Let M be a semi‐finite von Neumann algebra with normal state space S ( M ) . For any ϕ ∈ S ( M ) , let M ϕ : = { x ∈ M : x ϕ = ϕ x } be the centralizer of ϕ with center Z ( M ϕ ) . We show that for ϕ , ψ ∈ S ( M ) , the following are equivalent. ϕ = ψ . Z ( M ψ ) ⊆ Z ( M ϕ ) and ϕ | Z ( M ϕ ) = ψ | Z ( M ϕ ) . ϕ , ψ have the same distances to all the closed faces of S ( M ) .

A theorem of Dorronsoro from the 1980s quantifies the fact that real‐valued Sobolev functions on Euclidean spaces can be approximated by affine functions almost everywhere, and at all sufficiently small scales. We prove a variant of Dorronsoro's theorem in Heisenberg groups: functions in horizontal Sobolev spaces can be approximated by affine functions which are independent of the last variable. As

We prove that if G is a finite simple group of Lie type and S 1 , ⋯ , S k are subsets of G satisfying ∏ i = 1 k | S i | ⩾ | G | c for some c depending only on the rank of G , then there exist elements g 1 , ⋯ , g k such that G = ( S 1 ) g 1 ⋯ ( S k ) g k . This theorem generalizes an earlier theorem of the authors and Short.

We present a simple uniqueness argument for a collection of McKean–Vlasov problems that have seen recent interest. Our first result shows that, in the weak feedback regime, there is global uniqueness for a very general class of random drivers. By weak feedback we mean the case where the contagion parameters are small enough to prevent blow‐ups in solutions. Next, we specialise to a Brownian driver

For a collection G = { G 1 , ⋯ , G s } of not necessarily distinct graphs on the same vertex set V , a graph H with vertices in V is a G ‐transversal if there exists a bijection ϕ : E ( H ) → [ s ] such that e ∈ E ( G ϕ ( e ) ) for all e ∈ E ( H ) . We prove that for | V | = s ⩾ 3 and δ ( G i ) ⩾ s / 2 for each i ∈ [ s ] , there exists a G ‐transversal that is a Hamilton cycle. This confirms a conjecture

It is a classical fact that for any ε > 0 , a random permutation of length n = ( 1 + ε ) k 2 / 4 typically contains an increasing subsequence of length k . As a far‐reaching generalization, Alon conjectured that a random permutation of this same length n is typically k ‐universal , meaning that it simultaneously contains every pattern of length k . He also made the simple observation that for n = O

We consider a double phase Robin problem with a Carathéodory nonlinearity. When the reaction is superlinear but without satisfying the Ambrosetti–Rabinowitz condition, we prove an existence theorem. When the reaction is resonant, we prove a multiplicity theorem. Our approach is Morse theoretic, using the notion of homological local linking.

A virtual Schottky group is a Kleinian group K containing a Schottky group G as a finite index normal subgroup. These groups correspond to those groups of automorphisms of closed Riemann surfaces which can be realized at the level of their lowest uniformizations. In this paper, we provide a geometrical structural decomposition of K . When K / G is an abelian group, an explicit free product decomposition

We study the existence, nonexistence and multiplicity of positive solutions for a general class of degenerate nonlocal problems. The nonexistence results are provided in two different situations: under monotonicity assumptions involving the nonlinearity and without monotonicity conditions. A multiplicity result of positive solutions with ordered L p ‐norms complement some recent results obtained in

We prove that the signature bound for the topological 4‐genus of 3‐strand torus knots is sharp, using McCoy's twisting method. We also show that the bound is off by at most 1 for 4‐strand and 6‐strand torus knots, and improve the upper bound on the asymptotic ratio between the topological 4‐genus and the Seifert genus of torus knots from 2/3 to 14/27.

We revisit the geometry of involutions in groups of finite Morley rank. The focus is on specific configurations where, as in PGL 2 ( K ) , the group has a subgroup whose conjugates generically cover the group and intersect trivially. Our main result is the subtle yet strong statement that in such configurations the conjugates of the subgroup may not cover all strongly real elements. As an application

We study existence of linear isometric embedding of ℓ p m into S ∞ , for 1 ⩽ p < ∞ , and unique operator space structure on two‐dimensional Banach spaces. For p ∈ ( 2 , ∞ ) ∪ { 1 } , we show that indeed ℓ p 2 does not embed isometrically into S ∞ . This verifies a guess of Pisier and broadly generalizes the main result of Gupta and Reza (Houston J. Math . 44 (2018) 1205–1212). We also show that S 1

Up to Morita equivalence, every quasi‐hereditary algebra is the dual algebra of a directed bocs or a coring. From the bocs, an exact Borel subalgebra is obtained. In this paper a characterisation of exact Borel subalgebras arising in this way is given.

We construct non‐trivial L‐equivalence between curves of genus one and degree five, and between elliptic surfaces of multisection index five. These results give the first examples of L‐equivalence for curves (necessarily over non‐algebraically closed fields) and provide a new bit of evidence for the conjectural relationship between L‐equivalence and derived equivalence.

We obtain a double exponential bound in Brauer's generalisation of van der Waerden's theorem, which concerns progressions with the same colour as their common difference. Such a result has been obtained independently and in much greater generality by Sanders. Using Gowers' local inverse theorem, our bound is quintuple exponential in the length of the progression. We refine this bound in the colour

Let A be the twisted commutative algebra freely generated by d indeterminates of degree 1. We show that the regularity of an A ‐module can be bounded from the first ⌊ 1 4 d 2 ⌋ + 2 terms of its minimal free resolution. This partially extends results of Church and Ellenberg from the d = 1  case.

We show that given generators for subgroups G and H of S n , if G is primitive then generators for N H ( G ) may be computed in quasipolynomial time, namely 2 O ( log 3 n ) . The previous best known bound was simply exponential.

In this note, we show that a closed oriented contact manifold is obtained from the standard contact sphere of the same dimension by contact surgeries on isotropic and coisotropic spheres. In addition, we observe that all closed oriented contact manifolds admit symplectic caps.

The theory of mixed multiplicities of filtrations by m ‐primary ideals in a ring is introduced in [Cutkosky, P. Sarkar and H. Srinivasan, Trans. Amer. Math. Soc. 372 (2019) 6183–6211]. In this paper, we consider the positivity of mixed multiplicities of filtrations. We show that the mixed multiplicities of filtrations must be nonnegative real numbers and give examples to show that they could be zero

Here we survey the systole growth of arithmetic locally symmetric spaces under congruence covering and give a simple proof for the best possible Gromov constant for several important classes of symmetric spaces.

We prove a singular Brascamp–Lieb inequality, stated in Theorem 1, with a large group of involutive symmetries.

Pach showed that every d + 1 sets of points Q 1 , … , Q d + 1 ⊂ R d contain linearly sized subsets P i ⊂ Q i such that all the transversal simplices that they span intersect. We show, by means of an example, that a topological extension of Pach's theorem does not hold with subsets of size C ( log n ) 1 / ( d − 1 ) . We show that this is tight in dimension 2, for all surfaces other than S 2 . Surprisingly