L6

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. 

Adam Jones will explore the relationship between the category of smooth representations of a semisimple p-adic Lie group G and the module category over its associated pro-p Iwahori-Hecke algebra via the canonical invariance adjunction. This relationship is well understood in characteristic 0, in fact it yields a category equivalence equivalence, but in characteristic p it is very mysterious and largely defies understanding. We will explore methods of constructing an appropriate subcategory of Hecke modules which is well behaved under the adjunction, and which can be shown to contain all parabolic inductions. He will give examples of this yielding results when G has rank 1, and more recently when G = SL_3 in certain cases.

Recent foundational work by Ben-Bassat, Kelly, and Kremnitzer describes a model for derived analytic geometry as homotopical geometry relative to the infinity category of simplicial commutative complete bornological rings. In this talk, Rhiannon Savage will discuss a representability theorem for derived stacks in these contexts and will set out some new foundations for derived smooth geometry. Rhiannon will also briefly discuss the representability of the derived moduli stack of non-linear elliptic partial differential equations by an object we call a derived C∞-bornological affine scheme.

In this talk, I will present a representation-theoretical approach to constructing a non-commutative analogue of the classical Laplace transform on Lie groups. I will begin by discussing the motivations for such a generalization, emphasizing its connections with harmonic analysis, probability theory, and the study of evolution equations on non-commutative spaces. I will also outline some of the key challenges that arise when extending the Laplace transform to the setting of Lie groups, including the non-commutativity of the group operation and the complexity of its dual space.

The main part of the talk will focus on an explicit construction of the Laplace transform in the framework of connected, simply connected nilpotent Lie groups. This construction relies on Kirillov’s orbit method, which provides a powerful bridge between the geometry of coadjoint orbits and the representation theory of nilpotent groups.

As an application, I will describe an operator-theoretic analogue of the classical Müntz–Szász theorem, establishing a density result for a family of generalized polynomials in associated with the group setting. This result highlights the strength of the representation-theoretical approach and its potential for solving classical approximation problems in a non-commutative context.