University College London

We develop the asymptotic formulas for correlations  
\[ \sum_{n\le x}f_1(P_1(n))f_2(P_2(n))\cdot \dots \cdot f_m(P_m(n))\]

where $f_1,\dots,f_m$ are bounded ``pretentious" multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences: first, we characterize all multiplicative functions $f:\mathbb{N}\to\{-1,+1\}$ with bounded partial sums. This answers a question of Erd{\"o}s from $1957$ in the form conjectured by Tao. Second, we show that if the average of the first divided difference of multiplicative function is zero, then either $f(n)=n^s$ for $\operatorname{Re}(s)<1$ or $|f(n)|$ is small on average. This settles an old conjecture of K\'atai. Third, we discuss applications to the study of sign patterns of $(f(n),f(n+1),f(n+2))$ and $(f(n),f(n+1),f(n+2),f(n+3))$ where $f:\mathbb{N}\to \{-1,1\}$ is a given multiplicative function. If time permits, we discuss multidimensional version of some of the results mentioned above.
 

Structure-preserving signatures are an important cryptographic primitive that is useful for the design of modular cryptographic protocols. In this work, we show how to bypass most of the existing lower bounds in the most efficient Type-III bilinear group setting. We formally define a new variant of structure-preserving signatures in the Type-III setting and present a number of fully secure schemes with signatures half the size of existing ones. We also give different constructions including constructions of optimal one-time signatures. In addition, we prove lower bounds and provide some impossibility results for the variant we define. Finally, we show some applications of the new constructions.

In dimension three, convex surface theory implies that every tight contact structure on a connected sum M # N can be constructed as a connected sum of tight contact structures on M and N. I will explain some examples showing that this is not true in any dimension greater than three.  The proof is based on a recent higher-dimensional version of a classic result of Eliashberg about the symplectic fillings of contact manifolds obtained by subcritical surgery. This is joint work with Paolo Ghiggini and Klaus Niederkrüger.

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I will discuss some new results on the Iwasawa theory for the $3$-dimensional symmetric square Galois representation of a modular form, using the Euler system of Beilinson-Flach elements I constructed in joint work with Kings, Lei and Loeffler.

 We describe how cross-kernel matrices, that is, kernel matrices between the data and a custom chosen set of `feature spanning points' can be used for learning. The main potential of cross-kernel matrices is that (a) they provide Nyström-type speed-ups for kernel learning without relying on subsampling, thus avoiding potential problems with sampling degeneracy, while preserving the usual approximation guarantees and the attractive linear scaling of standard Nyström methods and (b) the use of non-square matrices for kernel learning provides a non-linear generalization of the singular value decomposition and singular features. We present a novel algorithm, Ideal PCA (IPCA), which is a cross-kernel matrix variant of PCA, showcasing both advantages: we demonstrate on real and synthetic data that IPCA allows to (a) obtain kernel PCA-like features faster and (b) to extract novel features of empirical advantage in non-linear manifold learning and classification.

We consider three nonparametric hypothesis testing problems: (1) Given samples from distributions p and q, a homogeneity test determines whether to accept or reject p=q; (2) Given a joint distribution p_xy over random variables x and y, an independence test investigates whether p_xy = p_x p_y, (3) Given a joint distribution over several variables, we may test for whether there exist a factorization (e.g., P_xyz = P_xyP_z, or for the case of total independence, P_xyz=P_xP_yP_z).

We present nonparametric tests for the three cases above, based on distances between embeddings of probability measures to reproducing kernel Hilbert spaces (RKHS), which constitute the test statistics (eg for independence, the distance is between the embedding of the joint, and that of the product of the marginals). The tests benefit from years of machine research on kernels for various domains, and thus apply to distributions on high dimensional vectors, images, strings, graphs, groups, and semigroups, among others. The energy distance and distance covariance statistics are also shown to fall within the RKHS family, when semimetrics of negative type are used. The final test (3) is of particular interest, as it may be used in detecting cases where two independent causes individually have weak influence on a third dependent variable, but their combined effect has a strong influence, even when these variables have high dimension.

I will explain some recent joint work with Georgios Dimitroglou Rizell in which we use moduli spaces of holomorphic discs with boundary on a monotone Lagrangian torus in ${\mathbb C}^n$ to prove that all such tori are smoothly isotopic when $n$ is odd and at least 5