Quantum computers are designed based on quantum mechanics principle, they are most suitable to solve the Schrodinger equation, and linear PDEs (and ODEs) evolved by unitary operators.  It is important to  to explore whether other problems in scientific computing, such as ODEs, PDEs, and  linear algebra that arise in both classical and quantum systems which are not unitary evolution,  can be handled by quantum computers.  

We will present a systematic way to develop quantum simulation algorithms for general differential equations. Our basic framework is dimension lifting, that transfers non-autonomous ODEs/PDEs systems to autonomous ones, nonlinear PDEs to linear ones, and linear ones to Schrodinger type PDEs—coined “Schrodingerization”—with uniform evolutions. Our formulation allows both qubit and qumode (continuous-variable) formulations, and their hybridizations, and provides the foundation for analog quantum computing which are easier to realize in the near term. We will also discuss  dimension lifting techniques for quantum simulation of stochastic DEs and PDEs with fractional derivatives. 

Modern deep learning methods provide the state-of-the-art in image reconstruction in most areas of computational imaging. However, such techniques are very data hungry and in a number of key imaging problems access to ground truth data is challenging if not impossible. This has led to the emergence of a range of self-supervised learning algorithms for imaging that attempt to learn to image without ground truth data. 

In this talk I will review some of the existing techniques and look at what is and might be possible in self-supervised imaging.

The Bogomolov-Miyaoka-Yau inequality for minimal compact complex surfaces of general type was proved in 1977 independently by Miyaoka, using methods of algebraic geometry, and by Yau, as an outgrowth of his proof of the Calabi conjectures. In this talk, we outline our program to prove the conjecture that symplectic 4-manifolds with $b^+>1$ obey the Bogomolov-Miyaoka-Yau inequality. Our method uses Morse theory on the gauge theoretic moduli space of non-Abelian monopoles, where the Morse function is a Hamiltonian for a natural circle action and natural two-form.  We shall describe generalizations of Donaldson’s symplectic subspace criterion (1996) from finite to infinite dimensions. These generalized symplectic subspace criteria can be used to show that the natural two-form is non-degenerate and thus an almost symplectic form on the moduli space of non-Abelian monopoles. This talk is based on joint work with Tom Leness and the monographs https://arxiv.org/abs/2010.15789  (to appear in AMS Mathematical Surveys and Monographs), https://arxiv.org/abs/2206.14710 and https://arxiv.org/abs/2410.13809. 

 In the first part of the talk, after revisiting some classical models for dilute polymeric fluids, we show that thermodynamically 
consistent models for non-isothermal flows of such fluids can be derived in a very elementary manner. Our approach is based on identifying the 
energy storage mechanisms and entropy production mechanisms in the fluid of interest, which in turn leads to explicit formulae for the Cauchy 
stress tensor and for all the fluxes involved. Having identified these mechanisms, we first derive the governing system of nonlinear partial 
differential equations coupling the unsteady incompressible temperature-dependent Navier–Stokes equations with a 
temperature-dependent generalization of the classical Fokker–Planck equation and an evolution equation for the internal energy. We then 
illustrate the potential use of the thermodynamic basis on a rudimentary stability analysis—specifically, the finite-amplitude (nonlinear) 
stability of a stationary spatially homogeneous state in a thermodynamically isolated system.

In the second part of the talk, we show that sequences of smooth solutions to the initial–boundary-value problem, which satisfy the 
underlying energy/entropy estimates (and their consequences in connection with the governing system of PDEs), converge to weak 
solutions that satisfy a renormalized entropy inequality. The talk is based on joint results with Miroslav Bulíček, Mark Dostalík, Vít Průša 
and Endré Süli.