

Research paper thumbnail of Classifier Construction in Boolean Networks Using Algebraic Methods

We investigate how classifiers for Boolean networks (BNs) can be constructed and modified under c... more We investigate how classifiers for Boolean networks (BNs) can be constructed and modified under constraints. A typical constraint is to observe only states in attractors or even more specifically steady states of BNs. Steady states of BNs are one of the most interesting features for application. Large models can possess many steady states. In the typical scenario motivating this paper we start from a Boolean model with a given classification of the state space into phenotypes defined by high-level readout components. In order to link molecular biomarkers with experimental design, we search for alternative components suitable for the given classification task. This is useful for modelers of regulatory networks for suggesting experiments and measurements based on their models. It can also help to explain causal relations between components and phenotypes. To tackle this problem we need to use the structure of the BN and the constraints. This calls for an algebraic approach. Indeed we ...

Research paper thumbnail of Maximum number of modes of Gaussian mixtures

Information and Inference: A Journal of the IMA, 2019

Gaussian mixture models are widely used in Statistics. A fundamental aspect of these distribution... more Gaussian mixture models are widely used in Statistics. A fundamental aspect of these distributions is the study of the local maxima of the density or modes. In particular, it is not known how many modes a mixture of kkk Gaussians in ddd dimensions can have. We give a brief account of this problem’s history. Then, we give improved lower bounds and the first upper bound on the maximum number of modes, provided it is finite.

Research paper thumbnail of Competitive equilibrium always exists for combinatorial auctions with graphical pricing schemes

ArXiv, 2021

We show that a competitive equilibrium always exists in combinatorial auctions with anonymous gra... more We show that a competitive equilibrium always exists in combinatorial auctions with anonymous graphical valuations and pricing, using discrete geometry. This is an intuitive and easy-to-construct class of valuations that can model both complementarity and substitutes, and to our knowledge, it is the first class besides gross substitutes that have guaranteed competitive equilibrium. We prove through counter-examples that our result is tight, and we give explicit algorithms for constructive competitive pricing vectors. We also give extensions to multi-unit combinatorial auctions (also known as product-mix auctions). Combined with theorems on graphical valuations and pricing equilibrium of Candogan, Ozdagar and Parillo, our results indicate that quadratic pricing is a highly practical method to run combinatorial auctions.

Research paper thumbnail of Theorems of Brion, Lawrence, and Varchenko on rational generating functions for cones

We discuss and give elementary proofs of results of Brion and of Lawrence-Varchenko on the lattic... more We discuss and give elementary proofs of results of Brion and of Lawrence-Varchenko on the lattice-point enumerator generating functions for polytopes and cones. This largely expository note contains a new proof of Brion's Formula using irrational decompositions, and a generalization of the Lawrence-Varchenko formula.

Research paper thumbnail of Face-subgroups of permutation polytopes

In [Baumeister, H., Nill, Paffenholz, On permutation polytopes, Adv. Math. 222 (2009), 431-452 / ... more In [Baumeister, H., Nill, Paffenholz, On permutation polytopes, Adv. Math. 222 (2009), 431-452 / arXiv:0709.1615] we conjectured a characterization of subgroups H of a permutation group G so that, on the level of permutation polytopes, P(H) is a face of P(G). Here we present the embarrassingly simple proof of this conjecture.

Research paper thumbnail of Examples and counterexamples for Perles' conjecture

The combinatorial structure of a d-dimensional simple convex polytope can be reconstructed from i... more The combinatorial structure of a d-dimensional simple convex polytope can be reconstructed from its abstract graph [Blind & Mani 1987, Kalai 1988]. However, no polynomial/efficient algorithm is known for this task, although a polynomially checkable certificate for the correct reconstruction was found by [Joswig, Kaibel & Koerner 2000]. A much stronger certificate would be given by the following characterization of the facet subgraphs, conjectured by M. Perles: ``The facet subgraphs of the graph of a simple d-polytope are exactly all the (d-1)-regular, connected, induced, non-separating subgraphs'' [Perles 1970]. We give examples for the validity of Perles conjecture: In particular, it holds for the duals of cyclic polytopes, and for the duals of stacked polytopes. On the other hand, we identify a topological obstruction that must be present in any counterexample to Perles' conjecture; thus, starting with a modification of ``Bing's house'', we construct explic...

