It is known that a flexible realization of a graph in the plane exists if and only if the graph has a so called NAC-coloring, which is an edge coloring by two colors such that every cycle is either monochromatic or it has at least two blue and two red edges. We prove that a minimally rigid graph has a NAC-coloring, and hence also a flexible realization, if and only if the graph is not a 2-tree. The main ingredience of the proof is the characterization of the graphs with n vertices and 2n-3 edges that have no stable cut by Le and Pfender, where a stable cut is a set of vertices inducing no edges such that its removal disconnects the graph. We strengthen to all flexible graphs the result by Chen and Yu stating that every graph with at most 2n-4 edges has a stable cut. We also study the bounds on the number of NAC-colorings for various classes of graphs. This is joint work with K. Clinch, D. Garamvölgyi, J. Haslegrave, T. Huynh and A. Nixon.

We describe an energy-based approach for defining the ``rigidity order'' of a bar and joint framework. We also describe higher order (multivariate) generalizations of the second derivative test from calculus. Together, this gives a new proof of Connelly's theorem, that the lack of a "second order flex" implies rigidity. It gives a new proof of V. Alexandrov's theorem, that when the dimension of non-trivial infinitesimal flexes equals 1, then the lack of a "kth order flex" implies rigidity. When Alexandrov's theorem applies, the rigidity order can be computed just using a sequence of linear algebra computations.

A graph is k-realizable if any set of distances realizable in R^n for some n can be realized in R^k. The pairs of points where the distances are specified are given by the edges of the graph. I will discuss a (speculative) conjecture about k-realizability and universal rigidity and provide some evidence for it.

A bipartite planar graph with n vertices has at most 2n edges. As a generalization of this well-known result, Kalai, Nevo and Novik conjectured that for a simplicial complex that is PL-embeddable in Euclidean 2k-space and has a (k+1)-colorable 1-skeleton, the number of k-dimensional faces is at most twice the number of (k-1)-dimensional faces. I will present a proof of this conjecture based on the multigraded strong Lefschetz property of the Stanley-Reisner ring. I will also mention a relation to bipartite rigidity.

In the last 20 years, there has been a lot of research into forced-symmetric and forced-periodic rigidity. In this talk, I will discuss a new result that characterises conditions for forced-symmetric rigidity in the Euclidean plane with respect to the orientation-reversing wallpaper group ℤ^2⋊C_s, also known as pm or **, under a fixed-lattice representation. This involves infinite frameworks that are both periodic and reflectionally symmetric. The result fills a key gap, as symmetric rigidity has not previously been studied for orientation-reversing wallpaper groups. From the proof, characterisations for the wallpaper groups pg and cm also follow. This is joint work with Eleftherios Kastis and Bernd Schulze.

We define a generic rigidity matroid for k-volumes of a simplicial complex in R^d, and prove that for 2 ≤ k ≤ d−1 it has the same rank as the classical generic d-rigidity matroid on the same vertex set (namely, the case k = 1, and in contrast with the k = d case). We conjecture a characterization for the bases of this matroid in terms of d-rigidity of the 1-skeleton of the complex and a combinatorial Hall condition on incidences of edges in k-faces. Joint work with Alan Lew, Yuval Peled and Orit Raz.

Modern machine learning increasingly relies on advanced mathematical tools to handle structured data, geometric constraints, and complex statistical models. Recently, neural networks with algebraic activation functions have gained attention for their inherent mathematical structures, interpretability and theoretical guarantees. In this short lightning talk, I will introduce the geometry of parameter spaces, showing how “geometrically rigid” parameter configurations can stabilize training and reduce degeneracies.

I will present results about how ML degree of a hypersurface arrangement changes when we add generic hypersurfaces. In particular, when the hypersurface arrangement is a hyperplane arrangement, we obtain results relating computing critical points of log likelihood equations to sampling interior points in the complement of the hyperplane arrangement.

