Tue, 26 Nov 2019

12:00 - 13:15
L4

The probability distribution of stress-energy measurement outcomes in QFT

Chris Fewster
(York)
Abstract

Measurement outcomes in quantum theory are randomly distributed, and local measurements of the energy density of a QFT exhibit nontrivial fluctuations even in a vacuum state. This talk will present recent progress in determining the probability distribution for such measurements. In the specific case of 1+1 dimensional CFT, there are two methods (one based on Ward identities, the other on "conformal welding") which can lead to explicit closed-form results in some cases. The analogous problem for the free field in 1+3 dimensions will also be discussed.

Tue, 14 May 2019

12:00 - 13:15
L4

Local operators in integrable quantum field theories

Henning Bostelmann
(York)
Abstract


Integrable models provide simplified examples of quantum field theories with self-interaction. As often in relativistic quantum theory, their local observables are difficult to control mathematically. One either tries to construct pointlike local quantum fields, leading to possibly divergent series expansions, or one defines the local observables indirectly via wedge-local quantities, losing control over their explicit form.

We propose a new, hybrid approach: We aim to describe local quantum fields; but rather than exhibiting their n-point functions and verifying the Wightman axioms, we establish them as closed operators affiliated with a net of von Neumann algebras. This is shown to work at least in the Ising model.

 

Thu, 30 Nov 2017
16:00
L6

A Galois counting problem

Sam Chow
(York)
Abstract

We count monic quartic polynomials with prescribed Galois group, by box height. Among other things, we obtain the order of magnitude for  quartics, and show that non-quartics are dominated by reducibles. Tools include the geometry of numbers, diophantine approximation, the invariant theory of binary forms, and the determinant method. Joint with Rainer Dietmann.

Thu, 09 Jun 2016
17:30
L6

Finitary properties for a monoid arising from the model theory of $S$-acts

Victoria Gould
(York)
Abstract

*/ /*-->*/ A {\em monoid} is a semigroup with identity. A {\em finitary property for monoids} is a property guaranteed to be satisfied by any finite monoid. A good example is the maximal condition on the lattice of right ideals: if a monoid satisfies this condition we say it is {\em weakly right noetherian}. A monoid $S$ may be represented via mappings of sets or, equivalently and more concretely, by {\em (right) $S$-acts}. Here an $S$-act is a set $A$ together with a map $A\times S\rightarrow A$ where $(a,s)\mapsto as$, such that

for all $a\in A$ and $s,t\in S$ we have $a1=a$ and $(as)t=a(st)$. I will be speaking about finitary properties for monoids arising from model theoretic considerations for $S$-acts.

 

Let $S$ be a monoid and let $L_S$ be the first-order language of $S$-acts, so that $L_S$ has no constant or relational symbols (other than $=$) and a unary function symbol $\rho_s$ for each $s\in S$. Clearly $\Sigma_S$ axiomatises the class of $S$-acts, where

\[\Sigma_S=\big\{ (\forall x)(x\rho_s \rho_t=x\rho_{st}):s,t\in S\big\}\cup\{ (\forall x)(x\rho_1=x)

\}.\]

 

Model theory tells us that $\Sigma_S$

has a model companion $\Sigma_S^*$ precisely when the class

${\mathcal E}$ of existentially closed $S$-acts is axiomatisable and

in this case, $\Sigma_S^*$ axiomatises ${\mathcal E}$. An old result of Wheeler tells us that $\Sigma_S^*$ exists if and only if for every finitely generated right congruence $\mu$ on $S$, every finitely generated $S$-subact of $S/\mu$ is finitely presented, that is, $S$ is {\em right coherent}. Interest in right coherency also arises from other considerations such as {\em purity} for $S$-acts.

Until recently, little was known about right coherent monoids and, in particular, whether free monoids are (right) coherent.

I will present some work of Gould, Hartmann and Ru\v{s}kuc in this direction: specifically we answer positively the question for free monoids.

 

Where $\Sigma_S^*$ exists, it is known to be

stable, and is superstable if and only if $S$ is weakly right noetherian.

By using an algebraic description of types over $\Sigma_S^*$ developed in the 1980s by Fountain and Gould,

we can show that $\Sigma_S^*$ is totally

transcendental if and only if $S$ is weakly right noetherian and $S$ is {\em ranked}. The latter condition says that every right congruence possesses a finite Cantor-Bendixon rank with respect to the {\em finite type topology}.

Our results show that there is a totally transcendental theory of $S$-acts for which Morley rank of types does not coincide with $U$-rank, contrasting with the corresponding situation for modules over a ring.

Thu, 20 Feb 2014

16:00 - 17:00
L6

From quadratic polynomials and continued fractions to modular forms

Paloma Bengoechea
(York)
Abstract
Zagier studied in 1999 certain real functions defined in a very simple way as sums of powers of quadratic polynomials with integer coefficients. These functions give the even parts of the period polynomials of the modular forms which are the coefficients in Fourier expansion of the kernel function for Shimura-Shintani correspondence. He conjectured for these sums a representation in terms of a finite set of polynomials coming from reduction of binary quadratic forms and the infinite set of transformations occuring in a continued fraction algorithm of the real variable. We will prove two different such representations, which imply the exponential convergence of the sums.

For Logic Seminar: Note change of time and location!

Thu, 06 Feb 2014

16:30 - 17:30
L5

Hartmanis-Stearns conjecture and Mahler's method

Evgeniy Zorin
(York)
Abstract
Hartmanis-Stearns conjecture states that any number that can be computed in a real time by a multitape Turing machine is either rational or transcendental, but never irrational algebraic. I will discuss approaches of the modern transcendence theory to this question as well as some results in this direction.

Note: Change of time and (for Logic) place! Joint with Number Theory (double header)

Tue, 29 Nov 2011

17:00 - 18:20
L3

Hardy-Steklov operators in Lebesgue spaces on the semi-axis

Elena Ushakova
(York)
Abstract

The talk presents a collection of results about mapping properties

of the Hardy-Steklov operator

$(Hf)(x)=\int_{a(x)}^{b(x)} f(y) dy$ in weighted Lebesgue spaces on the

semi-axis. In particular, the explicit boundedness and compactness criteria

for the operator are given and a number of applications are obtained. A

part of the results is based on a joint paper with Prof. V.D. Stepanov

Thu, 23 Jun 2011

16:00 - 17:00
L3

Linear Combinations of L-functions

Chris Hughes
(York)
Abstract

If two L-functions are added together, the Euler product is destroyed.

Thus the linear combination is not an L-function, and hence we should

not expect a Riemann Hypothesis for it. This is indeed the case: Not

all the zeros of linear combinations of L-functions lie on the

critical line.

However, if the two L-functions have the same functional equation then

almost all the zeros do lie on the critical line. This is not seen

when they have different functional equations.

We will discuss these results (which are due to Bombieri and Hejhal)

during the talk, and demonstrate them using characteristic polynomials

of random unitary matrices, where similar phenomena are observed. If

the two matrices have the same determinant, almost all the zeros of

linear combinations of characteristic polynomials lie on the unit

circle, whereas if they have different determinants all the zeros lie

off the unit circle.

Subscribe to York