Isabella Novik's research while affiliated with University of Washington Seattle and other places
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Publications (69)
Kalai conjectured that if P is a simplicial d-polytope that has no missing faces of dimension \(d-1\), then the graph of P and the space of affine 2-stresses of P determine P up to affine equivalence. We propose a higher-dimensional generalization of this conjecture: if \(2\le i\le d/2\) and P is a simplicial d-polytope that has no missing faces of...
A conjecture of Kalai asserts that for $d\geq 4$, the affine type of a prime simplicial $d$-polytope $P$ can be reconstructed from the space of affine $2$-stresses of $P$. We prove this conjecture for all $d\geq 5$. We also prove the following generalization: for all pairs $(i,d)$ with $2\leq i\leq \lceil \frac d 2\rceil -1$, the affine type of a s...
A conjecture of Kalai asserts that for $d\geq 4$, the affine type of a prime simplicial $d$-polytope $P$ can be reconstructed from the space of affine $2$-stresses of $P$. We prove this conjecture for all $d\geq 5$. We also prove the following generalization: for all pairs $(i,d)$ with $2\leq i\leq \lceil \frac d 2\rceil-1$, the affine type of a si...
We define a certain merging operation that given two $d$-polytopes $P$ and $Q$ such that $P$ has a simplex facet $F$ and $Q$ has a simple vertex $v$ produces a new $d$-polytope $P\hspace{0.1em}\triangleright Q$ with $f_0(P)+f_0(Q)-(d+1)$ vertices. We show that if for some $1\leq i\leq d-1$, $P$ and $Q$ are $(d-i)$-simplicial $i$-simple $d$-polytope...
The result of Padrol (Discret Comput Geom 50(4):865–902, 2013) asserts that for every d≥4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 4$$\end{document}, there e...
A conjecture of Kalai from 1994 posits that for an arbitrary 2 ≤ k ≤ ⌊d/2⌋, the combinatorial type of a simplicial d-polytope P is uniquely determined by the (k − 1)-skeleton of P (given as an abstract simplicial complex) together with the space of affine k-stresses on P. We establish the first non-trivial case of this conjecture, namely, the case...
Kalai conjectured that if $P$ is a simplicial $d$-polytope that has no missing faces of dimension $d-1$, then the graph of $P$ and the space of affine $2$-stresses of $P$ determine $P$ up to affine equivalence. We propose a higher-dimensional generalization of this conjecture: if $2\leq i\leq d/2$ and $P$ is a simplicial $d$-polytope that has no mi...
We survey several old and new problems related to the number of simplicial spheres, the number of neighborly simplicial spheres, the number of centrally symmetric simplicial spheres that are cs-neighborly, and the transversal numbers of hypergraphs that arise from simplicial spheres.
A conjecture of Kalai from 1994 posits that for an arbitrary $2\leq k\leq \lfloor d/2 \rfloor$, the combinatorial type of a simplicial $d$-polytope $P$ is uniquely determined by the $(k-1)$-skeleton of $P$ (given as an abstract simplicial complex) together with the space of affine $k$-stresses on $P$. We establish the first non-trivial case of this...
The result of Padrol asserts that for every $d\geq 4$, there exist $2^{\Omega(n\log n)}$ distinct combinatorial types of $\lfloor d/2\rfloor$-neighborly simplicial $(d-1)$-spheres with $n$ vertices. We present a construction showing that for every $d\geq 5$, there are at least $2^{\Omega(n^{\lfloor (d-1)/2\rfloor})}$ such types.
In 1987, Stanley conjectured that if a centrally symmetric Cohen--Macaulay simplicial complex $\Delta$ of dimension $d-1$ satisfies $h_i(\Delta)=\binom{d}{i}$ for some $1\leq i\leq d-1$, then $h_j(\Delta)=\binom{d}{j}$ for all $j\geq i$. This note proves Stanley's conjecture.
In 1995, Jockusch constructed an infinite family of centrally symmetric 3-dimensional simplicial spheres that are cs-2-neighborly. Here we generalize his construction and show that for all d≥3 and n≥d+1, there exists a centrally symmetric d-dimensional simplicial sphere with 2n vertices that is cs-⌈d/2⌉-neighborly. This result combined with work of...
In 1995, Josckusch constructed an infinite family of centrally symmetric (cs, for short) triangulations of $3$-spheres that are cs-$2$-neighborly. Recently, Novik and Zheng extended Jockusch's construction: for all $d$ and $n>d$, they constructed a cs triangulation of a $d$-sphere with $2n$ vertices, $\Delta^d_n$, that is cs-$\lceil d/2\rceil$-neig...