Given a linear subspace L of a complex vector space V, the dimension of the amoeba of L depends only on the matroid of L. In particular, the matroid of the amoeba of L is the matroid induced by the submodular function 2r - 1, where r is the rank function of the matroid of L. The rigidity matroid in dimension 2 is constructed in the same way. Can this be generalized to higher order rigidity? This is based on ongoing work with Dario Antolini.

We consider count matroids M_{k,l}(G) of a graph G, in which independence is defined by a sparsity count involving the parameters k and l. We give tight lower bounds, for each pair (k,l), that show that if G is sufficiently highly connected, then G−e has maximum rank for all e∈E(G), and M_{k,l}(G) is connected. These bounds unify and extend several previous results, including theorems of Nash-Williams and Tutte (k=l), and Lovász and Yemini (k=2, l=3). We use this results to generalize Whitney's celebrated result on the graphic matroid of G (which corresponds to the case of k=l=1) to all count matroids: if G is highly connected, depending on k and l, then the count matroid M_{k,l}(G) uniquely determines G. Joint results with D. Garamvölgyi and T. Jordán.

Let $N$ be a linear representation (i.e, matrix) of a matroid whose ground set $S$ includes a finite, distinguished subset $P$. We give function $L(N)$ that, unlike what we know of other Tutte functions and work like the Hopf algebra variants of Krajewski, Moffatt and Tanasa, has values in an \emph{anti-commutative} algebra. Let deletion and contraction be limited to $e\not\in P$. Then, the values are in the exterior algebra generated by $P_\alpha \coprod P_\beta$. The construction relies on concrete minor operations to establish consistent signs of the constituent terms so that, with suitable accounting for sequential orderings of set elements, $L(N)=L(N\setminus e)+L(N/e)$ in the exterior algebra. Our construction is derived from the structure of the equilibrium equations for linear electrical networks, and of their generalization to multi-dimensional elastic frameworks. Further, the construction does not require orthogonality for the spaces that generalize spaces of feasible currents and voltages, or of forces and displacements. Hence $L$ will be defined on equal rank pairs $(N_\alpha, N_\beta)$ (where originally, $N_\alpha=N_\beta=N$). We take the Tutte identities those for Welsh and Kayibi's linking polynomial of matroid pairs. With $N_\alpha\neq N_\beta$, we can derive the digraph all-minors matrix tree theorem by taking $P$ to be the set of vertices. We so get ratio of common basis expansion solutions for linear electrical and other linear systems with multi-terminal amplifiers (where a voltage or force at one place is a multiple of current or displacement at a different place). To incorporate resistance ($r_e$), conductance ($g_e$), elasticity coefficient, etc. parameters, we use parametrized Tutte function theory for which $L(N)=r_e L(N\setminus e)+ g_e L(N/e)$; the term for common basis $B$ includes $\prod _{e\in B} g_e \prod _{e\not\in B\cup P} r_e$.

We show that every rigid graph has a tree-like structure, which provides information on its global rigidity properties. Joint work with Soma Villányi.

The famous Erdős distinct distances problem asks the following: how many distinct distances must exist between a set of n points in the plane? There are many generalisations of this question that ask one to consider different spaces and metrics, or larger structures of points. We bring these problems into a common framework using the concept of g-rigidity. Specifically, if G=(V,E) is a (hyper)graph and g is a map assigning polynomial measurements to the (hyper)edges of G, our main results describe sharp lower bounds for the number of different edge-labellings that are achievable from realisations where all vertices are restricted to a point set P. This allows us to obtain results for pseudo-Euclidean metrics, Lp metrics, dot-product problems, matrix completion problems, and symmetric tensor completion problems. This talk is presenting joint work with Nora Frankl, Samuel Mansfield, Anthony Nixon, Jonathan Passant and Audie Warren.

