Lior Silberman

Associate Professor

Research Classification

Research Interests

Analysis on manifolds
Automorphic forms
Group Theory
Homogenous dynamics
Metric geometry
Number theory
Representation Theory

Relevant Thesis-Based Degree Programs



Master's students
Doctoral students
Postdoctoral Fellows

Mathematics, almost any field

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Graduate Student Supervision

Doctoral Student Supervision

Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.

The Topology of Representation Varieties (2016)

The goal of this thesis is to understand the topology of representation varieties. To be more precise, let G be a complex reductive linear algebraic group and let K ⊂ G be a maximal compact subgroup. Given a finitely generated nilpotent group Γ, we consider the representation spaces Hom(Γ,G) and Hom(Γ,K) endowed with the compact-open topology. Our main result shows that there is a strong deformation retraction of Hom(Γ,G) onto Hom(Γ,K). We also obtain a strong deformation retraction of the geometric invariant theory quotient Hom(Γ,G)//G onto the ordinary quotient Hom(Γ,K)/K. Using these deformations, we then describe the topology of these spaces.

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Master's Student Supervision

Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.

Presenting higher categories and weak functors via multi-opetopic nerves and terminal coalgebras (2021)

In this thesis, we construct a convenient presentation of weak n-categories for 0
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Sup-norm Problem of Certain Eigenfunctions on Arithmetic Hyperbolic Manifolds (2015)

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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Geometric retracts of Siegel's upper half space (2013)

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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Escape of Mass on Hibter Modular Varieties (2012)

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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