Forthcoming events in this series
What the L! The surprising world of L-functions
Abstract
L-functions have become a vital part of modern number theory over the past century, allowing comparisons between arithmetic objects with seemingly very different properties. In the first part of this talk, I will give an overview of where they arise, their properties, and the mathematics that has developed in order to understand them. In the second part, I will give a sketch of the beautiful result of Herbrand-Ribet concerning the arithmetic interpretations of certain special values of the Riemann zeta function, the prototypical example of an L-function.
When shifted primes do not occur in difference sets
Abstract
Let $[N] = \{1,..., N\}$ and let $A$ be a subset of $[N]$. A result of Sárközy in 1978 showed that if the difference set $A-A = \{ a - a’: a, a’ \in A\}$ does not contain any number which is one less than a prime, then $A = o(N)$. The quantitative upper bound on $A$ obtained from Sárközy’s proof has be improved subsequently by Lucier, and by Ruzsa and Sanders. In this talk, I will discuss my work on this problem. I will give a brief introduction of the iteration scheme and the Hardy-Littlewood method used in the known proofs, and our major arc estimate which leads to an improved bound.
Erdős' primitive set conjecture
Abstract
A subset of the integers larger than 1 is called $\textit{primitive}$ if no member divides another. Erdős proved in 1935 that the sum of $1/(n \log n)$ over $n$ in a primitive set $A$ is universally bounded for any choice of $A$. In 1988, he famously asked if this universal bound is attained by the set of prime numbers. In this talk we shall discuss some recent progress towards this conjecture and related results, drawing on ideas from analysis, probability, & combinatorics.
On Serre's Uniformity Conjecture
Abstract
Given a prime p and an elliptic curve E (say over Q), one can associate a "mod p Galois representation" of the absolute Galois group of Q by considering the natural action on p-torsion points of E.
In 1972, Serre showed that if the endomorphism ring of E is "minimal", then there exists a prime P(E) such that for all p>P(E), the mod p Galois representation is surjective. This raised an immediate question (now known as Serre's uniformity conjecture) on whether P(E) can be bounded as E ranges over elliptic curves over Q with minimal endomorphism rings.
I'll sketch a proof of this result, the current status of the conjecture, and (time permitting) some extensions of this result (e.g. to abelian varieties with appropriately analogous endomorphism rings).
What is Arakelov Geometry?
Abstract
Arakelov geometry studies schemes X over ℤ, together with the Hermitian complex geometry of X(ℂ).
Most notably, it has been used to give a proof of Mordell's conjecture (Faltings's Theorem) by Paul Vojta; curves of genus greater than 1 have at most finitely many rational points.
In this talk, we'll introduce some of the ideas behind Arakelov theory, and show how many results in Arakelov theory are analogous—with additional structure—to classic results such as intersection theory and Riemann Roch.
Cartier Operators
Abstract
Given a morphism of schemes of characteristic p, we can construct a morphism from the exterior algebra of Kahler differentials to the cohomology of De Rham complex, which is an isomorphism when the original morphism is smooth.
Relative decidability via the tilting correspondence
Abstract
The goal of the talk is to present a proof of the following statement:
Let (K,v) be an algebraic extension of (Q_p,v_p) whose completion is perfectoid. We show that K is relatively decidable to its tilt K^♭, i.e. if K^♭ is decidable in the language of valued fields, then so is K.
In the first part [of the talk], I will try to cover the necessary background needed from model theory and the theory of perfectoid fields.
From Chabauty's Method to Kim's Non-Abelian Chabauty's Method
Abstract
In 1941, Chabauty gave a way to compute the set of rational points on specific curves. In 2004, Minhyong Kim showed how to extend Chabauty's method to a bigger class of curves using anabelian methods. In the talk, I will explain Chabauty's method and give an outline of how Kim extended those methods.
