Colloquia Spring 2020
20/1 in Pascal
Tobias Ekholm (Uppsala University)
Skeins on Branes
The HOMFLY polynomial is an invariant of knots in the 3-sphere allowing one to distinguish different knots. It is a two variable polynomial which is defined combinatorially, via so called skein-relations. We give a geometric interpretation of the coefficients of the polynomial as a count of certain holomorphic curves associated to the knot. One of the variables in the HOMFLY accounts for the area or homological degree of the curves, the other for their Euler characteristic. This is in line with predictions by Ooguri and Vafa based on topological string theory. The proof embodies a new method to define invariant counts of holomorphic curves with Lagrangian boundary. This is a mathematically rigorous incarnation of the fact that boundaries of open topological strings create line defects in Chern-Simons theory as described by Witten.
17/2 in Pascal
Olle Häggström (Chalmers/GU)
AI Alignment, Embedded Agency and Decision Theory
The term artificial intelligence (AI) had not yet been coined in the days of Alan Turing. Nevertheless, he did foresee the field, and famously predicted that machines would eventually become so capable as to surpass human general intelligence, in which case he suggested that "we should have to expect the machines to take control". The (small but growing) research area known as AI Alignment takes this ominous prediction as a starting point, and aims to work out how to instil the AI with goals that lead to a good outcome (for humans) despite their taking control. Attempts to solve AI Alignment lead to many intriguing philosophical and mathematical questions involving, e.g., the notion of embedded agency, and the fundamentals of decision theory.
9/3 in Pascal
Laura Mancinska (University of Copenhagen)
Harnessing Quantum Entanglement
Entanglement is one of the key features of quantum mechanics. It lies at the heart of most cryptographic applications of quantum technologies and is necessary for computational speed-ups. However, given a specific information processing task, it is challenging to find the best way to harness entanglement and we are yet to uncover the full range of its potential applications.
We will see that the so-called nonlocal games provide a rigorous mathematical framework for studying entanglement and the advantage that it can offer. On the one hand, we will take a closer look at specific applications of entanglement, including protocols for certifying proper functioning of untrusted quantum devices. While on the other hand, we will attempt to gain a better understanding of the mathematical structure of entanglement by considering a restricted class nonlocal games. This class will give rise to a natural quantum relaxation for the notion of graph isomorphism.
20/4 via Zoom
Tom Britton (Stockholm University)
Mathematical modelling of infectious disease outbreaks like covid-19 (video from the seminar at Chalmers Play)
Abstract: Mathematical models for the spread of infectious diseases are used to: better understand spreading mechanisms, determine if a big outbreak is likely to occur and how big it will be, determine if a disease will become endemic, and investigate how various preventive measures can reduce spreading hopefully preventing a major pandemic outbreak or make an endemic disease vanish. Making inference is harder than usual in that the basic events, transmissions, are rarely observed but instead proxies like onset of symptoms are recorded, and also by the fact that these events are dependent rather than independent (as is usually the case).
In the talk I will give an overview of the area with particular focus on emerging outbreaks, including illustrations from the current coronavirus outbreak.
Colloquia Autumn 2020
19/10 via Zoom
Marija Cvijovic (Chalmers/GU)
Why Gilgamesh, Dorian Gray and Willy Wonka should have studied math
Abstract: Aging is a process, which has intrigued people since ancient times. It remains one of the central mysteries of biology, where both why we age and how we age are not known. Aging is a complex and regulated biological process with remarkable individual variation. Our ability to fully understand the complexity of the networks and their branches involved in the ageing process by experimental methods is limited. Mathematical methods provide a means to understand this complexity. In the first part of this talk, I will give a brief overview of existing mathematical models describing various aspects of ageing in multiple organisms. In the second part, I will talk about the work done in my group and how we combine mathematical modelling, yeast genetics and biophysics setting new standards and laying the groundwork for novel strategies to improve our understanding of the ageing process as a whole.
23/10 via Zoom
Andreas Rosén (Chalmers/GU)
Cauchy's singular integral: the story
Abstract: The Cauchy integral from complex analysis, placing the pole z on the curve, is the most famous example of a singular integral operator, where the integral is not absolutely convergent but needs to be evaluated in the principal value sense for it to make sense. A conjecture formulated by Alberto Calderón was that this Cauchy singular integral is bounded on L^2 on any Lipschitz regular curve. Just like all other questions that Calderón posed, this had a great impact on mathematics. From the 1970s, after the first proof in 1982 and well into the 2000s, this Calderón conjecture continued to enrich real variable analysis. My aim with this talk is to explain some core ideas in a non-technical way, and to tell some of the stories from behind the scene.