Understanding quantum many-body systems using tools from approximation theory

Understanding quantum many-body systems using tools from approximation theory

  Quantum many-body systems have been a subject of intense study both for the condensed-matter physicists and the quantum complexity theorists. A major challenge has been to understand the general properties of two key states of these systems: the ground states (that describe physics at very low temperatures) and Gibbs states (that are suited for high temperatures). In this talk, I will discuss some new insights that have been gained in this area of research by invoking methods from approximation theory for functions. In particular, it will be highlighted that approximations to functions based on the Chebyshev polynomials better capture various important properties of these systems, in comparison to existing information theoretic or physics-based methods. Some examples of these properties include entanglement-entropy, correlation length and the tensor-network representations.
 

The Logical Structure of Quantum Computational Advantage

The Logical Structure of Quantum Computational Advantage

  Quantum computation is a novel paradigm of computation based on quantum mechanical first principles in order to solve important problems beyond the abilities of any classical machine. However, confusion about precisely how and for which problems quantum computers offer an advantage stymies both fundamental understanding and technological progress. What are the key features of quantum theory that enable quantum advantages in communicational and computational tasks? Nonlocality is a special case of contextuality; both notions are concepts of foundational quantum theory that crystallise ways in which quantum behaviours can transcend classical constraints. They have recently emerged as promising hypothetical origins of quantum advantage. Many early proofs of contextuality used formal logical paradoxes to probe the classical-quantum boundary. Richer connections between contextuality and logic have recently been uncovered creating an opportunity to gain the deep, structural understanding of quantum advantage needed to design quantum algorithms and optimally efficient quantum hardware. I will introduce nonlocality and contextuality with an emphasis on their underlying logical structure and review results of quantum information that provide persuasive evidence of their usefulness as computational resources in restricted computational models. I will then present evidence of the power of paradoxes to drive the full capabilities of quantum computation and outline how a logical perspective can contribute to tackling important problems in quantum computer science. This talk will be a high-level overview and no prior knowledge of quantum computation will be assumed.
 

Applied Analysis Day

Applied Analysis Day

  Please contact Frithjof Lutscher (the organizer) for more details.
 

Harmonic Analysis on GL(n) over Finite Fields.

Harmonic Analysis on GL(n) over Finite Fields.

Abstract:

There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio:

Trace(ρ(g)) / dim(ρ),

for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.

In 2015, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank.

Rank suggests a new organization of representations based on the very few “small” ones. This stands in contrast to Harish-Chandra’s “philosophy of cusp forms”, which is (since the 60s) the main organization principle, and is based on the (huge collection) of “LARGE” representations.

This talk will discuss the notion of rank for the group GL(n) over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks.

This is joint work with Roger Howe (Yale and Texas A&M). The numerics for this work was carried by Steve Goldstein (Madison).

 

Posets, metric spaces, and topological data analysis

Posets, metric spaces, and topological data analysis

Abstract: Traditional TDA is the analysis of homotopy invariants of systems of spaces V(X) that arise from finite metric spaces X, via distance measures. These spaces can be expressed in terms of posets, which are barycentric subdivisions of the usual complexes V(X). The proofs of stability theorems in TDA are sharpened considerably by direct use of poset techniques. Expanding the domain of definition to extended pseudo metric spaces enables the construction of a realization functor on diagrams of spaces, which has a right adjoint $Y \mapsto S(Y)$, called the singular functor. The realization of the Vietoris-Rips system $V(X)$ for an ep-metric space $X$ is the space itself. The counit of the adjunction defines a map $\eta: V(X) \to S(X)$, which is a sectionwise weak equivalence - the proof uses simplicial approximation techniques. This is the context for the Healy-McInnes UMAP construction, which will be discussed if time permits. UMAP is non-traditional: clusters for UMAP are defined by paths through sequences of neighbour pairs, which can be a highly efficient process in practice.
 

Quantitative Approaches to Social Justice

Quantitative Approaches to Social Justice

Abstract: Civil rights leader, educator, and investigative journalist Ida B. Wells said that "the way to right wrongs is to shine the light of truth upon them." This talk will demonstrate how quantitative and computational approaches can shine a light on social injustices and help build solutions to remedy them. We will present quantitative social justice projects on topics ranging from diversity in art museums to equity in criminal sentencing to affirmative action, health care access, and other fields. The tools engaged include data mining, crowdsourcing, data cleaning, clustering, hypothesis testing, statistical modeling, Markov chains, data visualization, and more. We hope that this talk leaves you informed about the breadth of social justice applications than one can tackle using data science tools.
 

Distinguished Women in Math

Associativity, commutativity and units: a higher-dimensional ballet.

Distinguished Women in Math

Associativity, commutativity and units: a higher-dimensional ballet.

Abstract: Associativity, commutativity and unit laws are axioms we typically learn about early on, in the context of numbers. We might then take them for granted until we meet non-commutative situations, such as multiplication of matrices, or symmetry groups. In higher dimensions we start to encounter non-associativity and non-unitality as well, but there is more nuance: rather than associativity simply being true or not true, there are shades of grey, where associativity holds up to isomorphism, equivalence, or just some sort of map. In this talk I will describe how those familiar three families of axioms become the essence of all the interesting features of weak higher-dimensional category theory. Moreover, rather than being three different types of axioms they are inextricably related via a higher-dimensional version of distributivity. The ballet they present is one of ebb and flow, give and take, where rigidity for one ``dancer'' always needs to be offset by flexibility in another. I will show that the apparently mundane maths of high school has deep category theoretical insights embedded in it, if we care to look. I will not assume any prior knowledge of category theory, but prior interest in pure mathematics will help.