ALGEBRAIC AND GEOMETRIC COMBINATORICS ON
CONVEX POLYTOPESJanuary 16–20, 2017, Osaka, Japan

Gabriele Balletti
On the maximal dual volume of a lattice polytope with one interior point
It is well known that the volume of a canonical Fano lattice polytope P (i.e alattice polytope with one interior lattice point) of some fixed dimension is bounded.
A sharp bound has been conjectured, but never been proved. In a joint work withA. Kasprzyk and B. Nill we prove that such sharp bound is valid (and sharp) whenthe volume of the dual P is considered. As a corollary we have a sharp bound forthe volume of reflexive polytopes.

Katy Beeler
Generalized (0, 1, 2)-polytopes
0/1-polytopes and half-integral polytopes are important polytopal families in linear programming and combinatorial optimization. While 0/1-polytopes have been
fairly well studied and, in particular, are known to satisfy the Hirsch conjecture, notmuch is know about half-integral polytopes. Linearly isomorphic to the half-integral
polytopes are the (0, 1, 2)-polytopes, polytopes with vertex-coordinates in {0, 1, 2}.
In this talk I will present different generalizations of (0, 1, 2)-polytopes and in particular, an enumeration of all (0, 1, a)-polytopes in dimension three (polytopes withvertex-coordinates in {0, 1, a}). Additionally, I will prove that the optimal boundfor the diameter of (0, 1, a)-polytopes is b 23 dc, for any dimension d.

Mónica Blanco
Sublattice index of lattice 3-polytopes
Joint work with Francisco Santos
We call a lattice d-polytope P primitive if P ∩ Zd is an affine lattice basis for Zd .
More generally, we call sublattice index of P the index, as a sublattice of Zd , of theaffine lattice generated by P ∩ Zd . I will show that the complete classification of
non-primitive lattice 3-polytopes is as follows:1• Non-primitive lattice 3-polytopes of width one can have any index, but areeasy to classify since all their lattice points lie in two segments.
• Non-primitive lattice 3-polytopes of width larger than one can only have
sublattice index 2, 3 or 5 and there are quadratically many for each numberof lattice points in P . More precisely: there is a unique polytope of index
5 (a “terminal tetrahedron” of volume 20), and those of indices 2 and 3fall, apart of a few exceptions with at most eight lattice points, under a few
infinite families projecting to a specific (very short) list of lattice polygons.
Finally, we also show that all primitive 3-polytopes, except for two specific “terminaltetrahedra”, contain a unimodular tetrahedron.

Federico Castillo
Ehrhart positivity
We call a polytope P positive, if its Ehrhart polynomial has all positive coefficients.
In the literature there are different examples using different techniques to provepositivity. We will survey some of these results. It turns out, through work ofDanilov, McMullen, and others, that there is an interpretation of the coefficients
relating to the normalized volume of faces. We try to make this relation more
explicit in the particular case of the regular permutohedron. The goal, long term,
is to prove positivity for all generalized permutohedra. This is joint work with Fu
Liu.

Simon Hampe
The intersection ring of matroids
We study a particular graded ring structure on the set of all loopfree matroids ona fixed labeled ground set, which occurs naturally in tropical geometry. The productis given by matroid intersection and the additive structure is defined by assigningto each matroid the indicator vector of its chains of flats. This ring has many
striking properties, which we will outline in this talk. First and foremost, Derksen’s
G-invariant, and thus many more matroid invariants (e.g., the Tutte polynomial),
induce linear maps from this ring. One can show that the matroids form the verticesof a polytope in the the ring, so many extremal problems in matroid theory can betranslated to optimization problems. The talk will also include a very brief excursion
into tropical geometry, demonstrating the geometric intuition behind these ideas.

Jan Hofmann2

The finiteness threshold width of lattice polytopes
Trying to classify lattice polytopes, it seems natural to try to enumerate all of themby fixing the dimension and the number of lattice points. In this talk we will see thiswork marvellously in dimensions 1 and 2, but fail miserably from dimension 3 on.
One possible remedy for this is the finiteness threshold width of lattice polytopes,
which is defined as the width w∞ (d) beyond which there are only finitely manyd-dimensional lattice polytopes with a given number of lattice points and width> w∞ (d). In this talk we will see that w∞ (d) is finite and w∞ (4) = 2. This is joint
work with Mónica Blanco, Christian Haase and Francisco Santos.

Florian Kohl
Transfer-Matrix Methods meet Ehrhart Theory
Counting (proper) colorings of graphs is a classical problem in combinatorics
with connections to many different fields. In this talk, we want to examine proper
k-colorings of graphs of the form G × Pn , where G is any graph, and where Pn is thepath graph on n nodes. There are two special cases, namely (1) where k is fixed but
not n, and (2) where n is fixed but not k. In (1), the number of colorings can be
determined using the transfer-matrix method, and in (2), the number of colorings
can be counted using the chromatic polynomial/Ehrhart theory. We will use thesymmetry of G×Pn to combine the two methods to get explicit formulas (dependingon k and n) counting the number of colorings and we will give a restricted versionof the famous reciprocity theorem for the chromatic polynomial. Furthermore, wewill describe the doubly asymptotic behavior of graphs G × Cn , where Cn is thecycle graph with n nodes as both n and k go to infinity. If time permits, we will
explain how a similar method can be used to determine Ehrhart polynomials of orderpolytopes of a special class of posets. This is joint work with Alexander Engstroem.

