Sums of Squares Proofs, by Dr Hamza Fawzi
“A polynomial that is a sum of squares of other polynomials can only take nonnegative values. This trivial observation is surprisingly powerful: many inequalities in mathematics have simple sum-of-squares proofs. I will discuss algorithms that can automatically search for sum-of-squares proofs for polynomial inequalities, and the extent to which they can be considered as “automatic proof machines””
The universality phenomenon, by Dr Roland Bauerschmidt
Many complex systems in mathematics and physics show universal behaviour, i.e., behaviour that is independent of the details of the system. I will illustrate this universality phenomenon in several examples, some well-understood, some still mysterious.
Being rigorously subjective, by Dr Sergio Bacallado
What is the chance that the sun will rise tomorrow? The problem of inductive inference has been the subject of intense research since before Laplace proposed an answer to this question. This talk will review work by Bruno de Finetti, W. E. Johnson, and others which provides a foundation for inductive inference, at once subjective and rigorous. We will prove a version of de Finetti’s theorem and characterise a prior distribution using Johnson’s sufficientness postulate.
Simple loops on surfaces , by Prof Ivan Smith
Wednesday 30th November, 6pm
One of the results for which Maryam Mirzakhani won the Fields medal was the statement that “a random simple closed curve on a surface of genus two separates it into two pieces with probability one seventh”. I will try and explain some of the ideas involved in making sense of that statement and some of the mathematics in her proof. The central theme is hyperbolic geometry, and the geometry of the space of surfaces.
Secular dynamics in astrophysics: from planet formation around binary stars to black hole mergers, by Dr Roman Rafikov
Many problems that emerged in astrophysics in the last two decades require application of classical methods of celestial dynamics for their solution. In this talk, following the legacy of John Couch Adams, I will describe one particular aspect of celestial mechanics – secular dynamics of gravitating systems – that has found wide applications in a variety of problems. I will illustrate this statement by focusing on several hot topics of modern astrophysics: (1) formation of planets within and around binary stars (so called Tatooines), which have been discovered by the Kepler mission, (2) origin of Hot Jupiters – giant planets in very tight orbits around their parent stars, and (3) production of compact black hole binaries that can merge releasing spectacular bursts of gravitational waves detectable by LIGO. These examples will clearly show that, despite their seeming simplicity, secular gravitational interactions can result in a rich variety of dynamical behaviors of celestial objects.
Use of Similarity Scalings in Fluid Mechanics, by Prof Andrew Woods
In this talk I will present some models of turbulent gravity currents and turbulent buoyant plumes as models of large scale geophysical flows, such as volcanic plumes and turbidity currents. I will discuss the use of similarity solutions to describe the behaviour of these flows and the structure of the turbulence. I will then show how similar approaches may be used to model flow in porous rocks for carbon sequestration, where similarity solutions of the second kind can also arise.
Quartics, Sextics and Beyond: A Problem of J.J. Sylvester, by Dr Maciej Dunajski
The classical invariant theory has been formulated in the second half of the 19th century. James Joseph Sylvester (A student at St John’s, and a Junior Wrangler in 1837) was one of the key players. After a quiet period lasting for most part of the 20th century, the theory has reappeared in algebraic geometry, representation theory and other areas of modern mathematics and physics.
Despite the recent developments, some basic problems left over from the 19th century remain open. I this talk, I shall give an elementary introduction to the subject (IA groups, and/or IA vectors and matrices should be more than enough to follow), and present a solution of a problem posed by J. J. Sylvester, and solved recently in a joint work with Roger Penrose (A student at St Johns, 1958).
The Adams Society is the mathematics society of St John’s college, Cambridge. We host numerous events each year including mathematical talks and socials.
Free admission to all talks. Refreshments are served in the foyer before each talk.
Our talks usually take place in the Fisher Building, St John’s College.
Get in touch through our committee members or email firstname.lastname@example.org