Omar Al - Khazali, (University of Colorado Boulder, USA)
Title: TBA
Abstract: TBA
Gabriel Currier, (University of British Columbia, CANADA)
Title: Unit distances from a restricted direction set
Abstract: The Erdos unit distance problem asks the following: given n points in the plane, what is the maximum possible number of pairs of these points that can be at distance 1 apart? We show a result of this type for restricted direction sets. That is, we show that a collection of n points in the plane can determine at most o(n^(4/3)) unit distances, if we count unit distances coming only from a collection of at most O(n^(1/3)) directions. This result uses tools from incidence geometry, graph theory, and additive combinatorics, and is joint work with Jozsef Solymosi.
Daniel Kalmanovich, (Ariel University, ISRAEL)
Title: On the Longest Edge Bisection process
Abstract: The Longest Edge Bisection (LEB) of a triangle is performed by joining the midpoint of its longest edge to the opposite vertex. Applying this procedure iteratively produces an infinite family of triangles. Surprisingly, a classical result of Stynes (1980) shows that for any initial triangle, the elements of this infinite family fall into finitely many similarity classes. While the variety of shapes is finite, a periodic subset of "fat" triangles, termed terminal quadruples, effectively dominates the final mesh structure. We prove that for every initial triangle, the portion of area occupied by these terminal quadruples tends to one at an exponential rate. By introducing the bisection graph and using spectral methods, we provide the precise distribution of triangles at every step, characterize triangles with a single terminal quadruple, and exhibit sequences with an unbounded number of quadruples. Furthermore, we classify the strongly connected components of the bisection graph, deducing that the graph is bipartite, and that its spectrum is $\{\pm2, \pm\sqrt{2}, \pm1, 0\}$. Studying the structure of the corresponding condensation graph allows us to classify the triangles that give rise to a diagonalizable process. Based on joint work with Yaar Solomon.
Nico Lorenz, (Ruhr-Universität Bochum, GERMANY)
Title: Representation Graphs of Quadratic Forms (joint with Marc Zimmermann)
Abstract: Let $q$ be a nondegenerate quadratic form on an $F$-vector space $V$ and $a \in F$. We consider the Cayley graph on $V$ with generating set $V_a = \{x \in V \setminus \{0\} \mid q(x) = a\}$, i.e. the graph with vertices $V$ in which two different vectors $x, y \in V$ are adjacent if and only if $q(x - y) = a$. This graph is called the \emph{representation graph of $q$ with respect to $a$}.
A first example of such graphs is the unit distance graph on the Euclidean plane. This graph is subject to research for over 70 years and of particular prominence is the Hadwiger-Nelson problem: determine its \emph{chromatic number}, i.e. the minimal number $k$ which is needed to assign one of $k$ colors to each vertex such that points of Euclidean distance 1 are not of the same color.
While most results about representation graphs in the literature were obtained using combinatorial techniques, we will show how algebraic methods enable us to reprove and extend known results. In particular, we will show how decompositions of quadratic forms and the orthogonal group can be used to determine graph invariants related to connectedness like \emph{clique number}, \emph{diameter} and \emph{girth}.
Mehdi Makhul, (London School of Economics (LSE), LONDON)
Title: On sets with many $d+1$-rich hyperplanes
Abstract: Let $S$ be a set of $n$ points in $\mathbb{R}^d$, not all lying on a single hyperplane. It is known that the number of hyperplanes that intersect $S$ in exactly $(d+1)$ points is bounded above by $O(n^d)$ for sufficiently large $n$, depending on $d$, with equality attained by cosets of subgroups of elliptic normal curves or by the smooth points of rational acnodal curves of degree $d+1$. In this work, we describe the structural properties of point configurations that achieve, or come close to achieving, this bound in the asymptotic regime. Specifically, if $S \subset C$, where $C \subset \mathbb{R}^d$ is an irreducible algebraic curve of degree $r$, and if $S$ determines at least $c n^{d - 1/6}$ $(d+1)$-rich hyperplanes for some constant $c > 0$, then $C$ must have degree $d+1$ and is the complete intersection of $\binom{d}{2} - 1$ quadrics. Our approach combines techniques from web geometry with the Elekes-Szab\'{o} theorem, providing a unified framework for the structural analysis of extremal configurations.
Marta Pavelka, (University of Copenhagen, DENMARK)
Title: Underclosed clutters and their duals
Abstract: Interval graphs play a central role in combinatorics and commutative algebra, and several higher-dimensional analogues have appeared in recent years. In this talk, we compare two such classes, cointerval hypergraphs and underclosed complexes, and show that they are dual to each other through complementation. We then place these objects in the broader landscape of higher-dimensional chordality by proving that every underclosed clutter is chordal in the sense of Woodroofe. As an application, we answer a question of Dochtermann and Engström by showing that the relevant Alexander dual complexes are not only shellable but in fact vertex decomposable and lex shellable. This also yields linear quotients for the associated circuit ideals. This is joint work with Anton Dochtermann and Bennet Goeckner.
Ahmad Rafiqi, (American University of Sharjah (AUS), Sharjah, UNITED ARAB EMIRATES)
Title: Teichmuller polynomial of closed surface homeomorphisms and algebraic units.
