UCC Mathematics Seminar

 

Time and location: 4-5pm, Tuesday 21 January 2020, WGB G17

 

Speaker: Tuomas Sahlsten (University of Manchester)

 

Title:

Quantum Chaos on Random Surfaces

 

Abstract:

The study of Laplacian eigenfunctions on Riemannian manifolds has a long history and is motivated by many applications in physics. For instance, an interest from the quantum mechanics perspective is the behaviour of the eigenfunctions in the large eigenvalue limit; an example of which is the quantum unique ergodicity conjecture of Zeev Rudnick and Peter Sarnak and the subsequent related work by Elon Lindenstrauss. More recently, there has been growing interest in the study of eigenfunctions and their connection to the underlying geometric aspects of the manifold. In this talk I will present an approach to quantum chaos by trying to randomise the space (surface) and describe some recent results that have been developed jointly with Cliff Gilmore (Cork), Etienne Le Masson (Cergy) and Joe Thomas (Manchester). Here we establish probabilistic relationships between L^p norms of Laplacian eigenfunctions and the genus of hyperbolic surfaces using random surface techniques for Weil-Petersson probabilities in the moduli space of Riemann surfaces developed by the late Maryam Mirzakhani. This is motivated by the recent works of Bauerschmidt et al. on the spectral theory of large random regular graphs and level aspect delocalisation of holomorphic cusp forms.

UCC Mathematics Seminar

 

Time and location: 4-5pm, Tuesday 14 January 2020, WGB G17

 

Speaker: Nina Snigireva (University College Dublin)

 

Title:

Order continuous multilinear maps and polynomials

 

Abstract:

In this talk we will first review the notions of order convergence and order continuity in Banach lattices. Then we will address the question of what properties are preserved when a multilinear map is extended to the bidual and the role order continuity plays in this process. (This is joint work with C. Boyd and R. Ryan.)

UCC Mathematics Seminar

 

Time and location: 4-5pm, Tuesday 07 January 2020, WGB G.02

 

Speaker: Rolf Gohm (Aberystwyth University)

 

Title:

The probabilistic motivation to study semi-cosimplicial Hilbert spaces

 

Abstract:

De-Finetti-type theorems in probability theory can be analysed from the point of view of cosimplicial identities. What happens if we do similar arguments in the category of Hilbert spaces and isometries?

UCC Mathematics Seminar

 

Time and location: 5-6 pm, Tuesday 19 November 2019, WGB, G02

 

Speaker: Dmitri Zaitsev (Trinity College Dublin)

 

Title:

Effectiveness of multiplier ideal sheaves and triangular resolutions of singularities of holomorphic maps 

 

Abstract:

Multiplier ideals have been invented by Kohn as one of very few known bridges connecting singularities in PDE systems with those in geometry. However, effective control of the parameters in the PDE estimates via geometric invariants remains a major open problem. 

 

In my recent work with Sung Yeon Kim, we propose a new purely geometric tool of triangular resolutions that allows to control  effectiveness by means of a new set of geometric complexity invariants in terms of intersection multiplicities arising in our resolutions. 

UCC Mathematics Seminar

 

Time and location: 4-5pm, Tuesday 12 November 2019, WGB 304

 

Speaker: Eberhard Mayerhofer (University of Limerick)

 

Title:

Three essays on stopping

 

Abstract:

This talk is on my article "Three essays on stopping "(https://arxiv.org/abs/1909.13050), where the exponential distribution plays a crucial role, and where the focus is on particularly simple proofs of partly known facts. The audience is only required to have a basic idea about scalar diffusion processes and compound Poisson processes.

 

First, we give a closed-form formula for first passage time of a reflected Brownian motion with drift. This modifies a formula by Perry et al (2004). Second, we characterize all diffusion processes, where the maximum before a fixed draw-down is exponentially distributed. This complements the sufficient condition formulated by Lehoczky (1977). Third, we give a particularly simple proof for the fact that the maximum at a fixed drawdown threshold is exponentially distributed for any spectrally negative Lévy process, a result due to Mijatovic and Pistorius (2012).

