Topics will include some or all of the following: valuation theory, algebraic function fi elds, divisors, extensions of function fields, class groups, elliptic and hyperelliptic curves and their function fields. This course represents a basic introduction to function fields and algebraic curves over finite fields from a number theoretic perspective. This allows us to provide these very powerful tools to students at an early stage and hope that one day they can discover many more beautiful applications of these tools.Ī n Introduction to Global Function Fields Although this limits the strength of the theorems and the scope of their application, the trade off is that only a minimal prerequisite is required. The goal of this course is to introduce versions of these theorems involving only absolute values over the rational numbers. Without the Subspace Theorem, several elementary-to-state yet surprisingly-hard-to-prove results would have remained open. Roth's Theorem and especially its generalization, the Subspace Theorem, are among the milestones of diophantine geometry in the second half of the 20th century. Roth's Theorem, Subspace Theorem, and Some Applications Some basic notions on Fourier analysis or complex analysis are preferable (background material shall be provided). We invite the participants to discover a few notions regarding the zeros of the Riemann zeta function as well as some analytical tools to deduce the prime number theorem. First conjectured by Gauss, then proven by de la Vallée Poussin and Hadamard, building upon some groundbreaking ideas of Dirichlet and Riemann, this result is seminal in the field of analytic number theory. The Prime Number Theorem gives an estimate for the number of primes pi(x) up to an asymptotically large number x. Students from under-represented groups or those committed to EDI are encouraged to apply. In exemplary cases, upper-year undergraduate students will also be considered. The summer school is * free* and open to MSc students from a Canadian or North-West American University. The goal of this event is to promote positive cooperation and collaboration amongst students, while bringing light to equity, diversity, and inclusion issues within the study of Mathematics. This summer school will introduce students to Number Theory topics being researched throughout Alberta: algebraic number theory, diophantine geometry, and analytic number theory. You may immediately conclude that the next number after 10 is 12. If I show you the following list: 2, 4, 6, 8, 10. #Basic number theory fullLife is full of patterns, but often times, we do not realize as much as we should that mathematics too is full of patterns.
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