The Golod-Shafarevich Theorem: Endgame
Abstract
The principal ideal theorem (1930) guaranteed that any number field K would embed into a finite extension, called the Hilbert class field of K, in which every ideal of the original field became principal -- however the Hilbert class field itself will not necessarily have class number 1. The class field tower problem asked whether iteratively taking Hilbert class fields must stabilize after finitely many steps. In 1964, it was finally answered in the negative by Golod and Shafarevich who produced infinitely many examples and pioneered the framework that is still the most common setting for deciding when a number field will have an infinite class field tower.
In this talk, I will finish the proof of their cohomological result and thus fully justify how it settled the class field tower problem.
The Golod-Shafarevich Theorem
Abstract
The principal ideal theorem (1930) ascertained that any number field K embeds into a finite extension, called the Hilbert class field of K, in which every ideal of the original field became principal -- however the Hilbert class field itself will not necessarily have class number 1. The class field tower problem asked whether iteratively taking Hilbert class fields must stabilize after finitely many steps. In 1964, it was finally answered in the negative by Golod and Shafarevich who produced infinitely many examples and pioneered the framework that is still the most common setting for deciding when a number field will have an infinite class field tower.
In this talk, I will sketch the proof of their cohomological result and explain how it settled the class field tower problem.
Periods and the number Zagier forgot
Abstract
A particularly active area of research in modern algebraic number theory is the study of a class of numbers, called periods. In their simplest form, periods are integrals of rational functions over domains defined by rational in equations. They form a ring, which encompasses all algebraic numbers, logarithms thereof and \pi. They arise in the study of modular forms, cohomology and quantum field theory, and conjecturally have a sort of Galois theory.
We will take a whirlwind tour of these numbers, before discussing non-periods. In particular, we will sketch the construction of an explicit non-period, often forgotten about.
Cohomology of Varieties
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.
On sets of irreducible polynomials closed by composition
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).
16:00
The large sieve
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.
16:00
Curves and their fundamental groups
Abstract
I will describe a sketch of the proof of Grothendieck conjecture on fundamental groups.
16:00
Descent of a sum of Consecutive Cubes ... Twice!!
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.
16:00
Explicit Kummer coordinates for higher genus curves
Abstract
I will explain how to find an explicit embedding of the Kummer variety of a higher genus curve into projective space and discuss applications of such an embedding to the study of rational points on Jacobians of curves, as well as the original curves.
16:00
16:30
16:30
Torelli and Borel-Tits theorems via trichotomy
Abstract
Using the "trichotomy principle" by Boris Zilber I will give model theoretic proofs of appropriate versions of Torelli theorem and Borel-Tits theorem. The first one has interesting applications to anabelian geometry, I won't assume any prior knowledge in model theory.
16:30
Congruence and non-congruence level structures on elliptic curves: a hands-on tour of the modular tower
Abstract
16:30
Partition regularity of $x+y=z^2$ over $\mathbb{Z}/p\mathbb{Z}$
Abstract
Consider the following question. Given a $k$-colouring of the positive integers, must there exist a solution to $x+y=z^2$ with $x,y,z$ all the same colour (and not all equal to 2)? Using $10$ colours a counterexample can be given to show that the answer is "no". If one instead asks the same question over $\mathbb{Z}/p\mathbb{Z}$ for some prime $p$, the answer turns out to be "yes", provided $p$ is large enough in terms of the number of colours used. I will talk about how to prove this using techniques developed by Ben Green and Tom Sanders. The main ingredients are a regularity lemma, a counting lemma and a Ramsey lemma.
16:30
The degree zero part of the motivic polylogarithm and the Deligne-Beilinson cohomology
Abstract
Last year, G. Kings and D. Rossler related the degree zero part of the polylogarithm
on abelian schemes pol^0 with another object previously defined by V. Maillot and D.
Rossler. More precisely, they proved that the canonical class of currents constructed
by Maillot and Rossler provides us with the realization of pol^0 in analytic Deligne
cohomology.
I will show that, adding some properness conditions, it is possible to give a
refinement of Kings and Rossler’s result involving Deligne-Beilinson cohomology
instead of analytic Deligne cohomology.