SPEAKERS, TITLES, AND ABSTRACTS OF TALKS

Karim Adiprasito (adiprasito@math.fu-berlin.de, adiprasito@ihes.fr)
Many projectively unique polytopes
Abstract
I describe some new constructions for projectively unique polytopes based on PDE and oriented matroid theory, and use them to disprove some classical problems of Shephard, Perles, McMullen and Legendre.

Toma Albu (Toma.Albu@imar.ro)
Geodesics in glued Riemannian Manifolds
Abstract
In this talk we illustrate the Fermat's principle "angle of reflection of a ray of light in the physical space equals angle of incidence" in glued Riemannian manifolds.

Sheng Bau (Sheng.Bau@wits.ac.za)
Reductions of quadrangulations of the sphere

Jürgen Bokowski (juergen@bokowski.de)
Selected open and solved problems in computational synthetic geometry
Abstract
"Computational Synthetic Geometry" was the title of my Springer Lecture Notes Volume 1355 with Bernd Sturmfels from 1989. During the last 25 years many combinatorial structures such as abstract point line configurations in the sense of Branko Grünbaum’s book from 2009, (d-1)-spheres of questionable convex d-polytopes, or regular maps have been studied in view of their possible geometric realization. Selected open and solved problems from those areas will be presented for which very often oriented matroids have played an essential role.

Vasile Brînzănescu (Vasile.Brinzanescu@imar.ro)
On the relationship between D'Angelo q-type and Catlin q-type
Abstract
Joseph J. Kohn proved in 1979 that for a pseudoconvex domain in C^n with real-analytic boundary the subellipticity of the d(bar)-Neumann problem for (p; q) forms is equivalent to the property that all holomorphic varieties of complex dimension q have finite order of contact with the boundary of the domain. Soon afterwards, John D'Angelo introduced a quantitative measure for this order of contact, presently known as the D'Angelo q-type, Delta_q . David Catlin was working on extending Kohn's result to smooth pseudoconvex domains and introduced another order of contact. Catlin's notion is what became known as Catlin q-type, D_q. In this paper we relate Delta_q with D_q for q > 1 . (with Andreea Nicoara).

Cristian Nicolae Costinescu (ccostin@utcb.ro)
Cohomology with values in a special sheaf
Abstract
Firstly we present the basic properties of the cellular constant sheaves (introduced by the author in his PhD thesis). The purpose of this article is the calculation of the cohomology of a standard simplex with coefficients in a cellular constant sheaf; we give explicit formulas.

Gheorghe Crăciun (craciun@math.wisc.edu)
Persistence and Global Stability in Biological Interaction Networks

Jürgen Eckhoff (juergen.eckhoff@tu-dortmund.de)
The minimum number of triangles in graphs
Abstract
The problem of determining the smallest possible number of triangles (3-cliques) in a graph with n vertices and m edges is one of the oldest problems in extremal graph theory. Possibly because of the absence of a plausible conjecture, the problem is still open. I now am somewhat confident to have found a solution.

Rafael Espínola (espinola@us.es)
Diversities and fixed points

Boumediene Et-Taoui (boumediene.ettaoui@uha.fr)
Equi-isoclinic planes in Euclidean odd dimensional spaces and complex symmetric conference matrices of odd orders
Abstract
A n-set of equi-isoclinic planes in R^r is a set of n planes spanning R^r each pair of which has the same non-zero angle arccos\sqrt\lambda. Let v_\lambda(2; r) denote the maximum number of equi-isoclinic planes with angle arccos\sqrt\lambda in R^r. We prove that for any odd integer k ≥ 3 such that 2k = p^alpha + 1, p odd prime, alpha non-negative integer, v_1/(2k-2)(2; 2k - 1) = 2k - 1. The solution of this geometric problem is obtained by the construction of complex symmetric conference matrices of order 2k - 1. A complex n X n conference matrix Cn is a matrix with cii = 0 and |c_ij| = 1, i≠j that satisfies CC*= (n - 1)In.
Apart from their appearance in geometry complex conference matrices also have important applications in coding and quantum information theories.

Reinhardt Euler (Reinhardt.Euler@univ-brest.fr)
Modeling the Geometry of the Endoplasmic Reticulum Network
joint work with Laurent Lemarchand, Congping Lin and Imogen Sparkes
Abstract
We study the network geometry of the endoplasmic reticulum by graph theoretical and integer programming models. We determine plane graphs of minimal total edge length satisfying degree and angle constraints and we show that the optimal graphs are close to the ER network geometry. We use a binary linear program, that iteratively constructs an optimal solution, and a linear program, that iteratively exploits cutting planes within a branch-and-cut framework. All formulations were tested on real-life and randomly generated cases. The cutting plane approach turns out to be particularly efficient for the real-life testcases.

