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  1. ; (, Inventiones mathematicae)
    For and finite groups, does there exist a 3-manifold group with as a quotient but no as a quotient? We answer all such questions in terms of the group cohomology of finite groups. We prove non-existence with topological results generalizing the theory of semicharacteristics. To prove existence of 3-manifolds with certain finite quotients but not others, we use a probabilistic method, by first proving a formula for the distribution of the (profinite completion of) the fundamental group of a random 3-manifold in the Dunfield-Thurston model of random Heegaard splittings as the genus goes to infinity. We believe this is the first construction of a new distribution of random groups from its moments. 
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  2. ; ; (, Transactions of the American Mathematical Society)
    We construct a perverse sheaf related to the the matrix exponential sums investigated by Erdélyi and Tóth [Matrix Kloosterman sums, 2021, arXiv:2109.00762]. As this sheaf appears as a summand of certain tensor product of Kloosterman sheaves, we can establish the exact structure of the cohomology attached to the sums by relating it to the Springer correspondence and using the recursion formula of Erdélyi and Tóth. 
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  3. (, Journal d'Analyse Mathématique)
    We study p-adic hyper-Kloosterman sums, a generalization of the Kloosterman sum with a parameter k that recovers the classical Kloosterman sum when k = 2, over general p-adic rings and even equal characteristic local rings. These can be evaluated by a simple stationary phase estimate when k is not divisible by p, giving an essentially sharp bound for their size. We give a more complicated stationary phase estimate to evaluate them in the case when k is divisible by p. This gives both an upper bound and a lower bound showing the upper bound is essentially sharp. This generalizes previously known bounds in the case of ℤ/p. The lower bounds in the equal characteristic case have two applications to function field number theory, showing that certain short interval sums and certain moments of Dirichlet L-functions do not, as one might hope, admit square-root cancellation. 
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