Let $V$ consist of the $n$ choose 2 segments among the vertices of a convex $n$-gon. A subset of $V$ is called $(k+1)$-crossing-free if it does not contain a set of $k+1$ mutually crossing diagonals. Such free subsets form a simplicial complex that we call the $k$-associahedron, since the usual associahedron is (essentially) the case $k=1$. For arbitrary $k$ and $n\ge 2k$ this complex is known to be a shellable sphere of dimension $k(n−2k−1)$, and is conjectured to be polytopal (Jonsson 2003). Since the dimension of the $k$-associahedron happens to coincide with that of any abstract rigidity matrix of dimension $2k$ on $n$ elements, Pilaud and Santos (2010) conjectured that the facets in the complex, called $k$-triangulations, are generically isostatic graphs in dimension $2k$ for bar-and-joint rigidity. This conjecture was intended as a step towards the construction of the $k$-associahedron as a polytope (or at least as a complete fan) using as facet normals the rows of the corresponding rigidity matroid in a suitable embedding. Together with Luis Crespo I have explored this possibility in recent years and we have found that: A. Restricted to the moment curve, bar-and-joint rigidity coincides with another two classical forms of rigidity, Whiteley’s cofactor rigidity and Kalai’s hyperconnectivity. The former is particularly suitable to model k-triangulations since it is based on configurations of points in the plane. The latter connects rigidity and $k$-triangulations with the variety of Pfaffians of antisymmetric matrices of size $2k$, which allows us to prove that $k$-triangulations are indeed generically isostatic with respect to hypercopnnectivity, B. Suitable choices of points along the moment curve allow us to find realisations of the $k$-associahedron, or at least its fan, for all values of $(k,n)$ with $n≤13$, except for $(3,12)$ and $(3,13)$. C. We prove that no choice of points along the moment curve can provide realisations of the $k$-associahedron for any n≥2k+6 and k≥3. This includes the cases (3,12) and (3,13) mentioned above.

Classical results of Cauchy and Dehn imply that the 1-skeleton of a convex polyhedron P is rigid i.e. every continuous motion of the vertices of P in R^3 which preserves its edge lengths results in a polyhedron which is congruent to P. This result was extended to convex poytopes in R^d for all d\geq 3 by Whiteley, and to generic realisations of 1-skeletons of simplicial (d-1)-manifolds in R^d by Kalai for d\geq 4 and Fogelsanger for d\geq 3. We will generalise Kalai's result by showing that, for all \geq 4 and any fixed 1\leq k\leq d-3, every generic realisation p of the k-skeleton of a simplicial (d-1)-manifold S in R^d is volume rigid, i.e. every continuous motion of the vertices of (S,p) in R^d which preserves the volumes of its k-faces results in a realisation which is congruent to p. In addition, we conjecture that our result remains true for k=d-2 and verify this conjecture when d=4,5. This is joint work with James Cruickshank and Shin-Ichi Tanigawa.

The matroid characterizing the collection of edge lengths in a planar bar-and-joint framework due to Pollaczek-Geiringer, and rediscovered by Laman, is widely-celebrated. Less heralded is the fact that the edge directions (or bearings) of a planar framework define the same matroid. This result goes back to a 1987 book chapter by Whiteley and was reproved in 2003 by Martin using algebraic geometry. Their techniques give insight into an even less-studied matroid defined by angles between pairs of edges. In this talk, I will discuss some new results about that angle matroid, and describe some angle analogs of rigidity-theoretic concepts like circuits and the pure condition. This is based on work with Sean Dewar, Georg Grasseger, Anthony Nixon, William Sims, Meera Sitharam, and David Urizar. I will also report on new computational work, joint with Sophie Faris, to analyze all angle configurations on five vertices.

We study the maximum dimension d=d(n,p) for which an Erdos-Renyi G(n,p) random graph is d-rigid. Our main results reveal two different regimes of rigidity in G(n,p) separated at the critical probability p_c where the graph's minimum degree exceeds half its average degree. We find that if p