Gábor Fejes Tóth (fejes.toth.gabor@renyi.mta.hu)
Theorems of Erdös-Szekeres type for convex discs
Abstract
The fundamental theorem of Erdös and Szekeres states for every integer n ≥ 3 the existence of a smallest natural number f(n) such that any set of f(n) points in the plane in general position contains the vertices of a convex n-gon. The talk gives a survey on generalizations of this theorem for families of convex discs.
A family of convex discs is in convex position if no member of the family is contained in the convex hull of the others. The definition of "general position" can be extended for families of convex discs by assuming that any three members of the family are in convex position. This assumption implies that every suciently large family of mutually disjoint convex discs contains n members in convex position. Another way defining general position for families of convex disc is through the prop- erty that to every pair of the family there is a line that intersects the given two but no other members of the family. This property and sufficiently large cardinality of a family of convex discs implies the existence of n members in convex position without the assumption that the discs are mutually disjoint. The talk is based on joint work with Tibor Bisztriczky,

Louis Funar (louis.funar@ujf-grenoble.fr)
Tameness for manifolds of non-positive curvature
Abstract
We will discuss about various tameness conditions like QSF, geometric simple connectivity, semistability for non compact spaces and discrete groups, along with their metric refinements. We will give then some conjectural characterization of manifolds allowing complete CAT(0) metrics.

Jin-ichi Itoh (j-itoh@kumamoto-u.ac.jp)
Total torsion and curvature of open curves
Abstract
The total torsion (resp. curvature) of smooth curves in E^3 is defined as the integral of the absolute value of torsion (resp. curvature). This notion is extended to piecewise smooth curves. We study the infimum of the total torsion (resp. curvature) in a certain set of curves, where the endpoints, the osculating planes (resp. tangent directions) at the end points and the length are all prescribed. We show how the infimum is calculated from the boundary data and also discuss about the integral of square root of the sum of squared curvature and torsion, if possible.

János Kincses (kincses@math.u-szeged.hu)
On the Helly dimension of Hanner polytopes
Abstract
In the 50s O. Hanner introduced and investigated the convex bodies K with the (3, 2) intersection property (any 3 translates of K intersect whenever any 2 of them intersect). He proved that such a body must be a polytope and for each facet F of P it is true that P = conv(F ∪ −F). Later, Hansen and Lima (1981) gave the complete list of Hanner polytopes: they are exactly those polytopes which can be obtained from segments by repeated direct and l_1-sum. In this talk we present a method to calculate the Helly dimension of some Hanner polytopes. For the direct sum of convex sets, which is the natural operation for the Helly dimension, it is well known that
him(K1 ⊕ K2 ⊕ ... ⊕ Kn) = max_i himKi.
However, the Helly dimension of the l1-sum of convex sets is not determined by the Helly dimension of the summands only. Some years ago we gave sharp lower and upper bounds for the Helly dimension of the l_1-sum of convex sets:
himK1 + himK2 ≤ himK1 ⊗ K2,
himK1 ⊗ K2 ≤ min{(dimK1 + 1)(himK2 + 1) − 1, (himK1 + 1)(dimK2 + 1) − 1},
Now we improve slightly these bounds and we shall give the complete list of low Helly dimensional Hanner polytopes.

Peter Knorr (peter_knorr@gmx.net)
Regarding the lower bound of non-traceable simple 3-polytopes
Abstract
Prof. Zamfirescu once discovered a simple 3-polytopal graph of 88 vertices without any spanning path, for which no smaller example was found in decades. This lecture deals with a lower bound for the size of a graph with these properties and shows the strategies necessary to show that any of these graphs must have more than 52 vertices.

Wlodzimierz Kuperberg (kuperwl@auburn.edu)
Variations on the Hadwiger theme
Abstract
The (original) Hadwiger number of a convex body K is the maximum number H(K) of mutually non-overlapping translates of K, each touching K. Without losing the affine invariant nature of the Hadwiger number, several generalizations and modifications are proposed. One can replace translates of K with its t-homothetic images with a positive or negative t or a combination of both types. One can even consider a variation with t → ∞ or t → -∞ (in a certain sense) or a combination of both. Another possibility is to consider mutually non-overlapping t-homothetic images of K contained in K and touching the boundary of K from inside - a dual counterpart to the previous notions (here we should assume 0 < |t|≤2). Some examples will be shown in which the optimal arrangements are tight.