We extend several $g$-type theorems for connected, orientable homology manifolds without boundary to manifolds with boundary. As applications of these results we obtain K\"uhnel-type bounds on the Betti numbers as well as on certain weighted sums of Betti numbers of manifolds with boundary. Our main tool is the completion $\hat\Delta$ of a manifold...
In 1995, Jockusch constructed an infinite family of centrally symmetric $3$-dimensional simplicial spheres that are cs-$2$-neighborly. Here we generalize his construction and show that for all $d\geq 4$ and $n\geq d$, there exists a centrally symmetric $(d-1)$-dimensional simplicial sphere with $2n$ vertices that is cs-$\lfloor d/2\rfloor$-neighbor...
We introduce and investigate |$d$|-convex union representable complexes: the simplicial complexes that arise as the nerve of a finite collection of convex open sets in |${\mathbb{R}}^d$| whose union is also convex. Chen, Frick, and Shiu recently proved that such complexes are collapsible and asked if all collapsible complexes are convex union repre...
Stanley proved that for any centrally symmetric simplicial d-polytope P with d≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 3$$\end{document}, g2(P)≥d2-d\docum...
This paper is a survey of recent advances as well as open problems in the study of face numbers of centrally symmetric simplicial polytopes and spheres. The topics discussed range from neighborliness of centrally symmetric polytopes and the Upper Bound Theorem for centrally symmetric simplicial spheres to the Generalized Lower Bound Theorem for cen...
We introduce and investigate $d$-convex union representable complexes: the complexes that arise as the nerve of a finite collection of convex open sets in $\mathbb R^d$ whose union is also convex. Chen, Frick, and Shiu recently proved that such complexes are collapsible and asked if all collapsible complexes are convex union representable. We dispr...
A simplicial complex of dimension $d-1$ is said to be balanced if its graph is $d$-colorable. Juhnke-Kubitzke and Murai proved an analogue of the generalized lower bound theorem for balanced simplicial polytopes. We establish a generalization of their result to balanced triangulations of closed homology manifolds and balanced triangulations of orie...
What is the maximum number of vertices that a centrally symmetric 2-neighborly polytope of dimension $d$ can have? It is known that the answer does not exceed $2^d$. Here we provide an explicit construction showing that it is at least $2^{d-1}+2$.
This paper is a survey of recent advances as well as open problems in the study of face numbers of centrally symmetric simplicial polytopes and spheres. The topics discussed range from neighborliness of centrally symmetric polytopes and the upper bound theorem for centrally symmetric simplicial spheres to the generalized lower bound theorem for cen...
We introduce a notion of cross-flips: local moves that transform a balanced (i.e., properly (d+1)-colored) triangulation of a combinatorial d-manifold into another balanced triangulation. These moves form a natural analog of bistellar flips (also known as Pachner moves). Specifically, we establish the following theorem: any two balanced triangulati...
We resolve a conjecture of Kalai asserting that the $g_2$-number of any simplicial complex $\Delta$ that represents a connected normal pseudomanifold of dimension $d\geq 3$ is at least as large as ${d+2 \choose 2}m(\Delta)$, where $m(\Delta)$ denotes the minimum number of generators of the fundamental group of $\Delta$. Furthermore, we prove that a...
Let $\Delta$ be a triangulated homology ball whose boundary complex is $\partial\Delta$. A result of Hochster asserts that the canonical module of the Stanley--Reisner ring of $\Delta$, $\mathbb F[\Delta]$, is isomorphic to the Stanley--Reisner module of the pair $(\Delta, \partial\Delta)$, $\mathbb F[\Delta,\partial \Delta]$. This result implies t...
The 2015 Oberwolfach meeting “Geometric and Algebraic Combinatorics” was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highli...
We study face numbers of simplicial complexes that triangulate manifolds (or even normal pseudomanifolds) with boundary. Specifically,
we establish a sharp lower bound on the number of interior edges of a simplicial normal pseudomanifold with boundary in terms
of the number of interior vertices and relative Betti numbers. Moreover, for triangulatio...
Let $M$ be a closed triangulable manifold, and let $\Delta$ be a
triangulation of $M$. What is the smallest number of vertices that $\Delta$ can
have? How big or small can the number of edges of $\Delta$ be as a function of
the number of vertices? More generally, what are the possible face numbers
($f$-numbers, for short) that $\Delta$ can have? In...
A simplicial (d − 1)-dimensional complex K is called balanced if the graph of K (i. e., the 1-dimensional skeleton) is d-colorable. Here we discuss some recent results as well as several open questions on face numbers of balanced manifolds and pseudomanifolds; we also present constructions of balanced manifolds (with and without boundary) with few...