Abstract: A surface homeomorphism is typically pseudo-Anosov, and has an algebraic integer associated to it. The mapping torus M of such a homeomorphism is a fibered hyperbolic 3-manifold. M fibers in other ways, and the other fibrations also have algebraic integers associated to them. McMullen (2000) defined a multivariate polynomial invariant that encodes the defining polynomials for all nearby fibrations. We compute this polynomial invariant for closed surface homeomorphisms, and deduce a partial answer to a 40-year-old question regarding the nature of the algebraic integers that arise as stretch-factors.
Nil Şahin, (Bilkent University, Ankara, TURKEY)
Title and Abstract: Şahin_title_abstract
Fedor Selyanin, (HSE, RUSSIA)
Title: Negligible and thin polytopes
Abstract: The h*-polynomial of a lattice polytope encodes the number of lattice points in its integer dilations. The local h*-polynomial (or ℓ*-polynomial) arises naturally in the Katz–Stapledon decomposition formulas for the h*-polynomial in case of polyhedral subdivisions. A polytope P is called thin if ℓ*(P; 1) = 0.
According to the global Kouchnirenko's theorem, an affine hypersurface {f = 0} ⊂ Cn with a convenient Newton polytope P ⊂ Rn≥0 and non-degenerate coefficients has the homotopy type of a bouquet of ν(P) spheres of dimension n − 1. Here, ν(P) is a certain alternating sum of volumes, known as the Newton number. A convenient polytope P is called negligible if ν(P) = 0.
Following the paper arXiv:2507.03661, we will classify negligible polytopes as certain Cayley sums, called Bk-polytopes, using the Furukawa–Ito classification of dual-defective sets. By employing a generalization of the Katz–Stapledon decomposition formulas, we will show that for any convenient polytope P, the inequality ℓ*(P; 1) ≤ ν(P) holds. Consequently, negligible polytopes are thin.
Tanja Stojadinović, (University of Belgrade, SERBIA)
Title and Abstract: Stojadinoviç_title_and_abstract
Gökçen Dilaver Tunç, (Hacettepe University, Ankara, TURKEY)
Title and Abstract: Dilaver_title_and_abstract
Mathieu Vallée (Université Libre de Bruxelles, BELGIUM)
Title:
Totally equimodular matrices: decomposition and triangulation.
Abstract:
One way of obtaining strong combinatorial min-max theorems is to study if a system of linear inequalities Ax≤b is totally dual integral (TDI). The MaxFlow-MinCut theorem of Ford and Fulkerson is one fine example. In that specific case, the involved matrix A is totally unimodular (TU), meaning al its nonzero subdeterminants are ±1, and that implies that the system is TDI. In fact, since the matrix is TU, the system satisfies an even stronger property: it is box-TDI, that is, it remains TDI when we impose bounds to the variables. Totally equimodular matrices appear in this very context, as a generalization of totally unimodular matrices.
A full row rank matrix is equimodular (or unimodular) when all its nonzero subdeterminants of maximal size have the same absolute value, this value is called the equideterminant of the matrix. A matrix is totally equimodular (TE) if all its submatrices are equimodular when they are of full row rank. When a matrix is TU, it is TE, and all the involved equideterminants equal 1. Systems of linear inequalities defined by TE matrices are box-TDI.
We identify a matrix with the set of vectors composed of its rows. A set of vectors is TE if the associated matrix is TE. After giving several examples of TE matrices, I will give a first result: a decomposition theorem for TE matrices with full row rank. Incidentally, we obtain that systems defined by TE matrices with 0,±1 entries are totally dual dyadic. We will then leverage this decomposition theorem to analyze the cones generated by linearly independent TE sets of vectors, such cones are called te-cones. We first provide their Hilbert basis, which is the minimal set of integer vectors of the cone that generates all its integer points with nonnegative combinations. Finally, we will find a unimodular regular Hilbert triangulation for te-cones in (almost) all cases of the decomposition.
This is a joint work with Patrick Chervet and Roland Grappe (10.1007/s10107-026-02347-z).
Lyuhui Wu, (Sorbonne Université, IMJ-PRG, FRANCE)
Title: Convex analysis and Monge–Ampère equations on tropical varieties
Abstract: This talk is based on joint works with Omid Amini and Matthieu Piquerez.
In the first part of the talk, we study convex functions on polyhedral spaces. We define a convex
function on a polyhedral space as a continuous function that admits a local affine support function at
each point. This class of convex functions turns out to coincide with the class introduced by Botero-
Burgos-Sombra. We present several convex-analytic results, including a regularization theorem stating
that every convex function on a polyhedral space can be uniformly approximated by piecewise linear
convex functions. In the second part of the talk, we introduce the Monge–Ampère measure for functions on a
tropical variety. For a smooth function $f$ on $\mathbb{R}^n$, the local density of the Monge–Ampère measure
of $f$ equals to the determinant of the Hessian matrix of $f$. Motivated by the ideas of Chambert-Loir-Ducros,
Gubler- Künnemann and others, we define the Monge–Ampère measure for a piecewise smooth function using
intersection numbers of Chow rings of star fans of the tropical variety. We will then extend this definition
for any convex function on the tropical variety. If time permits, I will discuss the failure of the tropical
analogue of Calabi-Yau theorem on any tropical variety which is of dimension at least 2 and satisfies
some genericity condition.