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Abstract: Lattices are fundamental objects in physics, mathematics and computer science. Starting from a cubic lattice, say, we can perturb the structure by linear transformations (shearing, stretching, rotating) to obtain a whole family of lattices. I will discuss the resulting "space of lattices", the dynamics of group actions on this space, natural probability measures, as well as some fascinating applications to long-standing problems in various areas of mathematics and mathematical physics. My plan is to tell you about kinetic transport in crystals and quasicrystals (the Lorentz gas), pseudo-random properties of simple

arithmetic sequences, knapsack problems, diameters of random Cayley graphs and (time permitting) subtle lattice point counting problems in hyperbolic geometry.

Bio: Jens Marklof is Professor of Mathematical Physics at the University of Bristol, specialising in dynamical systems and ergodic theory, quantum chaos, and the theory of automorphic forms. Marklof received his PhD in 1997 from the University of Ulm, and held research fellowships at Princeton University, Hewlett-Packard, the Isaac Newton Institute in Cambridge, the Institut des Hautes Etudes Scientifique and the Laboratoire de Physique Theorique et Modeles Statistiques near Paris. He delivered a plenary address at the

International Congress of Mathematical Physics in Prague 2009, and was an invited section speaker at the International Congress of Mathematicians in Seoul 2014. Major awards include a 2010 LMS Whitehead Prize and a five-year ERC Advanced Grant. In 2015 Marklof was elected a Fellow of the Royal Society, the UK's national academy of sciences. From November 2023 he will serve a two-year term as President of the London Mathematical Society.