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This talk will be in two parts. I will outline joint work with Daryl Cooper concerning the space of holonomies of properly convex real projective structures on manifolds whose fundamental group satisfies a few natural properties. This generalises previous work by Benoist for closed manifolds. A key example, computed with Joan Porti, is used to illustrate the main results.
This talk will be in two parts. The first part will be introductory, and will address the question: Given an ordinary differential equation (ODE) with certain physical/geometric properties (for example a preserved measure, first and/or second integrals), how can one preserve these properties under discretization? The second part of the talk will cover some more recent work, and address the question: How can one deduce hard to find properties of an ODE from its discretization?
In the first half of the talk we shall introduce the notion of Lie superalgebras, and then give a quick outline of the classification of finite-dimensional complex simple Lie superalgebras. In the second part of the talk we shall discuss the representation theory of these Lie superalgebras and explain the irreducible character problem in the BGG category. Our main focus will be on our computation of the irreducible characters for two of the exceptional Lie superalgebras. This part is based on recent joint works with C.-W. Chen, L. Li, and W. Wang.
Welded tangles are knotted surfaces in R^4. Bar-Natan and Dancso described a class of welded tangles which have “foamed vertices” where one allows surfaces to merge and split. The resulting welded tangled foams carry an algebraic structure, similar to the planar algebras of Jones, called a circuit algebra. In joint work with Dancso and Halacheva we provide a one-to-one correspondence between circuit algebras and a form of rigid tensor category called “wheeled props.” This is a higher dimensional version of the well-known algebraic classification of planar algebras as certain pivotal categories.
The asymptotic approach is suggested for the description of interacting surface and internal oceanic solitary waves. This approach allows one to describe a stationary moving wave patterns consisting of two plane solitary waves moving at an angle to each other. The results obtained within the approximate asymptotic theory is validated by comparison with the exact two-soliton solution of the Kadomtsev-Petviashvili equation. The suggested approach is equally applicable to a wide class of non-integrable equations too. As an example, the asymptotic theory is applied to the description of wave patterns in the 2D Benjamin-Ono equation describing internal waves in the infinitely deep ocean containing a relatively thin one of the layers.
‘Stubborn conjectures concerning rewriting systems, geodesic normal forms and geodetic graphs’
Adam Piggott (Australian National University)
‘Which groups have polynomial geodesic growth?’
Murray Elder (University of Technology Sydney)
Traditional Indigenous marriage rules have been studied extensively since the mid 1800s. Despite this, they have historically been cast aside as having very little utility. Here, I will walk through some of the interesting mathematics of the Gamilaraay system and show that, instead, they are in fact a very clever construction. Indeed, the Gamilaraay system dynamically trades off kin avoidance to minimise incidence of recessive diseases against pairwise cooperation, as understood formally through Hamilton’s rule.
The theory of complex representations of p-adic groups can feel very technical and unwelcoming, but its central role in the conjectural local Langlands correspondence has pushed us to pursue its understanding.
In this talk, I will aim to share the spirit of, and open questions in, the representation theory of G, through the lens of restricting these representations to maximal compact open subgroups.
Our point of departure: the Bruhat-Tits building of G, a 50-year-old combinatorial and geometric object that continues to reveal secrets about the structure and representation theory of G today.
In 1983, Gerd Faltings proved the Mordell conjecture stating that curves of genus at least two have only finitely many rational points. This can be understood as the statement that most polynomial equations (in a precise sense)
f(x,y) = 0
of degree at least 4 have at most finitely many solutions. However, the effective version of this problem, that of constructing an algorithm for listing all rational solutions, is still unresolved. To get a sense of the difficulty, recall how long it took to prove that there are no solutions to
xⁿ + yⁿ = 1
other than the obvious ones. In this talk, I will survey some of the recent progress on an approach to this problem that proceeds by encoding rational solutions into arithmetic principal bundles and studying their moduli in a manner reminiscent of geometric gauge theory.