Past Junior Number Theory Seminar

30 January 2017
16:00
Abstract

We outline what we expect from a good cohomology theory and introduce some of the most common cohomology theories. We go on to discuss what properties each should encode and detail attempts to fit them into a common framework. We build evidence for this viewpoint through several worked number theoretic examples and explain how many of the key conjectures in number theory fit into this theory of motives.

  • Junior Number Theory Seminar
24 October 2016
16:00
Giacomo Micheli
Abstract

Let S be a set of monic degree 2 polynomials over a finite field and let C be the compositional semigroup generated by S. In this talk we establish a necessary and sufficient condition for C to be consisting entirely of irreducible polynomials. The condition we deduce depends on the finite data encoded in a certain graph uniquely determined by the generating set S. Using this machinery we are able both to show examples of semigroups of irreducible polynomials generated by two degree 2 polynomials and to give some non-existence results for some of these sets in infinitely many prime fields satisfying certain arithmetic conditions (this is a joint work with A.Ferraguti and R.Schnyder). Time permitting, we will also describe how to use character sum techniques to bound the size of the graph determined by the generating set (this is a joint work with D.R. Heath-Brown).

  • Junior Number Theory Seminar
10 October 2016
16:00
Aled Walker
Abstract

The large sieve is a powerful analytic tool in number theory, with many beautiful and diverse applications. In its most general form it resembles an approximate Bessel's inequality, and this clear modern theory rests on the combined effort of countless mathematicians in the mid-twentieth century -- Linnik, Roth, Selberg, Montgomery, Vaughan, and Bombieri, to name a few. However, it is hardly obvious to the beginner why this rather abstract inequality should be called 'large', or 'sieve'. In this introductory talk, aimed particularly at new graduate students, we discuss the rudimentary theory of the large sieve, some particular applications to sieving problems, and (at least one) proof. 

  • Junior Number Theory Seminar
9 May 2016
16:00
Vandita Patel
Abstract

Given an integer $d$ such that $2 \leq d \leq 50$, we want to
answer the question: When is the sum of
$d$ consecutive cubes a perfect power? In other words, we want to find all
integer solutions to the equation
$(x+1)^3 + (x+2)^3 + \cdots + (x+d)^3 = y^p$. In this talk, we present some
of the techniques used to tackle such diophantine problems.

 

  • Junior Number Theory Seminar

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