Delta-matroids are "type B" generalizations of matroids in the same way that maximal orthogonal Grassmannians are generalizations of Grassmannians. A delta-matroid analogue of the Tutte polynomial of a matroid is the interlace polynomial. We give a geometric interpretation for the interlace polynomial via the K-theory of maximal orthogonal Grassmannians. To do so, we develop a new Hirzebruch-Riemann-Roch-type formula for the type B permutohedral variety.

We prove a sharp stability result for the Brunn-Minkowski inequality for $A,B\subset\mathbb{R}^2$. Assuming that the Brunn-Minkowski deficit $\delta=|A+B|^{\frac{1}{2}}/(|A|^\frac12+|B|^\frac12)-1$ is sufficiently small in terms of $t=|A|^{\frac{1}{2}}/(|A|^{\frac{1}{2}}+|B|^{\frac{1}{2}})$, there exist homothetic convex sets $K_A \supset A$ and $K_B\supset B$ such that $\frac{|K_A\setminus A|}{|A|}+\frac{|K_B\setminus B|}{|B|} \le C t^{-\frac{1}{2}}\delta^{\frac{1}{2}}$. The key ingredient is to show for every $\epsilon>0$, if $\delta$ is sufficiently small then $|co(A+B)\setminus (A+B)|\le (1+\epsilon)(|co(A)\setminus A|+|co(B)\setminus B|)$.

We establish a connection between the algebraic geometry of the type B permutohedral toric variety and the combinatorics of delta-matroids. Using this connection, we compute the volume and lattice point counts of type B generalized permutohedra. Applying tropical Hodge theory to a new framework of "tautological classes of delta-matroids," modeled after certain vector bundles associated to realizable delta-matroids, we establish the log-concavity of a Tutte-like invariant for a broad family of delta-matroids that includes all realizable delta-matroids. Our results include new log-concavity statements for all (ordinary) matroids as special cases.

We prove structure theorems for o-minimal definable subsets S⊂ G of definable groups containing large multiplicative structures, and show definable groups do not have bounded torsion arbitrarily close to the identity. As an application, for certain models of n -step random walks X in G we show upper bounds ℙ(X∈ S)≤ n^-C and a structure theorem for the steps of X when ℙ(X∈ S)≥ n^-C' .

Let $X \subset \mathbb{C}^{n}$ be an algebraic variety, and let $\Lambda \subset \mathbb{C}^{n}$ be a discrete subgroup whose real and complex spans agree. We describe the topological closure of the image of $X$ in $\mathbb{C}^{n} / \Lambda $, thereby extending a result of Peterzil–Starchenko in the case when $\Lambda $ is cocompact. We also obtain a similar extension when $X\subset \mathbb{R}^{n}$ is definable in an o-minimal structure with no restrictions on $\Lambda $, and as an application prove the following conjecture of Gallinaro: for a closed semi-algebraic $X\subset \mathbb{C}^{n}$ (such as a complex algebraic variety) and $\exp :\mathbb{C}^{n}\to (\mathbb{C}^{*})^{n}$ the coordinate-wise exponential map, we have $\overline{\exp (X)}=\exp (X)\cup \bigcup _{i=1}^{m} \exp (C_{i})\cdot \mathbb{T}_{i}$ where $\mathbb{T}_{i}\subset (\mathbb{C}^{*})^{n}$ are positive-dimensional compact real tori and $C_{i}\subset \mathbb{C}^{n}$ are semi-algebraic.

For δ>0 sufficiently small and A⊂Zk with |A+A|≤(2k+δ)|A|, we show either A is covered by mk(δ) parallel hyperplanes, or satisfies |coˆ(A)∖A|≤ckδ|A|, where coˆ(A) is the smallest convex progression (convex set intersected with an affine sub-lattice) containing A. This generalizes the Freiman-Bilu 2k theorem, Freiman's 3|A|−4 theorem, and recent sharp stability results of the present authors for sumsets in Rk conjectured by Figalli and Jerison.

