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Let G be a linear algebraic group acting linearly on a vector space (or more generally, an affine variety) $ \mathcal {V} $ V , and let $ {\Bbbk [\mathcal {V}]^{G}} $ k [ V ] G be the corresponding algebra of invariant polynomial functions. A separating set $ S \subseteq {\Bbbk [\mathcal {V}]^{G}} $ S ⊆ k [ V ] G is a set of polynomials with the property that for all $ v,w \in \mathcal {V} $ v , w ∈ V , if there exists $ f \in {\Bbbk [\mathcal {V}]^{G}} $ f ∈ k [ V ] G separating $v$ and $w$, then there exists $ f \in S $ f ∈ S separating $v$ and $w$. In this article, we consider the action of $ G = \operatorname {GL}_2(\mathbb {C}) $ G = GL 2 ⁡ (C) on the $ \mathbb {C} $ C -vector space $ {\mathcal {M}}_2^n $ M 2 n of n-tuples of $ 2 \times 2 $ 2 × 2 matrices by simultaneous conjugation. Minimal generating sets $ S_n $ S n of $ \mathbb {C}[{\mathcal {M}}_2^n]^G $ C [ M 2 n ] G are well known and $ |S_n| = \frac 16(n^3+11n) $ | S n | = 1 6 (n 3 + 11 n). In recent work, Kaygorodov et al. [Kaygorodov I, Lopatin A, Popov Y. Separating invariants for $ 2 \times 2 $ 2 × 2 matrices. Linear Algebra Appl. 2018;559:114-124.] showed that for all $ n \geq ~1 $ n ≥ 1 , $ S_n $ S n is a minimal separating set by inclusion, i.e. that no proper subset of $ S_n $ S n is a separating set. This does not necessarily mean that $ S_n $ S n has minimum cardinality among all separating sets for $ \mathbb {C}[{\mathcal {M}}_2^n]^G $ C [ M 2 n ] G . Our main result shows that any separating set for $ \mathbb {C}[{\mathcal {M}}_2^n]^G $ C [ M 2 n ] G has cardinality $ \geq ~5n-5 $ ≥ 5 n − 5. In particular, there is no separating set of size $ \dim (\mathbb {C}[{\mathcal {M}}_2^n]^G) = 4n-3 $ dim ⁡ (C [ M 2 n ] G) = 4 n − 3 for $ n \geq ~3 $ n ≥ 3. Further, $ S_3 $ S 3 has indeed minimum cardinality as a separating set, but for $ n \geq ~4 $ n ≥ 4 there may exist a smaller separating set than $ S_n $ S n . We show that a smaller separating set does in fact exist for all $ n \geq ~5 $ n ≥ 5. We also prove similar results for the left–right action of $ \operatorname {SL}_2(\mathbb {C}) \times \operatorname {SL}_2(\mathbb {C}) $ SL 2 ⁡ (C) × SL 2 ⁡ (C) on $ {\mathcal {M}}_2^n $ M 2 n . [ABSTRACT FROM AUTHOR]

We prove effective equidistribution of horospherical flows in $\operatorname {SO}(n,1)^{\circ } / \Gamma $ when $\Gamma $ is geometrically finite and the frame flow is exponentially mixing for the Bowen–Margulis–Sullivan measure. We also discuss settings in which such an exponential mixing result is known to hold. As part of the proof, we show that the Patterson–Sullivan measure satisfies some friendly like properties when $\Gamma $ is geometrically finite. [ABSTRACT FROM AUTHOR]

We classify anticommutative algebras in a specific closed subvariety and describe all degenerations between these algebras. Anticommutative n -dimensional algebras of the third degeneration level are classified for any natural n using these results. [ABSTRACT FROM AUTHOR]

n-ary algebras of the first degeneration level are described. A detailed classification is given in the cases n = 2 , 3 . [ABSTRACT FROM AUTHOR]

