The topological Tverberg theorem asserts that if $r$ is a prime power then for any continuous map $f\colon \Delta_{(r-1)(d+1)}\rightarrow \mathbb{R}^d$ from the $(r-1)(d+1)$-dimensional simplex $\Delta_{(r-1)(d+1)}$ to $\mathbb{R}^d$ there exist $r$ pairwise disjoint faces of the simplex whose images have non-empty $r$-fold intersection. By refinement, the same conclusion holds if the simplex is replaced by any polytope of the same dimension. While this dimension is tight for simplices, recent work of Sober\'on and Zerbib shows that this need not be true for polytopes in general. Here we give topological Tverberg theorems for products of simplices. Each of these improves upon the $(r-1)(d+1)$-dimensional threshold, even while imposing the structural condition that the ``Tverberg faces''of the product are themselves the products of pairwise disjoint faces from each simplex factor. As before, refinement extends these results, and in particular their dimensional improvements, to products of arbitrary polytopes. As an example, if $d+1$ is a power of two then whenever $m\geq n\geq d+1$ and $m+n=3d+2$ we show that any continuous map $f\colon \Delta_m\times \Delta_n\rightarrow \mathbb{R}^d$ admits disjoint faces $\sigma_1,\sigma_2$ of $\Delta_m$ and $\tau_1,\tau_2$ of $\Delta_n$ such that $\cap_{i,j\in[2]} f(\sigma_i\times\tau_j)\neq \emptyset$. In the case of multilinear maps, our results imply partitions of grid-indexed point sets in $\mathbb{R}^d$ by specialized subsets with stronger intersection conclusions than given by Tverberg's original theorem. Lastly, we extend our results to van Kampen--Flores type theorems which impose dimensional restrictions on the faces of each product factor.
Let $\mu$ be a log-concave probability measure on $\mathbb R^n$ and let $f\colon\mathbb R^n\to\mathbb R^k$ be a polynomial mapping of degree at most $d$. We show that \[ \mu(f\in A) \le C\bigl(\lambda_k(A)\bigr)^{\frac{1}{k(d-1)+1}} \] for every Borel set $A\subset\mathbb R^k$ whenever the image measure $\mu\circ f^{-1}$ is absolutely continuous. The constant $C$ is independent of the dimension $n$, and the exponent $\frac{1}{k(d-1)+1}$ is sharp. This extends the scalar Carbery--Wright inequality and answers, in the log-concave setting, a question raised by Avni, Glazer, and Larsen. In addition, we show that the density of $\mu\circ f^{-1}$, whenever it exists, belongs to the Nikolskii--Besov space $B^{\frac{1}{k(d-1)+1}}_{1,\infty}(\mathbb R^k)$, with a dimension-free bound for the corresponding norm. A central difficulty in passing from scalar polynomials to vector-valued polynomial mappings is the lack of a suitable nondegeneracy parameter quantifying absolute continuity of $\mu\circ f^{-1}$, as the variance does in the scalar case. Natural candidates such as the covariance matrix or the Jacobian matrix either fail to characterize this property or do not lead to dimension-free estimates. We identify such a parameter and define it to be the covariance matrix of the vector formed by the monomials of degree up to $d^{k-1}$ in the normalized components of $f$. The dimension-free nature of our results allows us to extend Kusuoka's absolute continuity criterion for Gaussian polynomial random vectors to the log-concave setting. Moreover, in this setting, we obtain estimates relating convergence in distribution to convergence in total variation for polynomial random vectors.
For a convex body $K \subset \mathbb R^d$ let $\Delta(K)$ be the expected distance between two independent uniform points of $K$, and let $\theta(K)$ be the corresponding expectation for normalized surface measure on $\partial K$. The Zaporozhets-Tarasov conjecture asserts $\Delta(K) \le \theta(K)$. We prove this conjecture in case $d=2$. In addition, we give a six-vertex convex polytope in $\mathbb R^3$ for which the reverse strict inequality holds, and obtain counterexamples in every dimension $d \ge 3$ by taking products with segments. Finally, we show that $\theta(K)\ge \frac{\operatorname{per}K}{6}$ for every planar convex body.
