Trinity College Dublin

When it comes to the nonlinear heat equation u_t - \Delta u = u^p, a sharp condition for the stability of positive radial steady states was derived in the classical paper by Gui, Ni and Wang.  In this talk, I will present some recent joint work with Daniel Devine that focuses on a more general system of reaction-diffusion equations (which is also also known as the parabolic Henon-Lane-Emden system).  We obtain a sharp condition that determines the stability of positive radial steady states, and we also study the separation property of these solutions along with their asymptotic behaviour at infinity.

$$
\def\curl#1{\left\{#1\right\}}
\def\vek#1{\mathbf{#1}}
$$
lll-posed problems arise often in the context of scientific applications in which one cannot directly observe the object or quantity of interest. However, indirect observations or measurements can be made, and the observable data $y$ can be represented as the wanted observation $x$ being acted upon by an operator $\mathcal{A}$. Thus we want to solve the operator equation \begin{equation}\label{eqn.Txy} \mathcal{A} x = y, \end{equation} (1) often formulated in some Hilbert space $H$ with $\mathcal{A}:H\rightarrow H$ and $x,y\in H$. The difficulty then is that these problems are generally ill-posed, and thus $x$ does not depend continuously on the on the right-hand side. As $y$ is often derived from measurements, one has instead a perturbed $y^{\delta}$ such that ${y - y^{\delta}}_{H}<\delta$. Thus due to the ill-posedness, solving (1) with $y^{\delta}$ is not guaranteed to produce a meaningful solution. One such class of techniques to treat such problems are the Tikhonov-regularization methods. One seeks in reconstructing the solution to balance fidelity to the data against size of some functional evaluation of the reconstructed image (e.g., the norm of the reconstruction) to mitigate the effects of the ill-posedness. For some $\lambda>0$, we solve \begin{equation}\label{eqn.tikh} x_{\lambda} = \textrm{argmin}_{\widetilde{x}\in H}\left\lbrace{\left\|{b - A\widetilde{x}} \right\|_{H}^{2} + \lambda \left\|{\widetilde{x}}\right\|_{H}^{2}} \right\rbrace. \end{equation} In this talk, we discuss some new strategies for treating discretized versions of this problem. Here, we consider a discreditized, finite dimensional version of (1), \begin{equation}\label{eqn.Axb} Ax =  b \mbox{ with }  A\in \mathbb{R}^{n\times n}\mbox{ and } b\in\mathbb{R}^{n}, \end{equation} which inherits a discrete version of ill conditioning from [1]. We propose methods built on top of the Arnoldi-Tikhonov method of Lewis and Reichel, whereby one builds the Krylov subspace \begin{equation}
\mathcal{K}_{j}(\vek A,\vek w) = {\rm span\,}\curl{\vek w,\vek A\vek w,\vek A^{2}\vek w,\ldots,\vek A^{j-1}\vek w}\mbox{ where } \vek w\in\curl{\vek b,\vek A\vek b}
\end{equation}
and solves the discretized Tikhonov minimization problem projected onto that subspace. We propose to extend this strategy to setting of augmented Krylov subspace methods. Thus, we project onto a sum of subspaces of the form $\mathcal{U} + \mathcal{K}_{j}$ where $\mathcal{U}$ is a fixed subspace and $\mathcal{K}_{j}$ is a Krylov subspace. It turns out there are multiple ways to do this leading to different algorithms. We will explain how these different methods arise mathematically and demonstrate their effectiveness on a few example problems. Along the way, some new mathematical properties of the Arnoldi-Tikhonov method are also proven.

In this talk, we discuss an ongoing calculation of the
four-loop form factors in massless QCD. We begin by discussing our
novel approach to the calculation in detail. Of particular interest
are a new polynomial-time integration by parts reduction algorithm and
a new method to algebraically resolve the IR and UV singularities of
dimensionally-regulated bare perturbative scattering amplitudes.
Although not all integral topologies are linearly reducible for the
more non-trivial color structures, it is nevertheless feasible to
obtain accurate numerical results for the finite parts of the complete
four-loop form factors using publicly available sector decomposition
programs and bases of finite integrals. Finally, we present first
results for the four-loop gluon form factor Feynman diagrams which
contain three closed fermion loops.

The planar N=4 SYM is believed to be integrable. Following this thoroughly justified belief, its exact spectrum had been encoded recently into a quantum spectral curve (QSC). We can explicitly solve the QSC in various regimes; in particular, one can perform a highly-efficient weak coupling expansion.

I will explain how QSC looks like for the harmonic oscillator and then, using this analogy, introduce the QSC equations for the SYM spectrum. We will use these equations to compute a particular 6-loop conformal dimension in real time and then discuss explicit results (found up to 10-loop orders) as well as some general statements about the answer at any loop-order.

The problem of representing a (non-commutative) C*-algebra $A$ as the

algebra of sections of a bundle of C*-algebras over a suitable base

space may be viewed as that of finding a non-commutative Gelfand-Naimark

theorem. The space $\mathrm{Prim}(A)$ of primitive ideals of $A$, with

its hull-kernel topology, arises as a natural candidate for the base

space. One major obstacle is that $\mathrm{Prim}(A)$ is rarely

sufficiently well-behaved as a topological space for this purpose. A

theorem of Dauns and Hofmann shows that any C*-algebra $A$ may be

represented as the section algebra of a C*-bundle over the complete

regularisation of $\mathrm{Prim}(A)$, which is identified in a natural

way with a space of ideals known as the Glimm ideals of $A$, denoted

$\mathrm{Glimm}(A)$.

In the case of the minimal tensor product $A \otimes B$ of two

C*-algebras $A$ and $B$, we show how $\mathrm{Glimm}(A \otimes B)$ may

be constructed in terms of $\mathrm{Glimm}(A)$ and $\mathrm{Glimm} (B)$.

As a consequence, we describe the associated C*-bundle representation of

$A \otimes B$ over this space, and discuss properties of this bundle

where exactness of $A$ plays a decisive role.