 

 

 

UCC Mathematics Seminar

 

Time and location: 4-5pm, Tuesday 29 October 2019, WGB 405

 

Speaker: Gabriel Lord (Radboud University Nijmegen)

 

Title:

Numerics and a model for the stochastically forced vorticity equation

 

Abstract:

This talk will introduce the stochastically forced vorticity equation and a reduced stochastic differential equation model that we show captures the infinite dimensional behaviour. We start by introducing the background to stochastic differential equations before explaining the space-time forcing. The talk will discuss some of the issues in performing numerical simulations for these systems.

UCC Mathematics Seminar

 

Time and location: 4-5pm, Tuesday 22 October 2019, WGB G17

 

Speaker: Michel Schellekens (University College Cork)

 

Title:

Diyatropic Algorithms: Viewing entropy through the looking glass
(On Modular Analysis & Entropy Conservation)

 

Abstract:

Timing-modularity, the capacity to determine the time of an algorithm from the times of its parts, is not guaranteed. In general, the time of a part intricately depends on inputs passed on by other parts. Modularity, when it holds, drastically simplifies timing, for instance by enabling the derivation of recurrence equations for worst-case time, average case time or second moments. Little is known about the intrinsic properties of algorithms for which the time analysis is guaranteed to be modular. We study the question for comparison-based algorithms.

Standard algorithm analysis assumes a uniform input distribution. In the same vein, a modular analysis, deriving information of the whole from information on its parts, should assume that each of the algorithm's parts must, once again, operate over inputs that are uniformly distributed. This proves too strict a requirement. Fortunately modular analysis extends to algorithms that preserve uniform distributions ``locally". In this case, uniform distributions merely hold on parts of an input partition referred to as a ``global state". Algorithms preserving global states exhibit the tight distribution control which admits an elegant modular analysis. Those that do not, lie at the root of deep open problems in algorithm analysis.

Algorithms preserving global states can be designed based on the Modular Quantitative Analysis framework MOQA, which consists of: (a) modelling data structures as partially-ordered finite sets; (b) modelling data on these by topological sorts; (c) considering computation states as finite multisets of such data; (d) analysing algorithms by their induced transformations on states. In this view, an abstract specification of a sorting algorithm has input state given by any possible permutation of a finite set of elements (represented, according to (a) and (b), by a discrete partially-ordered set together with its topological sorts given by all permutations) and output state a sorted list of elements (represented, again according to (a) and (b), by a linearly-ordered finite set with its unique topological sort).

Series-parallel (SP-)orders form an important, computationally tractable class of data structures that support the model's computations. We introduce Mod-SP, the least MOQA-fragment sufficient to construct all topological sorts on SP orders via computations guaranteed to support modular timing. Mod-SP-computations, when made reversible through history-keeping, act as closed systems in which entropy is conserved, linking modularity of timing to entropy conservation of data. This sharpens traditional entropy preservation guaranteed by the second law of thermodynamics for reversible systems. For Mod-SP-computations entropy is neither created nor destroyed, merely transferred in modular fashion from one form (quantitative entropy) to another (positional entropy).

We establish an entropic duality theorem coupling each Mod-SP computation with a dual computation. The latter effects an increase in positional entropy proportional to the decrease in quantitative entropy effected by the original computation. We refer to algorithms satisfying this type of ``entropic coupling" as ``diyatropic" and show that all Mod-SP computations are diyatropic. This includes Insertion sort and the Heapify algorithm but not standard Heapsort's Selection Phase. Finally we show that Entropic Duality implies an Entropic Correspondence Theorem, conjectured by M. Fiore.

This work was completed during a Fulbright Scholarship (April 5 - Aug 31, 2019) at Stanford's Computer Science theory group, research host Don Knuth. The author is grateful for discussions with Don Knuth, Vaughan Pratt and Marcelo Fiore.