Shabnam Malik (shabnam.malik@gmail.com)
Hamiltonicity in Directed Toeplitz Graphs
Abstract
An (n X n) matrix A=(a_ij) is called a Toeplitz matrix if it has constant values along all diagonals parallel to the main diagonal. A directed Toeplitz graph is a digraph with Toeplitz adjacency matrix. In this talk I will discuss conditions for the existence of hamiltonian cycles in directed Toeplitz graphs.

Solomon Marcus (solomarcus@gmail.com)
Contemplating Tudor Zamfirescu as a man and as a mathematician

Luis Montejano (luis@matem.unam.mx)
Transversals to the family of the convex hull of k-sets
Abstract
What is the maximum positive integer $n$ such that every set of $n$ points in $\mathbb{R}^{d}$ has the property that the convex hulls of all $k$-set have a transversal $(d-\lambda )$-plane? In this paper, we investigate this and closely related questions. We define a special \emph{Kneser hypergraph} by using some topological results and the well-known $\lambda $-\emph{Helly property}. We relate our question with the chromatic number of the \emph{% Kneser hypergraph }and we establish a connection $(\lambda =1)$ with the so called Kneser% This problem is all connected with the Gale embeddings, the discrete version of the Rado’s Problem, and with cyclic polytopes.

Adriana Nicolae (adriana.nicolae@imar.ro)
Finding Lipschitz extensions continuously
Abstract
In this talk we focus on parameter dependence of extensions of Lipschitz mappings from the point of view of continuity in geodesic spaces of bounded curvature and in hyperconvex metric spaces.

János Pach (pach@cims.nyu.edu)
From distances to graphs and back Abstract
Take n>2 points in the plane such that the maximum distance between them is 1. At most how many pairs of points can be at unit distance? This question and its generalizations by Erdos and his collaborators have have motivated a lot of research in extremal graph and hypergraph theory. However, combinatorial and algebraic methods appear to be insufficient in solving many problems concerning graphs defined in geometric terms. We mention some problems and results that fall into this category. A sample question, partially solved by Konrad Swanepoel and myself: Given n points in d-space, what is the maximum number of "double-normal" pairs {p,q} with the property that all points lie between the hyperplanes that pass through p and q and are orthogonal to pq? For d=3, the answer is (1/4+o(1))n^2. We do not have an asymptotically tight solution in 4 dimensions.

D. Filip, A. Petruşel (petrusel@math.ubbcluj.ro)
Contributions of Tudor Zamfirescu to Fixed Point Theory

Michel Pocchiola (pocchiola@math.jussieu.fr)
A projective version of the Hadwiger's transversal theorem

Mihai Prunescu (mihai.prunescu@gmail.com, mihai.prunescu@imar.ro)
About a surprizing computer program of Matthias Müller
Abstract
Matthias Müller published in WWW a computer program that seems to decide the 3-SAT solvability in polynomial time. I make an analysis of what the program really does and I propose some improvements of the algorithm. In particular, I prove that a 3-SAT instance is solvable if and only if it avoids a maximal clique in the binom{n}{3}-partite graph of all possible clauses with n variables, with the relation "A and B are connected iff they do not conflict". It follows that the program always answers "true" for satisfiable 3-SAT instances. The question, if the program always answers "false" for unsolvable instances, remains open.

Alain Rivière (alain.riviere@u-picardie.fr)
Hausdorff dimension of the set of endpoints of typical convex surfaces
Abstract
We mainly prove that most d-dimensional convex surfaces Sigma have a set of endpoints of Hausdorff dimension at least d/3.
An endpoint is a point not lying in the interior of any shortest path in Sigma. ''Most'' means that the exceptions constitute a meager set, relatively to the usual Hausdorff-Pompeiu distance.
It is still an unsolved question, as much as we know, if this Hausdorff dimension can be greater than d/3, for a convex surface not necessary typical. Our result here is an estimate in this direction.

Edgardo Roldàn-Pensado (e.roldan@math.ucl.ac.uk)
Measure partitions with hyperplanes
Abstract
We study partitions of R^d obtained by successive cuts using hyperplanes with fixed directions. This generalises classical necklace splitting results. With similar methods we obtain a piecewise-linear version of the polynomial ham-sandwich theorem in the plane, and mass-partitioning results for chessboard colourings.

