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Pach showed that every d+1 sets of points Q1,…,Qd+1⊂Rd contain linearly sized subsets Pi⊂Qi 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(logn)1/(d−1). We show...
We prove a singular Brascamp–Lieb inequality, stated in Theorem 1, with a large group of involutive symmetries.
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 ⌊14d2⌋+2 terms of its minimal free resolution. This partially extends results of Church and Ellenberg from the d=1 case.
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,...
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...
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 show that given generators for subgroups G and H of Sn, if G is primitive then generators for NH(G) may be computed in quasipolynomial time, namely 2O(log3n). The previous best known bound was simply exponential.
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.
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.
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