The Hardy-Littlewood circle method provides a framework for combining Fourier analysis and analytic number theory to answer questions about counting the number of solutions to certain arithmetic equations. This talk provides a survey of the method, including applications both in analytic number theory and in the analysis of singular integrals and summations. The more modern improvements due to Vinogradov are included, and we briefly discuss their implications to both the analysis and the assumptions involved. This includes a discussion on the modern limitations and challenges with the circle method, and we conclude by discussing how higher order Fourier analysis can help replace the circle method in some cases.

We will have this on zoom as well.

https://unm.zoom.us/j/98232489187

Meeting ID: 982 3248 9187
Passcode: 235711