Let $f$ and $g$ be two real-valued functions defined on a compact convex subset $C$ of $\mathbb R^k$. Their _sup-convolution_ $f*g$ is given by the formula $$f*g(x)=\sup_{(y+z)/2=x}\frac{f(y)+f(z)}2.$$ That is, it is the largest average of $f(y)$ and $f(z)$ over all pairs $y,z$ that average to $x$. The sup-convolution of more functions is defined in a similar way. Sup-convolutions are interesting for many reasons. For instance, if a firm wishes to buy a certain amount $x$ of some resource from $n$ different suppliers and the price from each supplier is a function $p_i$ depends in a non-linear way on the quantity bought (which often happens in practice -- for example, there can be bulk deals), then working out how to minimize the total cost is asking to minimize the quantity $\sum_ip_i(x_i)$ over all $(x_1,\dots,x_n)$ such that $\sum_ix_i=x$. This can easily be transformed into a problem of working out the value of a sup-convolution at $x$. More relevant to this article is a close connection between sup-convolutions and sumsets. The _strict hypograph_ $\text{hyp}(f)$ of a function $f:\mathbb R^k\to\mathbb R$ is the set $\{(x,y:y

Given a compact convex domain $C\subset \mathbb{R}^k$ and bounded measurable functions $f_1,\ldots,f_n:C\to \mathbb{R}$, define the sup-convolution $(f_1\ast \ldots \ast f_n)(z)$ to be the supremum average value of $f_1(x_1),\ldots,f_n(x_n)$ over all $x_1,\ldots,x_n\in C$ which average to $z$. Continuing the study by Figalli and Jerison and the present authors of linear stability for the Brunn-Minkowski inequality with equal sets, for $k\le 3$ we find the optimal constants $c_{k,n}$ such that $$\int_C f^{\ast n}(x)-f(x) dx \ge c_{k,n}\int_C\text{co}(f)(x)-f(x) dx$$ where $\text{co}(f)$ is the upper convex hull of $f$. Additionally, we show $c_{k,n}=1-O(\frac{1}{n})$ for fixed $k$ and prove an analogous optimal inequality for two distinct functions. The key geometric insight is a decomposition of polytopal approximations of $C$ into hypersimplices according to the geometry of the set of points where $\text{co}(f)$ is close to $f$.

For any matroid $M$, we compute the Tutte polynomial $T_M(x,y)$ using the mixed intersection numbers of certain tautological classes in the combinatorial Chow ring $A^\bullet(M)$ arising from Grassmannians. Using mixed Hodge-Riemann relations, we deduce a strengthening of the log-concavity of the h-vector of a matroid complex, improving on an old conjecture of Dawson whose proof was announced recently by Ardila, Denham and Huh.

The classical Erd\H{o}s-Littlewood-Offord theorem says that for nonzero vectors $a_1,\dots,a_n\in \mathbb{R}^d$, any $x\in \mathbb{R}^d$, and uniformly random $(\xi_1,\dots,\xi_n)\in\{-1,1\}^n$, we have $\Pr(a_1\xi_1+\dots+a_n\xi_n=x)=O(n^{-1/2})$. In this paper we show that $\Pr(a_1\xi_1+\dots+a_n\xi_n\in S)\le n^{-1/2+o(1)}$ whenever $S$ is definable with respect to an o-minimal structure (for example, this holds when $S$ is any algebraic hypersurface), under the necessary condition that it does not contain a line segment. We also obtain an inverse theorem in this setting.

We introduce certain torus-equivariant classes on permutohedral varieties which we call"tautological classes of matroids"as a new geometric framework for studying matroids. Using this framework, we unify and extend many recent developments in matroid theory arising from its interaction with algebraic geometry. We achieve this by establishing a Chow-theoretic description and a log-concavity property for a 4-variable transformation of the Tutte polynomial, and by establishing an exceptional Hirzebruch-Riemann-Roch-type formula for permutohedral varieties that translates between K-theory and Chow theory.

We prove a sharp stability result concerning how close homothetic sets attaining near-equality in the Brunn-Minkowski inequality are to being convex. In particular, resolving a conjecture of Figalli and Jerison, we show there are universal constants $C_n,d_n>0$ such that for $A \subset \mathbb{R}^n$ of positive measure, if $|\frac{A+A}{2}\setminus A| \le d_n |A|$, then $|\operatorname{co}(A)\setminus A| \le C_n |\frac{A+A}{2}\setminus A|$ for $\operatorname{co}(A)$ the convex hull of $A$.

We compute the integral Chow ring of the quotient stack [(P-1)(n)/PGL(2)], which contains M(0,1 )as a dense open, and determine a natural Z-basis for the Chow ring in terms of certain ordered incidence strata. We further show that all Z-linear relations between the classes of ordered incidence strata arise from an analogue of the WDVV relations in A(center dot)((M) over bar (0,n)). Next we compute the classes of unordered incidence strata in the integral Chow ring of the quotient stack [sym(n)P(1)/PGL(2)] and classify all Z-linear relations between the strata via these analogues of WDVV relations. Finally, we compute the rational Chow rings of the complement of a union of unordered incidence strata.

