A lattice is defined as the set of all integer combinations of $n$ linearly independent vectors in $\R^n$. These geometrical objects possess a rich combinatorial structure that has attracted the attention of great mathematicians over the last two centuries. Lattices have an impressive number of applications in mathematics and computer science, from number theory and Diophantine approximation to complexity theory and cryptography. Over the last two decades, the computational study of lattices has witnessed several remarkable discoveries. Most notable are the development of the LLL algorithm by Lenstra, Lenstra and Lovasz and Ajtai's discovery of lattice-based cryptographic constructions. I propose to pursue these research directions and attempt to discover new connections between lattices and computer science. A particular focus will be put on applications in cryptography, as these can lead to many advances in the field and are also of great practical importance. I believe that the extraordinary properties of lattices have the potential to revolutionize many other areas of computer science such as complexity, cryptography, machine learning theory, quantum computation, and more. My scientific goals include obtaining stronger and more practical lattice-based cryptographic constructions, resolving important questions regarding the complexity of lattice problems, finding sub-exponential time algorithms for lattice problems and exploring some novel applications of lattices to areas such as Markov chains and machine learning theory.
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