Lauri Loiskekoski
Separation in the graph of a simple polytopeSeparation in graphs is related to graphs being expanders. A conjecture by Kalai
generalizes the planar separation theorem to simple polytopes. It states that thegraph of a simple d-polytope can be separated to two roughly equal parts by removd−2ing O(n d−1 ) vertices. We provide a counterexample to this conjecture. This is joint
work with Günter Ziegler.

Marie Meyer
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Polyhedral Constructions using the Laplacian Matrix
In this talk we will focus on two polyhedral constructions arising from the wellstudied Laplacian matrix of a graph. We will first look at a polyhedral proof of
the Matrix Tree Theorem presented by Dall and Pfeifle, which defines zonotopes
based on the columns of the Laplacian matrix of a graph. Then I will present a
new polyhedral construction, namely taking the convex hull of the columns of theLaplacian matrix to form a simplex. Finally I will discuss known properties of these
simplices according to graph type.

McCabe Olsen
Hilbert bases for Gorenstein lecture hall cones
Lecture hall partitions and related combinatorial objects have been the topic ofmuch recent study. The focus of this talk will be ongoing work in classifying theHilbert bases of certain Gorenstein lecture hall cones. In particular, we will discuss
the 1 mod k cones and `-sequence cones.

Benjamin SchröterSplit and k-Splits of the Hypersimplex
The study of regular subdivisions is an active field in mathematics. There aremany constructions for finest subdivisions, but only few for coarsest subdivisions.
Coarse subdivisions of the hypersimplex are of special interest in tropical geometry,
since tropical linear spaces are dual to such subdivisions. One class of coarsestsubdivisions are k-splits, invented by Sven Herrmann. I shall present a classificationof all ksplits for the hypersimplex and their strong relation to matroid theory. The
study of 2-splits leads to a new large class of matroids. As an application we give adimension bound for the moduli space of tropical linear spaces and a construction
for coarsest tropical linear spaces which are non realizable.

Hannah Sjöberg
Projections of the set of flag vectors of 4-polytopes
The set of f -vectors of 3-dimensional polytopes was fully described by Steinitz in1922. To characterize the set of f -vectors for higher dimensions is an open problem.
For dimensions higher than 3, the notion of flag vectors is useful. The flag vector
counts chains of faces. In this talk we will look at the cone of flag vectors of 4polytopes, projections of the set of flag vectors of 4-polytopes and generalizations
to higher dimensions.
4Liam Solus
Learning Bayesian Networks via Edge Walks on DAG Associahedra
The focus of this talk will be an application of convex polytopes to causal inference. Graphical models based on directed acyclic graphs (DAGs), also known as
Bayesian networks, are used to model complex cause-and-effect systems across a vast
number of research areas including computational biology, epidemiology, sociology,and environmental management. A DAG model is family of joint probability distributions over the nodes of a DAG G that entail a set of conditional independence (CI)relations encoded by the nonedges of G. A fundamental problem in causality is tolearn an unknown DAG G based only on a set of observed conditional independence
relations. Since multiple DAGs can encode the same set of CI relations, a property
termed Markov equivalence, the goal is to identify efficient algorithms that consistently recover a DAG within the correct Markov equivalence class. In this talk, wewill describe a pair of greedy algorithms for DAG model selection that operate viaedge walks on a family of generalized permutohedra called DAG associahedra. We
will present consistency guarantees for these algorithms, and compare them withthe more classical approaches to Bayesian model selection in both efficiency andstrength of satisfied identifiability assumptions.

Yusuke Suyama
Toric Fano varieties associated to building sets
We give a necessary and sufficient condition for the nonsingular projective toricvariety associated to the graphical building set of a finite simple graph to be Fano or
weak Fano in terms of the graph. Furthermore, we characterize building sets whose
associated toric varieties are Fano.

Akiyoshi Tsuchiya
Gorenstein simplices and the associated finite abelian groups
A lattice polytope is a convex polytope each of whose vertices has integer coordinates. It is known that a lattice simplex of dimension d corresponds a finite abeliansubgroup of (R/Z)d+1 . Conversely, given a finite abelian subgroup of (R/Z)d+1 suchthat the sum of all entries of each element is an integer, we can obtain a latticesimplex of dimension d. In this talk, we discuss a characterization of Gorenstein
simplices in terms of the associated finite abelian groups. Gorenstein polytopes are5

of interest in combinatorial commutative algebra, mirror symmetry and tropical geometry. In particular, we present complete characterizations of Gorenstein simplices
whose normalized volume equal p, p2 and pq, where p, q are prime integers.

Hailun Zheng
A characterization of simplicial manifolds with g2 ≤ 2
The celebrated low bound theorem states that any simplicial manifold of dimension ≥ 3 satisfies g2 ≥ 0, and equality holds if and only if it is a stacked sphere.
Furthermore, more recently, the class of all simplicial spheres with g2 = 1 wascharacterized by Nevo and Novinsky, by an argument based on rigidity theory forgraphs. In this talk, I will first define three different retriangulations of simplicial
complexes that preserve the homeomorphism type. Then I will show that all simplicial manifolds with g2 ≤ 2 can be obtained by retriangulating a polytopal spherewith a smaller g2 . This implies Nevo and Novinskys result for simplicial spheres ofdimension ≥ 4. More surprisingly, it also implies that all simplicial manifolds withg2 = 2 are polytopal spheres.

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