Tue, 10 Oct 2023

14:00 - 14:30
L4

A sparse hp-finite element method for the Helmholtz equation posed on disks, annuli and cylinders

Ioannis Papadopoulos
(Imperial)
Abstract

We introduce a sparse and very high order hp-finite element method for the weak form of the Helmholtz equation.  The domain may be a disk, an annulus, or a cylinder. The cells of the mesh are an innermost disk (omitted if the domain is an annulus) and concentric annuli.

We demonstrate the effectiveness of this method on PDEs with radial direction discontinuities in the coefficients and data. The discretization matrix is always symmetric and positive-definite in the positive-definite Helmholtz regime. Moreover, the Fourier modes decouple, reducing a two-dimensional PDE solve to a series of one-dimensional solves that may be computed in parallel, scaling with linear complexity. In the positive-definite case, we utilize the ADI method of Fortunato and Townsend to apply the method to a 3D cylinder with a quasi-optimal complexity solve.

Tue, 02 Nov 2021

15:30 - 16:30
L4

Gromov-Witten invariants of blow-ups

Qaasim Shafi
(Imperial)
Abstract
Gromov-Witten invariants play an essential role in mirror symmetry and enumerative geometry. Despite this, there are few effective tools for computing Gromov-Witten invariants of blow-ups. Blow-ups of X can be rewritten as subvarieties of Grassmann bundles over X. In joint work with Tom Coates and Wendelin Lutz, we exploit this fact and extend the abelian/non-abelian correspondence, a modern tool in Gromov-Witten theory. Combining these two steps allows us to get at the genus 0 invariants of a large class of blow-ups.   
Mon, 30 Nov 2020
14:15
Virtual

Application of a Bogomolov-Gieseker type inequality to counting invariants

Soheyla Feyzbakhsh
(Imperial)
Abstract

In this talk, I will work on a smooth projective threefold X which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda, such as the projective space P^3 or the quintic threefold. I will show certain moduli spaces of 2-dimensional torsion sheaves on X are smooth bundles over Hilbert schemes of ideal sheaves of curves and points in X. When X is Calabi-Yau this gives a simple wall crossing formula expressing curve counts (and so ultimately Gromov-Witten invariants) in terms of counts of D4-D2-D0 branes. This is joint work with Richard Thomas. 

Mon, 26 Feb 2018

14:15 - 15:15
L4

Coulomb branch, 3d Mirror symmetry, and Implosions

Amihay Hanany
(Imperial)
Abstract

3d N=4 supersymmetric gauge theories provide a method for constructing HyperK\”ahler singularities, known as the Coulomb branch.
This method is complementary to the more traditional way of construction using HyperK\”ahler quotients, known in physics as the “Higgs branch”.
Out of all possible gauge theories there is an interesting subclass of quiver varieties, where the Coulomb branch has been studied in some detail.
Some examples are moduli spaces of classical and exceptional instantons and closures of nilpotent orbits. An interesting feature of Coulomb and Higgs branches is the phenomenon of "3d mirror symmetry” where for a pair of gauge theories, the Higgs branch and Coulomb branch exchange.
There is a large class of “mirror pairs” which I will discuss in some detail.

A topic of recent interest is the notion of implosions. I will argue that there is a simple operation on the quiver which leads to implosion. In other words, given a quiver such that its Coulomb branch is moduli space A, a simple operation of the quiver (making a bouquet) provides the implosion of A.
This has been tested on closures of nilpotent orbits of A type and on nilpotent cones of orthogonal groups and found to agree with the expected results.
If time permits, I will discuss isometries of Coulomb branches

Thu, 04 May 2017

16:00 - 17:30
L4

Short-time near-the-money skew in rough fractional stochastic volatility models

Blanka Horvath
(Imperial)
Abstract

We consider rough stochastic volatility models where the driving noise of volatility has fractional scaling, in the “rough” regime of Hurst pa- rameter H < 1/2. This regime recently attracted a lot of attention both from the statistical and option pricing point of view. With focus on the latter, we sharpen the large deviation results of Forde-Zhang (2017) in a way that allows us to zoom-in around the money while maintaining full analytical tractability. More precisely, this amounts to proving higher order moderate deviation es- timates, only recently introduced in the option pricing context. This in turn allows us to push the applicability range of known at-the-money skew approxi- mation formulae from CLT type log-moneyness deviations of order t1/2 (recent works of Alo`s, Le ́on & Vives and Fukasawa) to the wider moderate deviations regime.

This is work in collaboration with C. Bayer, P. Friz, A. Gulsashvili and B. Stemper

Mon, 24 Oct 2016

15:45 - 16:45
L6

Band Surgeries and Crossing Changes between Fibered Links

Dorothy Buck
(Imperial)
Abstract

We characterize cutting arcs on ber surfaces that produce new ber surfaces,
and the changes in monodromy resulting from such cuts. As a corollary, we
characterize band surgeries between bered links and introduce an operation called
generalized Hopf banding. We further characterize generalized crossing changes between
bered links, and the resulting changes in monodromy.

This is joint work with Matt Rathbun, Kai Ishihara and Koya Shimokawa

Tue, 03 Nov 2015

15:45 - 16:45
L4

Poles of maximal order of Igusa zeta functions

Johannes Nicaise
(Imperial)
Abstract

Igusa's p-adic zeta function $Z(s)$ attached to a polynomial $f$ in $N$ variables is a meromorphic function on the complex plane that encodes the numbers of solutions of the equation $f=0$ modulo powers of a prime $p$. It is expressed as a $p$-adic integral, and Igusa proved that it is rational in $p^{-s}$ using resolution of singularities and the change of variables formula. From this computation it is immediately clear that the order of a pole of $Z(s)$ is at most $N$, the number of variables in $f$. In 1999, Wim Veys conjectured that the only possible pole of order $N$ of the so-called topological zeta function of $f$ is minus the log canonical threshold of $f$. I will explain a proof of this conjecture, which also applies to the $p$-adic and motivic zeta functions. The proof is inspired by non-archimedean geometry and Mirror Symmetry, but the main technique that is used is the Minimal Model program in birational geometry. This talk is based on joint work with Chenyang Xu.

Tue, 03 Feb 2015

15:45 - 16:45
L4

Homological projective duality

Richard Thomas
(Imperial)
Abstract
I will describe a little of Kuznetsov's wonderful theory of Homological projective duality, a generalisation of classical projective duality that relates derived categories of coherent sheaves on different algebraic varieties. I will explain an approach that seems simpler than the original, and some applications that occur in joint work with Addington, Calabrese and Segal.
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