Automorphic forms on an arithmetic Riemmanian manifold are certain smooth functions that possess extensive arithmetic symmetries underlying the manifold and satisfy certain intrinsic partial differential equations. These forms naturally give birth to certain complex meromorphic functions called L-functions, like the classical Riemann zeta functions. L-functions inherit deep arithmetic information from their automorphic forms. For instance, the location of the zeros of the L-functions, which is described by the famous Generalized Riemann Hypothesis, carries information on the distribution of the prime numbers. The Generalized Riemann Hypothesis is completely out of reach of the available technology. However, there are other fascinating questions regarding the growth of L-values that are more tractable.
The primary goal of this project, broadly, will be improving upon recent ground-breaking progress by Paul Nelson on the problems of estimating the growth of automorphic forms and their L-functions. Such problems can, philosophically, be categorized as a very soft version of the Generalized Riemann Hypothesis and their solutions will yield sophisticated arithmetic and analytic knowledge. For instance, non-trivial growth estimates of L-functions for symmetric square representations will have diverse applications in various equidistribution problems in mathematics and physics, e.g., quantum chaos. A concrete goal of this project is to explicate various estimates in the proof of Nelson that would effectively improve the existing bounds for L-functions, at least the low-degree ones.
The studentship is funded by Queen Mary and will cover home tuition fees, and provide an annual tax-free maintenance allowance for 3.5 years at the UKRI rate (£19,668 in 2022/23).
For international students interested in applying, please note that this studentship only covers home tuition fees and students will need to cover the difference in fees between the home and overseas basic rate. Tuition fee rates for 2023-24 are to be confirmed. Details on current (2022-23) tuition fee rates can be found at: https://www.qmul.ac.uk/postgraduate/research/funding_phd/tuition-fees/
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Fees and funding