A $(d-1)$-dimensional simplicial complex is called balanced if its underlying
graph admits a proper $d$-coloring. We show that many well-known face
enumeration results have natural balanced analogs (or at least conjectural
analogs). Specifically, we prove the balanced analog of the celebrated Lower
Bound Theorem for pseudomanifolds and characterize...
We introduce a notion of bipartite minors and prove a bipartite analog of
Wagner's theorem: a bipartite graph is planar if and only if it does not
contain $K_{3,3}$ as a bipartite minor. Similarly, we provide a forbidden minor
characterization for outerplanar graphs and forests. We then establish a
recursive characterization of bipartite $(2,2)$-La...
We develop a bipartite rigidity theory for bipartite graphs parallel to the
classical rigidity theory for general graphs, and define for two positive
integers $k,l$ the notions of $(k,l)$-rigid and $(k,l)$-stress free bipartite
graphs. This theory coincides with the study of Babson--Novik's balanced
shifting restricted to graphs. We establish bipar...
A notion of an $i$-banner simplicial complex is introduced. For various
values of $i$, these complexes interpolate between the class of flag complexes
and the class of all simplicial complexes. Examples of simplicial spheres of an
arbitrary dimension that are $(i+1)$-banner but not $i$-banner are constructed.
It is shown that several theorems for f...
We present explicit constructions of centrally symmetric 2-neighborly
d-dimensional polytopes with about 3^{d/2} = (1.73)^d vertices and of centrally
symmetric k-neighborly d-polytopes with about 2^{c_k d} vertices where c_k=3/20
k^2 2^k. Using this result, we construct for a fixed k > 1 and arbitrarily
large d and N, a centrally symmetric d-polyto...
The 2011 Oberwolfach meeting “Topological and Geometric Combinatorics” was organized by Anders Björner (Stockholm), Gil Kalai (Jerusalem), Isabella Novik (Seattle), and Günter M. Ziegler (Berlin). It covered a wide variety of aspects of Discrete Geometry, Topological Combinatorics, and Geometric Topology. Some of the highlights of the conference in...
We present explicit constructions of centrally symmetric polytopes with many
faces: first, we construct a d-dimensional centrally symmetric polytope P with
about (1.316)^d vertices such that every pair of non-antipodal vertices of P
spans an edge of P, second, for an integer k>1, we construct a d-dimensional
centrally symmetric polytope P of an arb...
We consider the convex hull B_k of the symmetric moment curve U(t)=(cos t,
sin t, cos 3t, sin 3t, ..., cos (2k-1)t, sin (2k-1)t) in R^{2k}, where t ranges
over the unit circle S= R/2pi Z. The curve U(t) is locally neighborly: as long
as t_1, ..., t_k lie in an open arc of S of a certain length phi_k>0, the
convex hull of the points U(t_1), ..., U(t...
A centrally symmetric $2d$-vertex combinatorial triangulation of the product
of spheres $\S^i\times\S^{d-2-i}$ is constructed for all pairs of non-negative
integers $i$ and $d$ with $0\leq i \leq d-2$. For the case of $i=d-2-i$, the
existence of such a triangulation was conjectured by Sparla. The constructed
complex admits a vertex-transitive actio...
We investigate the face numbers of simplicial complexes with Buchsbaum vertex links, especially pseudomanifolds with isolated singularities. This includes deriving Dehn-Sommerville relations for pseudomanifolds with isolated singularities and establishing lower bound theorems when the singularities are also homologically isolated. We give formulas...
The face ring of a simplicial complex modulo m generic linear forms is shown to have finite local cohomology if and only if the link of every face of dimension m or more is `nonsingular', i.e., has the homology of a wedge of spheres of the expected dimension. This is derived from an enumerative result for local cohomology of face rings modulo gener...
The socle of a graded Buchsbaum module is studied and is related to its local cohomology modules. This algebraic result is then applied to face enumeration of Buchsbaum simplicial complexes and posets. In particular, new necessary conditions on face numbers and Betti numbers of such complexes and posets are established. These conditions are used to...
A common generalization of two theorems on the face numbers of Cohen-Macaulay (CM, for short) simplicial complexes is established: the first is the theorem of Stanley (necessity) and Bjorner-Frankl-Stanley (sufficiency) that characterizes all possible face numbers of a-balanced CM complexes, while the second is the theorem of Novik (necessity) and...