We show that for every fixed $\varepsilon>0$, there exist arbitrarily large families of point-line pairs $(x_1,\ell_1),\ldots,(x_n,\ell_n)$ in $[0,1]^2$, with $x_i \in \ell_i$ for all $i$, and such that $\operatorname{dist}(x_i,\ell_j)\ge n^{-2/3-\varepsilon}$ for all $i \neq j$. Combined with a previous result of Cohen, the author, and Zakharov, this solves the minimal distance problem. The same construction also comes with an unexpected finite field consequence: for every $\varepsilon>0$, there exists a set of primes $q$ of positive relative density for which $\mathbb F_q^2$ contains an induced point-line matching of size $\gtrsim q^{3/2-\varepsilon}$. This disproves a conjecture of Hunter, the author, Verstra\"ete and Zhang.
A classical geometric result says that every nonzero cycle of the mod-$2$ incidence map from $d$-subsets to $(d-1)$-subsets of $[n]$ has support at least $d+1$, with equality attained by the boundary of a simplex on $d+1$ vertices. We prove an analogous result for the subspace lattice of $\mathbb{F}_q^n$, determining the minimum support size of a nonzero $d$-cycle over a field $K$ of characteristic $p \mid q+1$. Surprisingly, the boundary of a $(d+1)$-space is not always optimal. Shorter cycles occur for $d=1$, and for $d=2$ when $n\ge4$, and otherwise, the boundary of a $(d+1)$-space is shortest. For $d\ge4$, we prove a gap-stability result: every cycle with support less than $(2-10/q)$ times the minimum is a multiple of the boundary of a $d+1$ space. We also construct support-controlled cones, yielding a direct geometric analysis of the dimensions in which the homology groups of the subspace incidence complex vanish and explicit lower bounds on its coboundary expansion. The degree-$1$ expansion estimate is an ingredient in the stability theorem.
For a simplicial complex $\Delta$ and integers $n_i\ge 2$, set $A_{\Delta,\mathbf n}=\mathbb{k}[x_1,\ldots,x_m]/(I_\Delta+(x_1^{n_1},\ldots,x_m^{n_m}))$. We give an exact combinatorial formula for the canonical trace for arbitrary truncation exponents and for an arbitrary simplicial complex after deleting irrelevant ghost vertices. The formula extends the free-face formula of Gasanova--Herzog--Hibi--Moradi for square-zero flag face algebras and recovers, in the simplex-boundary case, a special case of their formula for monomial almost complete intersections. As a first consequence, we classify the nearly Gorenstein algebras in this family: on each connected component $C$ of $\Delta^{(1)}$, the induced complex is either the simplex $2^C$, with arbitrary exponents, or the boundary $\partial 2^C$, with every exponent equal to two. We also compute the Teter number on this nearly Gorenstein locus. For flag complexes the trace is generated by the free-face monomials for arbitrary exponents, and we characterize the equalities $\operatorname{tr}_A(\omega_A)=\mathfrak m_A^q$. In the square-zero one-dimensional case we isolate the additional contribution coming from triangle components.
The Grone--Merris--Bai theorem states that the Laplacian spectrum of a simple graph is majorized by its conjugate degree sequence. Recently, Zhang, Song, and Fan extended this result to simplicial complexes by establishing a majorization relation between the spectrum of the $(r-1)$-dimensional up-Laplacian and the conjugate $(r-1)$-degree sequence. In this paper, we characterize all equality cases in the partial-sum inequalities of this higher-dimensional majorization theorem. For every $r$-dimensional simplicial complex $X$ with $r\ge2$, we prove that \[ \sum_{i=1}^{q}\lambda_{r-1,i}(X) = \sum_{i=1}^{q}d_{r-1,i}^{\top}(X) \] if and only if \[ q\ge \max\{\operatorname{rank}B_r(X),\Delta_{r-1}(X)\}. \] Thus, unlike the graph case, equality can occur only after both sequences have exhausted all their nonzero terms. As consequences, equality in the first partial sum and equality between the entire sequences are both equivalent to $X$ containing a unique $r$-simplex. The proof is based on the local down-Laplacian decomposition and the equality case of the Ky Fan inequality.