Anticommutative Engel algebras of the first five degeneration levels are classified. All algebras appearing in this classification are nilpotent Malcev algebras. [ABSTRACT FROM AUTHOR]

We study orthogonal decompositions of symmetric and ordinary tensors using methods from linear algebra. For the field of real numbers we show that the sets of decomposable tensors can be defined by equations of degree 2. This gives a new proof of some of the results of Robeva and Boralevi et al. Orthogonal decompositions over the field of complex numbers had not been studied previously; we give an explicit description of the set of decomposable tensors using polynomial equalities and inequalities, and we begin a study of their closures. The main open problem that arises from this work is to obtain a complete description of the closures. This question is akin to that of characterizing border rank of tensors in algebraic complexity. We give partial results using in particular a connection with approximate simultaneous diagonalization (the so-called ASD property). [ABSTRACT FROM AUTHOR]

For each n > 2 we classify all n-dimensional algebras over an arbitrary infinite field which have the property that the n-dimensional abelian Lie algebra is their only proper degeneration.

We describe degenerations of three-dimensional Jordan superalgebras over C . In particular, we describe all irreducible components in the corresponding varieties. [ABSTRACT FROM AUTHOR]

Let G be the group GLr(C)×(C×)n. We conjecture that the finely-graded Hilbert series of a G orbit closure in the space of r-by-n matrices is wholly determined by the associated matroid. In support of this, we prove that the coefficients of this Hilbert series corresponding to certain hook-shaped Schur functions in the GLr(C) variables are determined by the matroid, and that the orbit closure has a set-theoretic system of ideal generators whose combinatorics are also so determined. We also discuss relations between these Hilbert series for related matrices, including their stabilizing behaviour as r increases. [ABSTRACT FROM AUTHOR]

We consider the covering map π : C n → T of a compact complex torus. Given an algebraic variety X ⊆ C n we describe the topological closure of π ( X ) in T . We obtain a similar description when T is a real torus and X ⊆ R n is a set definable in an o-minimal structure over the reals. [ABSTRACT FROM AUTHOR]

We describe all rigid algebras and all irreducible components in the variety of four-dimensional Leibniz algebras 𝔏𝔢𝔦𝔟4 over ℂ. In particular, we prove that the Grunewald–O'Halloran conjecture is not valid and the Vergne conjecture is valid for 𝔏𝔢𝔦𝔟4. [ABSTRACT FROM AUTHOR]

We prove a conjecture of Medvedev and Scanlon for en-domorphisms of connected commutative linear algebraic groups G de-fined over an algebraically closed field k of characteristic 0. That is, if ϕ: G → G is a dominant endomorphism, we prove that one of the following holds: either there exists a non-constant rational func-tion f εk(G) preserved by ϕ (i.e., f ° ϕ = f), or there exists a point x ε G(k) whose ϕ-orbit is Zariski dense in G. [ABSTRACT FROM AUTHOR]

For several objects of interest in geometric complexity theory, namely for the determinant, the permanent, the product of variables, the power sum, the unit tensor, and the matrix multiplication tensor, we introduce and study a fundamental SL-invariant function that relates the coordinate ring of the orbit with the coordinate ring of its closure. For the power sums we can write down this fundamental invariant explicitly in most cases. Our constructions generalize the two Aronhold invariants on ternary cubics. For the other objects we identify the invariant function conditional on intriguing combinatorial problems much like the well-known Alon–Tarsi conjecture on Latin squares. We provide computer calculations in small dimensions for these cases. As a main tool for our analysis, we determine the stabilizers, and we establish the polystability of all the mentioned forms and tensors (including the generic ones). [ABSTRACT FROM AUTHOR]