Joël Rouyer (Joel.Rouyer@ymail.com)
Geometry of most Alexandrov spaces
Abstract
An Alexandrov surface (with curvature bounded below) is a closed topological surface endowed with an intrinsic metric for which the Topogonov's Theorem holds (here, considered as an axiom). The space of all compact Alexandrov surfaces, endowed with the Gromov-Hausdorff metric, is a Baire space.
Classical examples of Alexandrov surfaces are convex surfaces and closed Riemannian surfaces. Typical properties of convex surfaces are for long studied, but the investigation of typical properties of Alexandrov surfaces is very recent.
After introducing Alexandrov surfaces and their space, I will present some of their typical properties, concerning endpoints, conical points, Gaussian curvature, simple closed geodesics, and farthest points.

Dominique Schmitt (Dominique.Schmitt@uha.fr)
Separation by convex pseudo-circles
Abstract
Let S be a finite set of n points in the plane in general position. We prove that every inclusion-maximal family of subsets of S separable by convex pseudo-circles has the same cardinal binom(n,0)+binom(n,1)+binom(n,2)+binom(n,3). This number does not depend on the configuration of S and is the same as the number of subsets of S separable by true circles. Buzaglo, Holzman, and Pinchasi showed that it is an upper bound for the number of subsets separable by (non necessarily convex) pseudo-circles.
Actually, we first count the number of elements in a maximal family of k-subsets of S separable by convex pseudo-circles, for a given k. We show that Lee's result on the number of k-subsets separable by true circles still holds for convex pseudo-circles. In particular, this means that the number of k-subsets of S separable by a maximal family of convex pseudo-circles is an invariant of S: It does not depend on the choice of the maximal family.
To prove this result, we introduce a graph that generalizes the dual graph of the order-k Voronoi diagram, and whose vertices are the k-subsets of S separable by a maximal family of convex pseudo-circles. In order to count the number of vertices of this graph, we first show that it admits a planar realization which is a triangulation. It turns out that these triangulations are the centroid triangulations Liu and Snoeyink conjectured to construct.

Rolf Schneider (rolf.schneider@math.uni-freiburg.de)
Recent Baire category results in convexity

J.M.S. Simões-Pereira (siper@mat.uc.pt)
Just to prove a result or to prove it and then to contemplate it?
Abstract
Sometimes applied math students like to materialize a result and see it, not only to know that it is true. As an example, let us look at the truly classic isomorphism between the group of the icosahedron rotations and the A5 permutation group.

Zdzisław Skupień (skupien@agh.edu.pl)
Doubly extremal graphs
Abstract
The topic stems from my recent publication: Z.S., Majorization and the minimum number of dominating sets, DAM 165 (2014) 295--302. Among n-vertex forests without edge/vertex isolates for n>9 and n≠ 38, there is a unique UDSS (union of disjoint subdivided stars) which has the largest [smallest] numder of efficent [total] dominating sets. Hence the two extremized domination counts, max-efficient and min-total, are dual in a sense.

David White (David.White@wits.ac.za)
The Decycling Numbers of Toeplitz Graphs
Abstract
I will be answering the question of whether or not the decycling numbers of the families of Toeplitz and Cayley graphs of order n satisfy the intermediate value property. In particular, I will outline a proof by construction that for each n ≥ 3, and for each r, with 0 ≤ r ≤ n - 2, there exists a Toeplitz graph of order n with decycling number r. Subject to time, I will also discuss a proof that the decycling numbers of connected Cayley graphs of order n satisfy the intermediate value property if and only if n = 4 or 6.

Liping Yuan (lpyuan88@yahoo.com)
Right Triple Convexity
Abstract
A set $M$ in $\mathds{R}$ is $rt$-convex if every pair of its points is included in a 3-point subset $\{x,y,z\}$ of $M$ satisfying $\angle xyz=\pi/2.$ We characterize $rt$-convex sets, and investigate $rt$-convexity for 2-connected polygonally connected sets, for 3-connected sets, for geometric graphs, and for finite sets. Furthermore, we find here for various sets the minimal number such that, by suitably adding to them that number of points, we get $rt$-convex sets.

Christina Zamfirescu (zamfichris@gmail.com)
Graphs, Applications and Questionnaires
Abstract
The presentation will start with a few remarks on the contribution of my brother, Tudor Zamfirescu, to my research. The first part will review results obtained in transformation digraphs, and their connection with intersection digraphs, with applications to complexity evaluation to chemical molecular structures. In the second part, I will deal with research done in cooperation with Dr Ioana Schiopu-Kratina from University of Ottawa, applying graph theoretical techniques to questionnaire design.