Differential equations in automorphic forms and an application to particle physics
2019-05
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Differential equations in automorphic forms and an application to particle physics
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2019-05
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Physicists such as Green, Vanhove, et al show that differential equations involving automorphic forms govern the behavior of gravitons. One particular point of interest is solutions to $(\Delta-\lambda)u=E_{\alpha} E_{\beta}$ on an arithmetic quotient of the exceptional group $E_8$. We establish that the existence of a solution to $(\Delta-\lambda)u=E_{\alpha}E_{\beta}$ on the simpler space $SL_2(\Z)\backslash SL_2(\R)$ for certain values of $\alpha$ and $\beta$ depends on nontrivial zeros of the Riemann zeta function $\zeta(s)$. Further, when such a solution exists, we use spectral theory to solve $(\Delta-\lambda)u=E_{\alpha}E_{\beta}$ on $SL_2(\Z)\backslash SL_2(\R)$ and provide proof of the meromorphic continuation of the solution. The construction of such a solution uses Arthur truncation, the Maass-Selberg formula, and automorphic Sobolev spaces.
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University of Minnesota Ph.D. dissertation. May 2019. Major: Mathematics. Advisor: Paul Garrett. 1 computer file (PDF); vi, 92 pages.
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Logan, Kimberly. (2019). Differential equations in automorphic forms and an application to particle physics. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/206231.
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