Rational Points on Varieties

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The research project lies in the area of pure mathematics, specifically algebraic geometry and number theory. Pure mathematics is highly abstract and bordering upon the esoteric, but fundamentally crucial for society through numerous applications, such as cryptography and data science. The project will make progress in the area of rational points on varieties, which is the modern study of Diophantine equations. The fundamental theme will be to study the existence and distribution of solutions to these equations, which are of interest to number theorists. Specific goals will be to prove new results on the failure of the Hasse principle for del Pezzo surfaces and also prove new cases of Manin's conjecture for del Pezzo surfaces and closely related varieties.This year I am focusing more on number theory; the main aim is to try to remove the restriction on the order of given group G of a theorem by David J. Wright, using the method developed by my supervisor and his collaborators. The theorem is about the asymptotic formula of the number of Galois extensions over a global function field for fixed Galois group G, which requires the order of G is not divisible by the character of the base field.