ALGEBRAIC AND GEOMETRIC COMBINATORICS ON
LATTICE POLYTOPES

July 30–August 3, 2018, Osaka, JapanPer Alexandersson
The integer decomposition property and Gelfand-Tsetlin polytopes
We examine the integer decomposition property (IDP) of Gelfand-Tsetlin polytopes. The integral property and the IDP have a nice interplay with a certain
partial order on the family of GT polytopes. Furthermore, in the case corresponding to standard Young tableaux, we can completely classify which polytopes areintegral, and in this case they also have a regular unimodular triangulation.
If there is time, we also discuss some large counterexamples to natural conjectures.

Elie Alhajjar
Numerical Semigroups and Kunz Polytopes
A numerical semigroup S is an additive submonoid of N whose complement isfinite. The cardinality of N0 \ S is called the genus of S and is denoted by g(S).
The first nonzero element of S is called the multiplicity of S and is denoted bym(S). In this talk, we focus on the number N (m, g) of numerical semigroups withparameters m and g. It is known that N (m, g) can be formulated as the numberof integer points in a certain family of rational polytopes and hence coincides witha quasi-polynomial in g of degree m − 2. We show that the leading coefficient isconstant and provide an interpretation for it. Moreover, we relate N (m, g) to thenumber M ED(m, g) of maximally embedded numerical semigroups with the same
parameters. We finish the talk by discussing the special case m = 4.

Gabriele Balletti
How connected are the skeletons of polytopes?
Connectivity of a graph measures how easy it is to disconnect a graph by deleting vertices. For example, we say that a cycle is 2-connected because you cannot
disconnect it by removing less than 2 vertices. This generalizes to the skeletons ofpolytopes: a result by Balinsky (1961) shows that you need to remove at least d1

vertices to disconnect the skeleton of a d-dimensional polytope. Barnette (1982) generalized this result to simplicial pseudomanifolds, and Athanasiadis (2011) provedthat you need to remove 2d vertices in order to disconnect the skeleton of a flag ddimensional simplicial pseudomanifold. Using commutative algebra tools, we show
a surprisingly easy proof of a generalization of those results.

Mónica Blanco
Lattice 3-polytopes: quantum jumps and interior points
For P and Q lattice 3-polytopes, the pair (P, Q) is a quantum jump if P ⊂ Q and
Q \ P contains exactly one lattice point. In this talk I will put together all we know
on quantum jumps in dimension 3, and which information this gives us on lattice
3-polytopes with interior lattice points.

Christopher Borger
The mixed volume and the mixed degree of a family of lattice polytopes
The mixed volume of n-dimensional lattice polytopes P1 , . . . , Pn can be seen asa generalization of the lattice volume of a single lattice polytope. It has a natural
algebraic interpretation as it equals to the generic number of solutions of a system
of Laurent polynomials f1 , . . . , fn with associated Newton polytopes P1 , . . . , Pn . Ina similar way the mixed degree is a quite recent proposal of a generalization of thelattice degree of a single polytope to a family of lattice polytopes. We give an insight
into the various questions regarding these two notions such as classifications in fixed
dimensions, relations among mixed volumes of certain families and deriving upperbounds for the volume of the Minkowski sum P1 + · · · + Pn in terms of the mixed
volume of P1 , . . . , Pn .

Federico Castillo
Deformation cones for polytopes
Given a lattice polytope, the set of all polytopes having the same (or a coarsening)normal fan is a polyhedral cone. This cone has appeared in different contexts, forexample it is closely related to the nef cone of the associated toric variety. In the caseof the regular permutohedron we get the cone of submodular functions. The purposeof this talk is to survey known results and show how to compute this deformationcones in further combinatorial examples. This is joint work with Fu Liu, Federico
Ardila, and Alex Postnikov.
2Daniel Cavey
Classification of Minimal Polygons with Specified Singularity Content
It is known that there are only finitely many mutation-equivalence classes with agiven singularity content, and each of these equivalence classes contains only finitely
many minimal polygons. In this talk I describe the notions of mutation and singularity content, and their relevance to the classification of del Pezzo surfaces admittinga toric degeneration.

Giulia Codenotti
The covering minima of lattice polytopes
The covering minima of a convex body were introduced by Kannan and Lovasz
to give a better bound in the flatness theorem, which states that lattice point freeconvex bodies cannot have arbitrarily large width. These minima are similar in flavor
to Minkowski’s successive minima, and on the other hand generalize the coveringradius of a convex body. I will speak about ongoing joint work with Francisco Santos
and Matthias Schymura, where we investigate extremal values of these covering
minima for lattice polytopes.