It is shown that the face ring of a pure simplicial complex modulo $m$ generic linear forms is a ring with finite local cohomology if and only if the link of every face of dimension $m$ or more is nonsingular. Comment: 8 pages
The face numbers of simplicial complexes without missing faces of dimension larger than $i$ are studied. It is shown that among all such $(d-1)$-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the componentwise minimal $f$-vector; and moreover, among all such 2-Cohen--Macaulay (2-CM) complexes, the same sphere...
We use Klee’s Dehn–Sommerville relations and other results on face numbers of homology manifolds without boundary to (i)prove
Kalai’s conjecture providing lower bounds on the f-vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii)verify Kühnel’s conjecture that
gives an upper bound on the middle Betti number o...
The face ring of a homology manifold (without boundary) modulo a generic system of parameters is studied. Its socle is computed and it is verified that a particular quotient of this ring is Gorenstein. This fact is used to prove that the sphere $g$-conjecture implies all enumerative consequences of its far reaching generalization (due to Kalai) to...
We define a centrally symmetric analogue of the cyclic polytope and study its facial structure. We conjecture that our polytopes provide asymptotically the largest number of faces in all dimensions among all centrally symmetric polytopes with n vertices of a given even dimension d=2k when d is fixed and n grows. For a fixed even dimension d=2k and...
The multiplicity conjecture of Herzog, Huneke, and Srinivasan is verified for the face rings of the following classes of simplicial complexes: matroid complexes, complexes of dimension one and two, and Gorenstein complexes of dimension at most four. The lower bound part of this conjecture is also established for the face rings of all doubly Cohen-M...
Certain necessary conditions on the face numbers and Betti numbers of sim- plicial complexes endowed with a proper action of a prime order cyclic group are established. A notion of colored algebraic shifting is dened and its properties are studied. As an application a new simple proof of the characterization of the flag face numbers of balanced Coh...
A short new proof of the fact that all shifted complexes are fixed by reverse lexicographic shifting is given. A notion of lexicographic shifting, Δlex—an operation that transforms a monomial ideal of S = K[x
i: i ∈ ℕ] that is finitely generated in each degree into a squarefree strongly stable ideal—is defined and studied. It is proved that (in con...
We show that there exist k-neighborly centrally symmetric d-dimensional polytopes
with 2(n + d) vertices, where
$$k(d,n)=\Theta\left(\frac{d}{1+\log ((d+n)/d)}\right).$$
We also show that this bound is tight.
The spindle pole body (SPB) is the microtubule organizing center of Saccharomyces cerevisiae. Its core includes the proteins Spc42, Spc110 (kendrin/pericentrin ortholog), calmodulin (Cmd1), Spc29, and Cnm67. Each was tagged with CFP and YFP and their proximity to each other was determined by fluorescence resonance energy transfer (FRET). FRET was m...
Necessary conditions on the face numbers of Cohen–Macaulay simplicial complexes admitting a proper action of the cyclic group of a prime order are given. This result is extended further to necessary conditions on the face numbers and the Betti numbers of Buchsbaum simplicial complexes with a proper -action. Adin's upper bounds on the face numbers o...
We define a set of invariants of a homogeneous ideal I in a polynomial ring called the symmetric iterated Betti numbers of I. We prove that for IΓ, the Stanley–Reisner ideal of a simplicial complex Γ, these numbers are the symmetric counterparts of the exterior iterated Betti numbers of Γ introduced by Duval and Rose, and that the extremal Betti nu...
Two new relatives of the Upper Bound Theorem are established. The first is a variant of the Strong Upper Bound Theorem for even-dimensional simplicial spheres. The second is an Upper Bound Theorem for odd-dimensional simplicial manifolds admitting a free involution.
We describe a new family of free resolutions for a monomial ideal I, generalizing Lyubeznik''s construction. These resolutions are cellular resolutions supported on the rooted complexes of the lcm-lattice of I. Our resolutions are minimal for the matroid ideal of a finite projective space.
We verify the Upper Bound Conjecture (UBC) for a class of odd-dimensional simplicial complexes that in particular includes all Eulerian simplicial complexes with isolated singularities. The proof relies on a new invariant of simplicial complexes -- a short simplicial h-vector.
We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids and oriented matroids. These are Stanley-Reisner ideals of complexes of independent sets, and of triangulations of Lawrence matroid polytopes. Our resolution provides a cellular realization of Stanley's formula for their Betti numb...
We obtain lower bounds on the coefficients of the cd -index of any (2k− 1)-dimensional simplicial manifold (or, more generally, any Eulerian Buchsbaum complex)Δ. These bounds imply that many of the coefficients of the cd -index of such Δ are positive and that[formula]χ denotes the reduced Euler characteristic andβ0 , β1, . . . are reduced Betti num...
