Objective
Understanding the unitary representations of a given group G is one of the most persistent problems
of mathematics. If G is the integers, the ensuing theory is that of the Fourier transform. If G is
the Heisenberg group, then the resulting representation theory is the theory of matrix models of
Quantum Mechanics. Accelerating, if G is the absolute Galois group of the rationals, the theory is
described by the Langlands program.
My goal is to understand the finite dimensional (f.d.) unitary representations of discrete groups
through the lens of how they can converge to the regular representation. I describe both weak and
strong forms of convergence and focus mainly on strong convergence. I first ask which groups have f.d. unitary representations that strongly converge to their regular
representation? What if we require representations to factor through permutation groups? These
questions are deep, wide-ranging, and push far beyond the state-of-the-art.
Next I ask to what extent strong convergence of f.d. unitary representations is generic, when we
have a way to randomize representations. In particular this applies to the fundamental groups of
closed surfaces, which are a test bed for the current program. In many cases randomization is the only tool we know to establish strong convergence, so we have
as yet no explicit examples of the phenomenon that the proposal in centered on! We present an
algebraic candidate that is intimately related to Selberg’s Eigenvalue Conjecture in automorphic
forms. Most of the above questions have spectacular consequences to spectral gaps of locally symmetric
spaces, a connection that I discovered with Hide. We do not understand this outside special cases
yet. I ambitiously aim to completely describe the connections between strong convergence of representations
of a lattice and their induced representations of the ambient Lie group.
Finally, we imagine what lies beyond strong convergence and whether random matrix theory can
take us there.
Keywords
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Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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HORIZON.1.1 - European Research Council (ERC)
MAIN PROGRAMME
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Topic(s)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Funding Scheme
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Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
HORIZON-ERC - HORIZON ERC Grants
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Call for proposal
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Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) ERC-2025-COG
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DH1 3LE DURHAM
United Kingdom
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