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Papers by Christian Haase
arXiv (Cornell University), Feb 8, 2023
arXiv (Cornell University), Jun 23, 2004
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 ...
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.
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.
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.
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.
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...
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.
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.
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.
Journal für die reine und angewandte Mathematik (Crelles Journal), 2009
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.
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.
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.
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.
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.
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.
arXiv (Cornell University), Feb 8, 2023
arXiv (Cornell University), Jun 23, 2004
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 ...
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.
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.
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.
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.
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...
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.
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.
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.
Journal für die reine und angewandte Mathematik (Crelles Journal), 2009
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.
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.
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.
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.
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.
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.