We show the compactly supported motive of the moduli stack of degree n rational curves on the weighted projective stack $${\mathcal {P}}(a,b)$$ is of mixed Tate type over any base field K with $$\hbox {char}(K) \not \mid a,b$$ and has class $${\mathbb {L}}^{(a+b)n+1}-{\mathbb {L}}^{(a+b)n-1}$$ in the Grothendieck ring of stacks. In particular, this improves upon the results of (Han and Park in Math Ann 375(3–4), 1745–1760, 2019) regarding the arithmetic invariant of the moduli stack $${\mathcal {L}}_{1,12n} :=\mathrm {Hom}_{n}({\mathbb {P}}^1, \overline{{\mathcal {M}}}_{1,1})$$ of stable elliptic fibrations over $${\mathbb {P}}^{1}$$ with 12n nodal singular fibers and a marked Weierstrass section.

We prove that the vanishings of the modified diagonal cycles of Gross and Schoen govern the $\mathbb{Z}$-linear relations between small $m$-diagonals $\text{pt}^{\{1,\ldots,n\}\setminus A}\times\Delta_A$ in the rational Chow ring of $X^n$ for $A$ ranging over $m$-element subsets of $\{1,\ldots,n\}$. Our results generalize to arbitrary symmetric classes in place of the diagonal in $X^m$, and with different types of inclusions $A^\bullet(X^m)_{\mathbb{Q}}^{S_m} \hookrightarrow A^\bullet(X^n)_{\mathbb{Q}}$. The combinatorial heart of this paper, which may be of independent interest, is showing the $\mathbb{Z}$-linear relations between elementary symmetric polynomials $e_k(x_{a_1},\ldots,x_{a_m}) \in \mathbb{Z}[x_1,\ldots,x_n]$ are generated by the $S_n$-translates of a certain alternating sum over the facets of a hyperoctahedron.

For $\delta>0$ sufficiently small and $A\subset \mathbb{Z}^k$ with $|A+A|\le (2^k+\delta)|A|$, we show either $A$ is covered by $m_k(\delta)$ parallel hyperplanes, or satisfies $|\widehat{\operatorname{co}}(A)\setminus A|\le c_k\delta |A|$, where $\widehat{\operatorname{co}}(A)$ is the smallest convex progression (convex set intersected with a sublattice) containing $A$. This generalizes the Freiman-Bilu $2^k$ theorem, Freiman's $3|A|-4$ theorem, and recent sharp stability results of the present authors for sumsets in $\mathbb{R}^k$ conjectured by Figalli and Jerison.

Given a collection of colored chain posets, we estimate the number of colored subsets of the boolean lattice which avoid all chains in the collection.

We compute the $GL_{r+1}$-equivariant Chow class of the $GL_{r+1}$-orbit closure of any point $(x_1, \ldots, x_n) \in (\mathbb{P}^r)^n$ in terms of the rank polytope of the matroid represented by $x_1, \ldots, x_n \in \mathbb{P}^r$. Using these classes and generalizations involving point configurations in higher dimensional projective spaces, we define for each $d\times n$ matrix $M$ an $n$-ary operation $[M]_\hbar$ on the small equivariant quantum cohomology ring of $\mathbb{P}^r$, which is the $n$-ary quantum product when $M$ is an invertible matrix. We prove that $M \mapsto [M]_\hbar$ is a valuative matroid polytope association. Like the quantum product, these operations satisfy recursive properties encoding solutions to enumerative problems involving point configurations of given moduli in a relative setting. As an application, we compute the number of line sections with given moduli of a general degree $2r+1$ hypersurface in $\mathbb{P}^r$, generalizing the known case of quintic plane curves.

Bollobás, Reed, and Thomason proved every 3‐uniform hypergraph ℋ with m edges has a vertex‐partition V ()= V 1 ⊔ V 2 ⊔ V 3 such that each part meets at least edges, later improved to 0.6 m by Halsegrave and improved asymptotically to 0.65 m + o ( m ) by Ma and Yu. We improve this asymptotic bound to , which is best possible up to the error term, resolving a special case of a conjecture of Bollobás and Scott.

In 1979, Shearer and Kleitman conjectured the existence of $\lfloor n/2 \rfloor+1$ orthogonal chain decompositions of the hypercube poset $Q_n$ and constructed two orthogonal chain decompositions. In this paper, we make the first nontrivial progress on this conjecture by constructing three orthogonal chain decompositions of $Q_n$ for all $n\ge 4$ with the possible exceptions $n=9,11,13,23$. To do this, we introduce the notion of "almost orthogonal symmetric chain decompositions." We explicitly describe three such decompositions of $Q_5$ and $Q_7$ and describe conditions which allow us to decompose products of hypercube posets into $k$ almost orthogonal symmetric chain decompositions given such decompositions of the original hypercube posets.