Research paper thumbnail of Polyhedral adjunction theory

In this paper we give a combinatorial view on the adjunction theory of toric varieties. Inspired ... more In this paper we give a combinatorial view on the adjunction theory of toric varieties. Inspired by classical adjunction theory of polarized algebraic varieties we define two convex-geometric notions: the Q-codegree and the nef value of a rational polytope P. We define the adjoint polytope P^(s) as the set of those points in P, whose lattice distance to every facet of P is at least s. We prove a structure theorem for lattice polytopes P with high Q-codegree. If P^(s) is empty for some s < 2/(dim(P)+2), then the lattice polytope P has lattice width one. This has consequences in Ehrhart theory and on polarized toric varieties with dual defect. Moreover, we illustrate how classification results in adjunction theory can be translated into new classification results for lattice polytopes.

Research paper thumbnail of Quasi-period collapse and GL_n(Z)-scissors congruence in rational polytopes

Quasi-period collapse occurs when the Ehrhart quasi-polynomial of a rational polytope has a quasi... more Quasi-period collapse occurs when the Ehrhart quasi-polynomial of a rational polytope has a quasi-period less than the denominator of that polytope. This phenomenon is poorly understood, and all known cases in which it occurs have been proven with ad hoc methods. In this note, we present a conjectural explanation for quasi-period collapse in rational polytopes. We show that this explanation applies to some previous cases appearing in the literature. We also exhibit examples of Ehrhart polynomials of rational polytopes that are not the Ehrhart polynomials of any integral polytope. Our approach depends on the invariance of the Ehrhart quasi-polynomial under the action of affine unimodular transformations. Motivated by the similarity of this idea to the scissors congruence problem, we explore the development of a Dehn-like invariant for rational polytopes in the lattice setting.

Research paper thumbnail of 3 an Inequality for Adjoint Rational Surfaces

Research paper thumbnail of Integral Affine Structures on Spheres Complete Intersections

Abstract. We extend our model for affine structures on toric Calabi-Yau hypersurfaces [HZ02] to t... more Abstract. We extend our model for affine structures on toric Calabi-Yau hypersurfaces [HZ02] to the case of complete intersections. 1.

Research paper thumbnail of Lattice polytopes in algebra, physics

Research paper thumbnail of Cayley decompositions of lattice polytopes and upper bounds for h*-polynomials

Journal für die reine und angewandte Mathematik (Crelles Journal), 2009

Research paper thumbnail of Algebraic hyperbolicity for surfaces in toric threefolds

arXiv: Algebraic Geometry, 2019

Adapting focal loci techniques used by Chiantini and Lopez, we provide lower bounds on the genera... more Adapting focal loci techniques used by Chiantini and Lopez, we provide lower bounds on the genera of curves contained in very general surfaces in Gorenstein toric threefolds. We illustrate the utility of these bounds by obtaining results on algebraic hyperbolicity of very general surfaces in toric threefolds.

Research paper thumbnail of Permutation Polytopes of Cyclic Groups

arXiv: Combinatorics, 2011

We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permu... more We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permutation groups, i.e., the convex hulls of cyclic groups of permutation matrices. We give formulas for their dimension and vertex degree. In the situation that the generator of the group consists of at most two orbits, we can give a complete combinatorial description of the associated permutation polytope. In the case of three orbits the facet structure is already quite complex. For a large class of examples we show that there exist exponentially many facets.

Research paper thumbnail of Integral affine structures on spheres III: complete intersections

arXiv: Algebraic Geometry, 2005

We extend our model for affine structures on toric Calabi-Yau hypersurfaces math.AG/0205321 to th... more We extend our model for affine structures on toric Calabi-Yau hypersurfaces math.AG/0205321 to the case of complete intersections.

Research paper thumbnail of Contemporary Mathematics Problems from the Cottonwood Room

This collection was compiled by Christian Haase and Bruce Reznick from problems presented at the ... more This collection was compiled by Christian Haase and Bruce Reznick from problems presented at the problem sessions, and submissions solicited from the participants of the AMS/IMS/SIAM summer Research Conference on Integer points in polyhedra.

Research paper thumbnail of An inequality for adjoint rational surfaces

arXiv: Algebraic Geometry, 2013

We generalize an inequality for convex lattice polygons -- aka toric surfaces -- to general ratio... more We generalize an inequality for convex lattice polygons -- aka toric surfaces -- to general rational surfaces.

Research paper thumbnail of Toric Newton-Okounkov functions with an application to the rationality of certain Seshadri constants on surfaces

We initiate a combinatorial study of Newton-Okounkov functions on toric varieties with an eye on ... more We initiate a combinatorial study of Newton-Okounkov functions on toric varieties with an eye on the rationality of asymptotic invariants of line bundles. In the course of our efforts we identify a combinatorial condition which ensures a controlled behavior of the appropriate Newton-Okounkov function on a toric surface. Our approach yields the rationality of many Seshadri constants that have not been settled before.