Multivariate cryptography is one of the main candidates to guarantee the security of communication in the presence of quantum computers. While there exist a large number of secure and efficient multivariate signature schemes, the number of practical multivariate encryption schemes is somewhat limited. In this paper we present our results on creating a new multivariate encryption scheme, which is an extension of the original SimpleMatrix encryption scheme of PQCrypto 2013. Our scheme allows fast en- and decryption and resists all known attacks against multivariate cryptosystems. Furthermore, we present a new idea to solve the decryption failure problem of the original SimpleMatrix encryption scheme.

Li-Nadler proposed a conjecture about traces of Hecke categories, which implies the semistable part of the Betti geometric Langlands conjecture of Ben-Zvi-Nadler in genus 1. We prove a Weyl group analogue of this conjecture. Our theorem holds in the natural generality of reflection groups in Euclidean or hyperbolic space. As a corollary, we give an expression of the centralizer of a finite order element in a reflection group using homotopy theory.

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We give a solution to the inverse problem (given a prescribed function, find a corresponding group) for large classes of speed, entropy, isoperimetric profile, return probability and Lp-compression functions of finitely generated groups. For smaller classes, we give solutions among solvable groups of exponential volume growth. As corollaries, we prove a recent conjecture of Amir on joint evaluation of speed and entropy exponents and we obtain a new proof of the existence of uncountably many pairwise non-quasi-isometric solvable groups, originally due to Cornulier and Tessera. We also obtain a formula relating the Lp-compression exponent of a group and its wreath product with the cyclic group forp in [1,2].

For a random walk on a finitely generated group G we obtain a generalization of a classical inequality of Ancona. We deduce as a corollary that the identity map on G extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being singletons. This provides new results for Martin compactifications of relatively hyperbolic groups.

We study large groups of birational transformations Bir(X), where X is a variety of dimension at least 3, defined over C or a subfield of C. Two prominent cases are when X is the projective space P^n, in which case Bir(X) is the Cremona group of rank n, or when X⊂P^{n+1} is a smooth cubic hypersurface. In both cases, and more generally when X is birational to a conic bundle, we produce infinitely many distinct group homomorphisms from Bir(X) to Z/2, showing in particular that the group Bir(X) is not perfect, and thus not simple. As a consequence, we also obtain that the Cremona group of rank n⩾3 is not generated by linear and Jonquières elements.

We construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree. We use them to define infinite graphical small cancellation presentations of groups. This technique allows us to provide examples of groups with exotic properties: • We construct the first examples of finitely generated coarsely non-amenable groups (that is, groups without Guoliang Yu’s Property A) that are coarsely embeddable into a Hilbert space. Moreover, our groups act properly on CAT(0) cubical complexes. • We construct the first examples of finitely generated groups, with expanders embedded isometrically into their Cayley graphs—in contrast, in the case of the Gromov monster expanders are not even coarsely embedded. We present further applications.

In this paper we generalize a result in [J. An, Z. Wang, On the realization of Riemannian symmetric spaces in Lie groups, Topology Appl. 153 (7) (2005) 10081015, showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of fixed points of involutions are also proved.

(1) Ker (f,) is a characteristic subgroup of T,(M),(2) the group G leaves Ker (f*) invariant. This assumption is needed in the last two statements of Corollary 6,(vi) of Theorems 8, Theorem 11 and Theorem 13. In Theorem 12, the corresponding assumption can be stated as follows: Let H be the subspace of H,(M, I?) defined by {PIP U ai, U... U ai, _,= 0 for all ii}. Then G leaves H invariant.