Information
From 01/01/2021-31/12/2025 I am the PI of ERC Starting Grant project UBIQGAP (grant no. 949143)
`The ubiquity of optimal spectral gaps'
Team members
Irving Calderón (PDRA)
Joe Thomas (PDRA)
Anitej Banerjee (Ph.D. Student)
Ewan Cassidy (Ph.D. Student)
[Click for an abridged abstract]
Spectral gap is a fundamental concept in mathematics, physics, and computer science as it governs the exponential rate at which a process converges towards its stationary state. It informs the spectral lines of hydrogen, how we shuffle cards, the behavior of semiconductors, and web search algorithms.
Moreover, some of the most prominent issues of contemporary mathematics, including the Ramanujan-Petersson conjecture and the Yang-Mills mass gap, revolve around spectral gap. This proposal seeks to investigate the nature of the spectral gap for hyperbolic surfaces and unitary representations of fundamental groups of surfaces.
In the former case, the spectral gap occurs in the spectrum of the Laplace-Beltrami operator on the surface, and in the latter, it occurs in the spectrum of a Hecke operator attached to the representation. The two main motifs of the proposal are ubiquity and optimality. Is the spectral gap ubiquitous? Does it exist for random surfaces and random representations? Is it easy to construct surfaces with a large spectral gap? In what cases can one prove that the spectral gap is close to optimal? The sharpest and most ambitious questions discussed in this proposal combine these two aspects and ask whether objects with (almost) optimal spectral gap appear with high frequency.
A main technical tool is the development of new formulas for integration over representation varieties of fundamental groups of surfaces. These integral formulas are of high independent interest. For example, I propose to establish estimates that extend important results in Voiculescu's Free Probability Theory from the context of free groups, to fundamental groups of closed compact surfaces, and beyond.
Moreover, some of the most prominent issues of contemporary mathematics, including the Ramanujan-Petersson conjecture and the Yang-Mills mass gap, revolve around spectral gap. This proposal seeks to investigate the nature of the spectral gap for hyperbolic surfaces and unitary representations of fundamental groups of surfaces.
In the former case, the spectral gap occurs in the spectrum of the Laplace-Beltrami operator on the surface, and in the latter, it occurs in the spectrum of a Hecke operator attached to the representation. The two main motifs of the proposal are ubiquity and optimality. Is the spectral gap ubiquitous? Does it exist for random surfaces and random representations? Is it easy to construct surfaces with a large spectral gap? In what cases can one prove that the spectral gap is close to optimal? The sharpest and most ambitious questions discussed in this proposal combine these two aspects and ask whether objects with (almost) optimal spectral gap appear with high frequency.
A main technical tool is the development of new formulas for integration over representation varieties of fundamental groups of surfaces. These integral formulas are of high independent interest. For example, I propose to establish estimates that extend important results in Voiculescu's Free Probability Theory from the context of free groups, to fundamental groups of closed compact surfaces, and beyond.
Team members
Irving Calderón (PDRA)
Joe Thomas (PDRA)
Anitej Banerjee (Ph.D. Student)
Ewan Cassidy (Ph.D. Student)
| Outputs of the program | ||
|---|---|---|
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Explicit spectral gap for Schottky subgroups of SL_2(Z)
(With Irving Calderón) Preprint | |
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Strongly convergent unitary representations of limit groups.
Michael Magee and Lars Louder Preprint, contains an appendix by Will Hide and Michael Magee | |
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Short geodesics and small eigenvalues on random hyperbolic punctured spheres.
Will Hide and Joe Thomas
Preprint |
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Quantum Unique Ergodicity for Cayley graphs of quasirandom groups.
Michael Magee and Joe Thomas Preprint |
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Extension of Alon's and Friedman's conjectures to Schottky surfaces.
Michael Magee and Frédéric Naud
Preprint |
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Random Unitary Representations of Surface Groups II: The large n limit.
Michael Magee
Preprint |
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The asymptotic statistics of random covering surfaces.
(With Doron Puder)
Forum of Mathematics, Pi, to appear. |
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Near optimal spectral gaps for hyperbolic surfaces.
(With Will Hide)
Annals of Mathematics, to appear. [One hour talk] [Quanta article] |
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A random cover of a compact hyperbolic surface has relative
spectral gap 3/16 - ϵ.
Michael Magee, Frédéric Naud, and
Doron Puder
Geometric and Functional Analysis (GAFA), 2022. |
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Core surfaces.
Michael Magee and Doron Puder
Geometriae Dedicata, 2022. |
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Random Unitary Representations of Surface Groups I: Asymptotic expansions.
Michael Magee
Communications in Mathematical Physics, 2022. |