Computing the Continuous Discretely

Sur le probléme des courbes gauches en Topologie

This is a survey on recent developments on unimodality, log-concavity and real-rootedness in combinatorics Stanley and Brenti have written extensive surveys of various techniques that can be used to prove real-rootedness, log-concavity or unimodality After a brief introduction, we will complement these surveys with a survey over some new techniques that have been developed, as well as problems and conjectures that have been solved This is a draft of a chapter to appear in Handbook of Enumerative Combinatorics, published by CRC Press

Unimodality, log-concavity, real-rootedness and beyond

This thesis develops the theory of sheaves and cosheaves with an eye towards applications in science and engineering. To provide a theory that is computable, we focus on a combinatorial version of sheaves and cosheaves called cellular sheaves and cosheaves, which are finite families of vector spaces and maps parametrized by a cell complex. We develop cellular (co)sheaves as a new tool for topological data analysis, network coding and sensor networks. A foundation for multi-dimensional level-set persistent homology is laid via constructible cosheaves, which are equivalent to representations of MacPherson's entrance path category. By proving a van Kampen theorem, we give a direct proof of this equivalence. A cosheaf version of the i'th derived pushforward of the constant sheaf along a definable map is constructed directly as a representation of this category. We go on to clarify the relationship of cellular sheaves to cosheaves by providing a formula that defines a derived equivalence, which in turn recovers Verdier duality. Compactly-supported sheaf cohomology is expressed as the coend with the image of the constant sheaf through this equivalence. The equivalence is further used to establish relations between sheaf cohomology and a herein newly introduced theory of cellular sheaf homology. Inspired to provide fast algorithms for persistence, we prove that the derived category of cellular sheaves over a 1D cell complex is equivalent to a category of graded sheaves. Finally, we introduce the interleaving distance as an extended pseudo-metric on the category of sheaves. We prove that global sections partition the space of sheaves into connected components. We conclude with an investigation into the geometry of the space of constructible sheaves over the real line, which we relate to the bottleneck distance in persistence.

https://repository.upenn.edu/cgi/viewcontent.cgi?article=3061&context=edissertations

Sheaves, Cosheaves and Applications

We give conjectures on the possible graded Betti numbers of Cohen�Macaulay modules up to multiplication by positive rational numbers. The idea is that the Betti diagrams should be non-negative linear combinations of pure diagrams. The conjectures are verified in the cases where the structure of resolutions is known, that is: for modules of codimension two, for Gorenstein algebras of codimension three and for complete intersections. The motivation for proposing the conjectures comes from the Multiplicity conjecture of Herzog, Huneke and Srinivasan

Graded betti numbers of Cohen-Macaulay modules and the multiplicity conjecture

/pdf/h-vectors-of-gorenstein-polytopes-1ilw5phr01.pdf

h-Vectors of Gorenstein polytopes

Minimal graded free resolutions are an important and central topic in algebra.
They are a useful tool for studying modules over finitely generated graded K-
algebras. Such a resolution determines the Hilbert series, the Castelnuovo-Mumford
regularity and other invariants of the module.
This thesis is concerned with the structure of minimal graded free resolutions. We
relate our results to several recent trends in commutative algebra.
The first of these trends deals with relations between properties of the Stanley-
Reisner ring associated to a simplicial complex and the Stanley-Reisner ring of its
Alexander dual.
Another development is the investigation of the linear part of a minimal graded
free resolution as defined by Eisenbud and Schreyer.
Several authors were interested in the problem to give lower bounds for the Betti
numbers of a module. In particular, Eisenbud-Koh, Green, Herzog and Reiner-
Welker studied the graded Betti numbers which determine the linear strand of a
minimal graded free resolution.
Bigraded algebras occur naturally in many research areas of commutative algebra.
A typical example of a bigraded algebra is the Rees ring of a graded ideal. Herzog
and Trung used this bigraded structure of the Rees ring to study the Castelnuovo-
Mumford regularity of powers of graded ideals in a polynomial ring. Conca, Herzog,
Trung and Valla dealt with diagonal subalgebras of bigraded algebras. Aramova,
Crona and De Negri studied homological properties of bigraded K-algebras.

On minimal graded free resolutions

http://sigproc.eng.cam.ac.uk/foswiki/pub/SPARS2015/Program/talk20_UlrichVon-der-Ohe_slides.pdf

A multivariate generalization of Prony's method

/pdf/generalized-alexander-duality-and-applications-4uvle38euf.pdf

Tim Römer

Papers

h-Vectors of Gorenstein polytopes

On minimal graded free resolutions

A multivariate generalization of Prony's method

Generalized Alexander duality and applications

Equivariant Hilbert series in non-noetherian polynomial rings