In this note we prove that if a simplicial complex K can be embedded geometrically in R
m
, then a certain linear system of equations associated with K possesses a small integral solution.
A new and more geometric proof is obtained of Stanley's lower bounds on the face numbers of centrally symmetric simple polytopes.
In this paper we prove the Upper Bound Conjecture (UBC) for some classes of (simplicial) homology manifolds: we show that the UBC holds for all odd-dimensional homology manifolds and for all 2k-dimensional homology manifolds Δ such that βk
(Δ)⩽Σ{βi
(Δ):i ≠k-2,k,k+2 and 1 ⩽i⩽2k-1}, where βi
(Δ) are reduced Betti numbers of Δ. (This condition is sati...
Citations
... More generally, if P has no missing faces of dimension ≥ d − i + 1, then the space of affine i-stresses of P determines P up to affine equivalence. [17] and Conjecture 1.2(1) was privately communicated to us and recorded in [31]. The second part of Conjecture 1.2 is a generalization of the first part: in addition to being stated for a general i, knowing the (i − 1)-skeleton is not part of the assumptions. ...
... log n/ . While we are still very far from being able to shed light on Conjecture 7.1, very recently Novik and Zheng [77] proved that for all d 5, sn.d; n/ 2 .n b.d 1/=2c / : (7.1) ...
... This dream was finally realized by Adiprasito in [2]. Other recent applications of linear and affine stresses to the fvector theory and especially to the lower-bound-type questions include [1,4,19,30], to name just a few; see also the results on the g-vectors of flag PL spheres in this paper. We encourage the reader to think of further potential applications of spaces of affine stresses. ...
... An efficient characterization of convex codes is unfortunately out of the question-recent work in [17] shows that recognizing convex codes is ∃R-hard. Nevertheless, researchers have used techniques from algebra [5,8,10], discrete geometry [1,3,11,16,18], and topology [2,4] to analyze many interesting families of codes and develop frameworks in which to test whether or not a code is convex. Some works also study codes that arise from "good covers" [2,4], collections of closed convex sets [3,9], and "nondegenerate" collections of convex sets [1,3]. ...
... While Kalai's conjecture is specifically about Tverberg depth, the sum of dimensions of depth regions can be computed for any depth measure, and thus the conjecture can be generalized to other depth measures. In fact, in a talk Kalai conjectured that the Cascade conjecture is true for Tukey depth, mentioning on his slides that 'this should be doable' [16]. In this work, we will prove the conjecture to be true for a family of depth measures that includes Tukey depth. ...
Reference: Enclosing Depth and Other Depth Measures
![[object Object]](https://i1.rgstatic.net/ii/profile.image/273662314020870-1442257588560_Q64/Francisco-Santos-29.jpg)
... The paper [4] provides far reaching generalizations of these algebraic results to face rings of much more general simplicial complexes such as normal pseudomanifolds. It is now more pressing than ever to compute the Hilbert functions of generic Artinian reductions of face rings and certain further quotients of these rings for the purpose of understanding complexes with singularities more complicated than those studied in [73,74]. 7. Numbers of neighborly polytopes and neighborly spheres As we saw in Section 3, the cyclic d -polytopes have the remarkable property of being bd=2c-neighborly. ...
... 1/-spheres with 2.d C 2/ vertices that are cs-bd=2c-neighborly. Recently, building on the work of Jockusch [39], Novik and Zheng [75] provided a complete answer: for all values of d 4 and n d , there exists a cs simplicial .d 1/-sphere with 2n vertices, d 1 n , that is cs-bd=2c-neighborly. ...
... There are tighter but more sophisticated bounds in the literature, see e.g.,[21,28]. ...
... This dream was finally realized by Adiprasito in [2]. Other recent applications of linear and affine stresses to the fvector theory and especially to the lower-bound-type questions include [1,4,19,30], to name just a few; see also the results on the g-vectors of flag PL spheres in this paper. We encourage the reader to think of further potential applications of spaces of affine stresses. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/281356929126411-1444092127822_Q64/Eran-Nevo.jpg)
... Recently some progress has been made on understanding the maximum possible number of vertices that a cs-2neighborly d -polytope can have. Although we still do not know the exact value, we now know it up to a factor of two: for d 2, there exists a cs d -polytope with 2 d 1 C 2 vertices that is cs-2-neighborly [70]; on the other hand, for d 3, no cs polytope with 2 d or more vertices can be cs-2-neighborly [53]. Written in terms of k.d; n/, this result says that, for d 3, k.d; n/ 2 for all n Ä 2 d 2 C 1 d while k.d; n/ D 1 for all n 2 d 1 d . ...