Brian Davis
Predicting the Integer Decomposition Property via Machine Learning
In this talk we describe the use of neural networks in approximating algebraic
properties associated to lattice simplices. In particular we predict the distributionof Hilbert basis elements in the fundamental parallelepiped, from which we detect the
Integer Decomposition Property (IDP). We discuss the results of the approximation
method when scanning very large test sets for examples of IDP simplices faster thanis possible when computing with Normaliz.

Laura Escobar
Wall-crossing phenomena for Newton-Okounkov bodies
A Newton-Okounkov body is a convex set associated to a projective variety,equipped with a valuation. These bodies generalize the theory of Newton polytopes. Work of Kaveh-Manon gives an explicit link between tropical geometry and
Newton-Okounkov bodies. We use this link to describe a wall-crossing phenomenon for Newton-Okounkov bodies. As an application we show how the wall-crossing
3formula for the tropicalization of Gr(2, n) is an instance of our phenomenon for
Newton-Okounkov bodies. Joint work with Megumi Harada.

Magda Hlavacek
Dehn-Sommerville equations for cubical polytopes
The classical Dehn-Sommerville equations, relating the face numbers of simplicial
polytopes, have an analogue for cubical polytopes. We show how to obtain these
relations by considering the zeta polynomials of the face lattices of cubical polytopes.
We also explore how these ideas connect to chain partitions of these face lattices.

Óscar Iglesias Valiño
The complete classification of empty lattice 4-simplices
In previous work we classified all empty 4-simplices of width at least three. We
here classify those of width two. There are 2 two-parameter families that projectto the second dilation of a unimodular triangle, 29 + 23 one-parameter familiesof them that project to hollow 3-polytopes, and 2282 individual ones that do not.
These families complete the classification of empty 4-simplices. This is joint workwith Francisco Santos.

Lukas Katthän
The lecture hall cone as a toric deformation
The lecture hall cone is a particular simplicial cone whose lattice points correspond to so-called Lecture Hall Partitions. The celebrated Lecture Hall Theorem of
Bousquet-Mélou and Eriksson states that the generating function of these partitions
factors in a very nice way. Over the years, several proofs of this result have beenfound, but it is still not considered to be well-understood from a geometric perspective. In my talk I am going to present a conjecture where the Ehrhart ring of the
Lecture Hall cone is realized as toric deformation of a simpler ring. In particular,
this conjecture would imply the Lecture Hall theorem.

Florian Kohl
Levelness of Order Polytopes
Since their introduction by Stanley order polytopes have been intriguing mathematicians as their geometry can be used to examine (algebraic) properties of finite
posets. In this talk, we follow this route to examine the levelness property of order4

polytopes. The levelness property was also introduced by Stanley and it generalizes
the Gorenstein property. This property has been recently characterized by Miyazaki
for the case of order polytopes. We provide an alternative characterization using
weighted digraphs. Using this characterization, we give a new infinite family of level
posets and show that determining levelness is in co-NP. This family can be used tocreate infinitely many examples illustrating that the levelness property can not becharacterized by the h∗ -vector. This is joint work with Christian Haase and Akiyoshi
Tsuchiya.

Yonggyu Lee
Geometrical structure of Tesler polytopes
Tesler polytope is the set of upper triangular matrices with non negative entries
satisfying certain equations called hook sum conditions. We will show that the Tesler
polytopes of positive hook sums are all combinatorially isomorphic to each otherand discuss about how to apply this result to the conjecture of Ehrhart positivityof Tesler polytope of hook sum (1, . . . , 1). This is joint work with Fu Liu.

Sebastian Manecke
Gram’s relation for cone valuations
We study linear relations of interior and exterior angle vectors of polytopes, where
we generalize the usual solid angle with simple cone valuations. For both interior
and exterior angles, we prove that only one linear relation exists for any cone angle,
one being Gram’s relation.
The uniqueness of the relation follows from a connection between angle sums
and the combinatorics of zonotopes. The angle sum vectors are given by the CoWhitney numbers of the lattice of flats associated to the zonotope. This shows thatsurprisingly the angle-sum vectors of zonotopes are independent of the notion ofangle used.
This is joint work with Spencer Backman and Raman Sanyal.

Andrés Vindas Meléndez
Fixed Subpolytopes of the Permutahedron
Motivated by the generalization of Ehrhart theory with group actions, this projectmakes progress towards obtaining the equivariant Ehrhart theory of the permutahedron. The fixed subpolytopes of the permutahedron are the polytopes that are fixedby acting on the permutahedron by a permutation. We prove some general results
5about the fixed subpolytopes. In particular, we compute their dimension, show thatthey are combinatorially equivalent to permutahedra, provide hyperplane and vertex
descriptions, and prove that they are zonotopes. Lastly, we obtain a formula for thevolume of these fixed subpolytopes, which is a generalization of Richard Stanley’s result of the volume for the standard permutahedron. This is joint work with Federico
Ardila (San Francisco State) and Anna Schindler (University of Washington).

