Lior Silberman

In this thesis, we construct a convenient presentation of weak n-categories for 0 = n = omega whose corresponding weak n-functors in particular do not strictly preserve units.The approach constructs the finite-dimensional higher categories inductively using multi-opetopic sets, analogous to the construction of Segal categories, and multi-simplicial nerves of Tamsamani.We prove that our construction is correct in dimension two by providing a fully faithful and weakly essentially surjective functor from the category of bicategories and pseudofunctors to our weak 2-categories.For the infinite-dimensional case, we realise the weak omega-categories as formal limits of their finite-dimensional truncations, using coalgebras over an appropriate endofunctor.We also prove that these weak omega-categories admit an equivalent characterisation as infinitary opetopic sets subject to constraints analogous to the finite-dimensional case.Finally, we specialise our construction to infinity-groupoids and prove that the coalgebraic structure induces a canonical functor from nice topological spaces that defines the Poincaré infinity-groupoid construction.We show that the Poincaré infinity-groupoid has the correct higher morphisms in all dimensions and retains the information about all homotopy groups of the space.Moreover, we show that this construction preserves and reflects weak equivalences.We conclude by proposing a construction that likely recovers a space from its Poincaré infinity-groupoid which conjecturally establishes a version of the Homotopy Hypothesis.

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We prove a power saving over the local bound for the L∞ norm of uniformly non-tempered Hecke-Maass forms on arithmetic hyperbolic manifolds of dimension 4and 5. We use accidental isomorphism and use the Hecke theory of the correspond-ing groups to show that if the automorphic form is non-tempered at positive densityof finite places then the Hecke eigenvalues are large; amplifying the saving comingfrom the non temperedness we get a power saving.

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The purpose of this thesis is to construct a codimension 1 cocompact Sp₂gℤ-equivariant strong deformation retract Wg of Siegel's upper half space hg . This yields a partial contribution towards the problem of constructing a strong spine for the real linear symplectic group Sp₂gℝ.

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Let F be a number field, G = PGL(2,F_∞), and K be a maximal compact subgroup of G. We eliminate the possibility of escape of mass for measures associated to Hecke-Maaß cusp forms on Hilbert modular varieties, and more generally on congruence locally symmetric spaces covered by G/K, hence enabling its application to the non-compact case of the Arithmetic Quantum Unique Ergodicity Conjecture. This thesis generalizes work by Soundararajan in 2010 eliminating escape of mass for congruence surfaces, including the classical modular surface SL(2,Z)\H², and follows his approach closely.First, we define M, a congruence locally symmetric space covered by G/K, and articulate the details of its structure. Then we define Hecke-Maass cusp forms and provide their Whittaker expansion along with identities regarding the Whittaker coefficients. Utilizing these identities, we introduce mock P-Hecke multiplicative functions and bound a key related growth measure following Soundararajan’s paper. Finally, amassing our results, we eliminate the possibility of escape of mass for Hecke-Maass cusp forms on M.

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