Past Number Theory Seminar

13 October 2016
16:00
Stanley Yao Xiao
Abstract

Let $F$ be a binary form of degree $d \geq 3$ with integer coefficients and non-zero discriminant. In this talk we give an asymptotic formula for the quantity $R_F(Z)$, the number of integers in the interval $[-Z,Z]$ representable by the binary form $F$.

This is joint work with C.L. Stewart.

  • Number Theory Seminar
16 June 2016
16:00
Roger Heath-Brown
Abstract

Let $a_1, \cdots, a_N$ be the sequence of y-smooth numbers up to x (i.e. composed only of primes up to y). When y is a small power of x, what can one say about the size of the gaps $a_{j+1}-a_j$? In particular, what about

$$\sum_1^N (a_{j+1}-a_j)^2?$$

  • Number Theory Seminar
9 June 2016
16:00
Joni Teräväinen
Abstract

When considering $E_k$ numbers (products of exactly $k$ primes), it is natural to ask, how they are distributed in short intervals. One can show much stronger results when one restricts to almost all intervals. In this context,  we seek the smallest value of c such that the intervals $[x,x+(\log x)^c]$ contain an $E_k$ number almost always. Harman showed that $c=7+\varepsilon$ is admissible for $E_2$ numbers, and this was the best known result also for $E_k$ numbers with $k>2$.

We show that for $E_3$ numbers one can take $c=1+\varepsilon$, which is optimal up to $\varepsilon$. We also obtain the value $c=3.51$ for $E_2$ numbers. The proof uses pointwise, large values and mean value results for Dirichlet polynomials as well as sieve methods.

  • Number Theory Seminar
2 June 2016
16:00
Rachel Newton
Abstract

Let $L/K$ be an extension of number fields and let $J_L$ and $J_K$ be the associated groups of ideles. Using the diagonal embedding, we view $L^*$ and $K^*$ as subgroups of $J_L$ and $J_K$ respectively. The norm map $N: J_L\to  J_K$ restricts to the usual field norm $N: L^*\to K^*$ on $L^*$. Thus, if an element of $K^*$ is a norm from $L^*$, then it is a norm from $J_L$. We say that the Hasse norm principle holds for $L/K$ if the converse holds, i.e. if every element of $K^*$ which is a norm from $J_L$ is in fact a norm from $L^*$. 

The original Hasse norm theorem states that the Hasse norm principle holds for cyclic extensions. Biquadratic extensions give the smallest examples for which the Hasse norm principle can fail. One might ask, what proportion of biquadratic extensions of $K$ fail the Hasse norm principle? More generally, for an abelian group $G$, what proportion of extensions of $K$ with Galois group $G$ fail the Hasse norm principle? I will describe the finite abelian groups for which this proportion is positive. This involves counting abelian extensions of bounded discriminant with infinitely many local conditions imposed, which is achieved using tools from harmonic analysis.

This is joint work with Christopher Frei and Daniel Loughran.

  • Number Theory Seminar
26 May 2016
16:00
Trevor Wooley
Abstract

The sub-convexity barrier traditionally prevents one from applying the Hardy-Littlewood (circle) method to Diophantine problems in which the number of variables is smaller than twice the inherent total degree. Thus, for a homogeneous polynomial in a number of variables bounded above by twice its degree, useful estimates for the associated exponential sum can be expected to be no better than the square-root of the associated reservoir of variables. In consequence, the error term in any application of the circle method to such a problem cannot be expected to be smaller than the anticipated main term, and one fails to deliver an asymptotic formula. There are perishingly few examples in which this sub-convexity barrier has been circumvented, and even fewer having associated degree exceeding two. In this talk we review old and more recent progress, and exhibit a new class of examples of Diophantine problems associated with, though definitely not, of translation-invariant type.

  • Number Theory Seminar
19 May 2016
16:00
Abstract

For $k \geq 3$ we give new values of $\rho_k$ such that
$$ \| \alpha p^k + \beta \| < p^{-\rho_k} $$
has infinitely many solutions in primes whenever $\alpha$ is irrational and $\beta$ is real. The mean
value results of Bourgain, Demeter, and Guth are useful for $k \geq 6$; for all $k$, the results also
depend on bounding the number of solutions of a congruence of the form

$$ \left\| \frac{sy^k}{q} \right\| < \frac{1}{Z} \ \ (1 \leq y \leq Y < q) $$

where $q$ is a given large natural number.

  • Number Theory Seminar
12 May 2016
16:00
Norbert A’Campo
Abstract
An implicite definition for the hyperbolic plane $H=H_I$ is in: ${\rm Spec}(\mathbb{R}[X]) = H_I \cup \mathbb{R}$. All geometric hyperbolic features will follow from this definition in an elementary way.
 
A second definition is $H=H_J=\{J \in {\rm End}(R^2) \mid J^2=-Id, dx \wedge dy(u,Ju) \geq 0 \}$. Working with $H=H_J$ allows to prove rather directly main theorems about Riemann surfaces.
  • Number Theory Seminar
5 May 2016
16:00
Giovanni Rosso
Abstract

In a recent work Andreata, Iovita, and Pilloni constructed the eigenvariety for cuspidal Siegel modular forms. This eigenvariety has the expected dimension (the genus of the Siegel forms) but it parametrizes only cuspidal forms. We explain how to generalize the construction to the non-cuspidal case. To be precise, we introduce the notion of "degree of cuspidality" and we construct an eigenvariety that parametrizes forms of a given degree of cuspidability. The dimension of these eigenvarieties depends on the degree of cuspidality we want to consider: the more non-cuspidal the forms, the smaller the dimension. This is a joint work with Riccardo Brasca.

  • Number Theory Seminar
28 April 2016
16:00
Andrew Ranicki
Abstract

The talk will be based on some of the material in the joint survey with Etienne Ghys

"Signatures in algebra, topology and dynamics"

http://arxiv.org/abs/1512.092582

In the 19th century Sturm's theorem on the number of roots of a real polynomial motivated Sylvester to define the signature of a quadratic form. In the 20th century the classification of quadratic forms over algebraic number fields motivated Witt to introduce the "Witt groups" of stable isomorphism classes of quadratic forms over arbitrary fields. Still in the 20th century the study of high-dimensional topological manifolds with nontrivial fundamental group motivated Wall to introduce the "Wall groups" of stable isomorphism classes of quadratic forms over arbitrary rings with involution. In our survey we interpreted Sturm's theorem in terms of the Witt-Wall groups of function fields. The talk will emphasize the common thread running through this developments, namely the notion of the localization of a ring inverting elements. More recently, the Cohn localization of inverting matrices over a noncommutative ring has been applied to topology in the 21st century, in the context of the speaker's algebraic theory of surgery.

 

  • Number Theory Seminar
10 March 2016
16:00
Igor Wigman
Abstract

We study the number of nodal domains of toral Laplace eigenfunctions. Following Nazarov-Sodin's results for random fields and Bourgain's de-randomisation procedure we establish a precise asymptotic result for "generic" eigenfunctions. Our main results in particular imply an optimal lower bound for the number of nodal domains of generic toral eigenfunctions.

  • Number Theory Seminar

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