Makoto Miura
Hibi toric varieties and mirror symmetry
A Hibi toric variety is the projective toric variety corresponding to the order polytope of a finite poset, which equips rich combinatorial descriptions inherited fromthe poset. The varieties also appear as the special fibers of certain toric degenerations of Grassmannians or some other Gorenstein Fano varieties. This fact is
helpful to study the geometry of complete intersection Calabi–Yau 3-folds in such
Fano varieities and to construct conjectural mirror manifolds for them via conifold
transitions.
Our aim in this talk is to see how well the combinatorics of Hibi toric varieties
works for the study of certain kinds of Calabi–Yau 3-folds and their mirror symmetry.

Louis Ng
Magic Counting with Inside-Out Polytope
In this paper, we investigate strong 4 × 4 pandiagonal magic squares (no entries
repeat; equal row sums, column sums, and diagonal sums, including the wraparound ones), strong 5 × 5 pandiagonal magic squares, weak 2 × n magic rectangles(repeating entries allowed; equal row sums and equal column sums), and 2 × n
magilatin rectangles (no entries repeat in a row/column; equal row sums and equalcolumn sums). The magic counts depend on the magic sum or the upper bound of theentries. We compute their counting quasipolynomials and the associated generating
functions by using a geometrical interpretation of the problems, considering themas counting the lattice points in polytopes with removed hyperplanes.

Jorge Alberto Olarte
Transversal valuated matroids and their presentation space
The tropical Stiefel map sends a matrix with tropical entries to the valuatedmatroid given by its maximal tropical minors. We say that a valuated matroid istransversal if it arises this way. We show that this is indeed a generalization of6

transversal matroids and that a valuated matroid is transversal if and only if the all
the facets of the polytopal complex dual to the corresponding tropical linear spaceare polytopes of transversal matroids. Given a transversal valuated matroid, wecall a matrix in the preimage under the tropical Stiefel map a presentation and thepreimage itself the space of presentations. The rows of any presentation are always
points in the corresponding tropical linear space. So we show how to choose points
in the tropical space such that they form the rows of a presentation and that the
space of presentations is a product of fans modulo permutation of rows.

McCabe Olsen
Level algebras and lecture hall polytopes
Given a family of lattice polytopes, a common question in Ehrhart theory is classifying which polytopes in the family are Gorenstein. A less common question isclassifying which polytopes in the family admit level semigroup algebras, a generalization of Gorenstein algebras. In this article, we consider these questions for
s-lecture hall polytopes, which are a family of simplices arising from lecture hallpartitions. We provide a complete characterization of the Gorenstein property for
lecture hall polytopes. We provide a different, more geometric characterization of
the Gorenstein property which applies to a large subfamily of lecture hall polytopes.
Additionally, we also give a complete characterization for the level property in termsof s-inversion sequences and demonstrate some consequences of the classification result. This is joint work with Florian Kohl.

Marta Panizzut
K3 polytopes and their quartic surfaces
The closure of the connected components of the complement of a tropical hypersurface are called regions. The have the structure of convex polyhedra and theymight be bounded or unbounded. A 3-dimensional polytope is a K3 polytope if itis the closure of the bounded region of a smooth tropical quartic surface.
In this talk we begin by studying properties of K3 polytopes. In particular we
exploit their duality to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Then we focus on quartic
surfaces that tropicalize to K3 polytopes, and we look at them through the lensesof Geometric Invariant Theory.
This is a joint work with Gabriele Balletti and Bernd Sturmfels.

Irem Portakal
7Torus actions on matrix Schubert varieties
Matrix Schubert Varieties were first studied by Fulton in 1990s. He gave a nicecombinatorial description of these varieties in terms of Rothe diagrams. Our aim
is to understand the natural effective torus action on these varieties. It turns outthat this can be achieved by bipartite graphs. In particular, for the toric case, we
describe the face structure of the polyhedral cone in terms of graphs.

Maren Ring
McMullen’s formulas for Ehrhart coefficientsThe Ehrhart polynomial counts the number of lattice points in dilates of a polytope and gives us a useful connection between volumes and discrete points. Determining the coefficients of this polynomial can be done via so-called McMullen’s
formulas (aka local formulas), which give the i-th coefficient as a weighted sum of thevolumes of the i-dimensional faces of the polytope. Based on choices of lattice cells
we construct a new local formula that provides us with additional knowledge about
Ehrhart coefficients and can be used for the exploitation of polyhedral symmetries.
This is joint work with A. Schürmann.