Let G be a countably infinite discrete group, let βG be the Stone–Čech compactification of G, and let ${G^{\rm{*}}} = \beta G \setminus G$. An idempotent $p \in {G^{\rm{*}}}$ is left (right) maximal if for every idempotent $q \in {G^{\rm{*}}}$, pq = p (qp = P) implies qp = q (qp = q). An idempotent $p \in {G^{\rm{*}}}$ is strongly right maximal if the equation xp = p has the unique solution x = p in G*. We show that there is an idempotent $p \in {G^{\rm{*}}}$ which is both left maximal and strongly right maximal. [ABSTRACT FROM PUBLISHER]

The moduli space of genus 3 translation surfaces with a single zero has two connected components. We show that in the odd connected component Hodd(4) the only G+(2,ℝ) orbit closures are closed orbits, the Prym locus ..., and Hodd(4) Together with work of Matheus--Wright, this implies that there are only finitely many nonarithmetic closed orbits (Teichmüller curves) in Hodd(4) outside of the Prym locus. [ABSTRACT FROM AUTHOR]

Let N be a quiver representation with non-zero admissible annihilator. In this paper, we prove the normality of the orbit closure $\bar {\mathcal {O}}_{N}$ when it is a hypersurface. The result thus gives new examples of normal orbit closures of quiver representations. [ABSTRACT FROM AUTHOR]

Let G be a countably infinite discrete group, let ßG be the Stone--Cech compactification of G, and let ... . The left (right) preordering on idempotents of ... is defined by ... . As any compact Hausdorff right topological semigroup, ... has right maximal idempotents. We show (in ZFC) that there are left maximal idempotents in ... . In the case G=ZG=Z, this is the answer to a long standing question. [ABSTRACT FROM AUTHOR]

We classify the orbit closures in the variety of complex, three-dimensional Novikov algebras and obtain the Hasse diagrams for the closure ordering of the orbits. We provide invariants which are easy to compute and which enable us to decide whether or not one Novikov algebra degenerates to another Novikov algebra. [ABSTRACT FROM AUTHOR]

We investigate the Zariski closure of the projective equivalence class of a matrix. New results are presented regarding the matrices in this variety and their matroids, and we give equations for the variety. We also discuss the K-polynomial of the closure of a projective equivalence class, and two other geometric invariants that can be obtained from this.

For a set Ω an unordered relation on Ω is a family R of subsets of Ω. If R is such a relation we let $\mathcal{G}(R)$ be the group of all permutations on Ω that preserve R, that is g belongs to $\mathcal{G}(R)$ if and only if x∈ R implies x∈ R. We are interested in permutation groups which can be represented as $G=\mathcal{G}(R)$ for a suitable unordered relation R on Ω. When this is the case, we say that G is defined by the relation R, or that G is a relation group. We prove that a primitive permutation group ≠Alt( Ω) and of degree ≥11 is a relation group. The same is true for many classes of finite imprimitive groups, and we give general conditions on the size of blocks of imprimitivity, and the groups induced on such blocks, which guarantee that the group is defined by a relation. This property is closely connected to the orbit closure of permutation groups. Since relation groups are orbit closed the results here imply that many classes of imprimitive permutation groups are orbit closed. [ABSTRACT FROM AUTHOR]

We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients in the context of the geometric complexity theory program to prove a variant of Valiant's algebraic analogue of the P ≠ NP conjecture. We also describe the precise separation of complexity classes that their program proposes to demonstrate. [ABSTRACT FROM AUTHOR]

Abstract: We revisit a technique of S. Lehr on automata and use it to prove old and new results in a simple way. We give a very simple proof of the 1986 theorem of Honkala that it is decidable whether a given -automatic sequence is ultimately periodic. We prove that it is decidable whether a given -automatic sequence is overlap-free (or squarefree, or cubefree, etc.). We prove that the lexicographically least sequence in the orbit closure of a -automatic sequence is -automatic, and use this last result to show that several related quantities, such as the critical exponent, irrationality measure, and recurrence quotient for Sturmian words with slope , have automatic continued fraction expansions if does. [Copyright &y& Elsevier]