Benjamin Schröter
Reconstruction of Matroid Polytopes
Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is reconstructable by its 1-skeleton, i.e., determined by its vertex-edge graph.
Joswig extended this result to special cubical polytopes and recently Doolittle etal. presented explicit sufficient conditions when a polytope is reconstructable fromits k-skeleton in terms of non-simple vertices. In contrast to all these results, aconstruction of Perles shows that for every d > 3 there are d-polytopes which are
not reconstructable from their (d − 3)-skeleta.
In my talk I will focus on matroid polytopes which are a particular class of 0/1polytopes. Matroids are fundamental combinatorial objects as they encode an abstract version of dependency and independency. Maximal independent sets satisfy
a basis-exchange. An edge in the basis-exchange graph corresponds to such an exchange and the basis-exchange graph is exactly the vertex-edge graph of the matroid
polytope. I show that the basis-exchange graph almost determines a matroid andhence a matroid polytope is reconstructable from its graph.
This is work in progress with Guillermo Pineda-Villavicencio.

8

Liam Solus
Real-rootedness, Unimodality, and Symmetric Decompositions of Polynomials
A central line of research in algebraic and geometric combinatorics concerns itself
with the investigation of when certain polynomials with nonnegative integer coefficients are unimodal; i.e. when their sequence of coefficients is unimodal. One way
to prove such a polynomial is unimodal is to show that it has only real roots. A
second way is to decompose it as a sum of unimodal polynomials whose unimodality
in turn implies that of the original polynomial. In this talk, we will combine these
two approaches and discuss how real-rootedness techniques play nicely with somedecompositions of polynomials into symmetric polynomials. We will then apply
these techniques to extend some results and answer some open questions on various families of Ehrhart h∗ -polynomials, local h-polynomials, and h-polynomials of
simplicial complexes.

Johanna Steinmeyer
No small Counterexamples to Perles’ Conjecture
In a celebrated theorem, Blind and Mani proved that the entire combinatorial
structure of a simple polytope is determined by its vertex-edge graph. Shortly
after, Kalai gave an elegant and constructive proof of the same result. Looking
to generalize this, Perles asked the following question: In the graph of a simple
d-polytope, is every (d-1)-regular, connected, induced, non-separating subgraph thegraph of a facet? This conjecture was disproved by Haase and Ziegler, but it remains
unknown how to locally characterize those subgraphs that correspond to the facetsof the polytope. We discuss a systematic approach to showing that certain classesof graphs can not yield a counterexample to Perles’ Conjecture when they occur as
subgraphs.

Yusuke Suyama
Toric Fano varieties associated to graph cubeahedra
The graph cubeahedron of a finite simple graph is a smooth polytope obtained
from a cube by truncating the faces corresponding to connected induced subgraphs
in increasing order of dimension. In this talk, we give a necessary and sufficientcondition for the nonsingular projective toric variety associated to the graph cubeahedron to be Fano or weak Fano in terms of the graph.

Bach Tran
9On k-normality and Regularity of Normal Projective Toric Varieties
We will give a bound for a very ample lattice polytope to be k-normal. Equivalently, we give a new combinatorial bound for the Castelnuovo-Mumford regularity
of normal projective toric varieties

Akiyoshi Tsuchiya
Cayley sums and Minkowski sums of 2-convex-normal lattice polytopes
In this talk, we discuss the integer decomposition property for Cayley sums andMinkowski sums of lattice polytopes. In particular, we focus on these constructions arising from 2-convex-normal lattice polytopes. Moreover, we discuss the levelproperty of Cayley sums and Minkowski sums.

Lena Walter
The Kingman Coalescent as a Density on a Space of Trees
Randomly pick n individuals from a population and trace their genealogy backwards in time until you reach the most recent common ancestor. The Kingman
n-coalescent is a probabilistic model for the tree one obtains this way. The common
definitions are stated from a stochastic point of view. The Kingman Coalescent canalso be described by a probability density function on a space of certain trees. For
the space of phylogenetic trees Ardila and Klivans showed, that this space is closely
related to the Bergman fan of the graphical matroid of the complete graph. Using
its fine fan structure, similar things can be said about the space we are considering. I
will describe the density and report on work in progress with Christian Haase about
relations of population genetics and algebraic geometry via the tropical connection.

Hailun Zheng
A lower bound theorem for centrally symmetric simplicial polytopes
An important invariant in the study of face numbers of simplicial d-polytopes
is the g-vector. The generalized lower bound theorem states that gi ≥ 0 for any
simplicial polytope and characterizes the case of equality. Much less is known for
centrally symmetric polytopes. A seminal work is established by Stanley thirty
years ago, where he proved that for any centrally
symmetric
simplicial d-polytopedd
P with d ≥ 3 and 1 ≤ i ≤ d/2, gi (P ) ≥ i − i−1 . In this talk, I will introduce
the rigidity theory of frameworks, and show how to apply this machinery to givea characterization of centrally symmetric d-polytopes with d ≥ 4 that satisfy g2 =d− d. This is joint work with Steve Klee, Eran Nevo and Isabella Novik.
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