Any closed 3-manifold can be obtained by glueing two handle bodies along their boundary. For a fixed such glueing, any other differs by changing the glueing map by an element in the mapping class group. Beginning with an idea of Dunfield and Thurston, we can use a random walk on the mapping class group to construct random 3-manifolds. I will report on recent work on the structure of such manifolds, in particular in view of tower of coverings and their topological growth: Torsion homology growth, the minimal degree of a cover with positive Betti number, expander families. I will in particularly explain the connection to some open questions about the mapping class group.

The cobordism hypothesis gives a correspondence between the
framed local topological field theories with values in C and a fully
dualizable objects in C.  Changing framing gives an O(n) action on the
space of local TFTs, and hence by the cobordism hypothesis it gives a
(homotopy coherent) action of O(n) on the space of fully dualizable
objects in C.  One example of this phenomenon is that O(3) acts on the
space of fusion categories.  In fact, O(3) acts on the larger space of
finite tensor categories.  I'll describe this action explicitly and
discuss its relationship to the double dual, Radford's theorem,
pivotal structures, and spherical structures.  This is part of work in
progress joint with Chris Douglas and Chris Schommer-Pries.
 

Large-scale homology computations are often rendered tractable by discrete Morse theory. Every discrete Morse function on a given cell complex X produces a Morse chain complex whose chain groups are spanned by critical cells and whose homology is isomorphic to that of X. However, the space-level information is typically lost because very little is known about how critical cells are attached to each other. In this talk, we discretize a beautiful construction of Cohen, Jones and Segal in order to completely recover the homotopy type of X from an overlaid discrete Morse function.

The projects include melting of the paste, fluid flow, particle segregation, baking, shrinkage and material properties of baked electrode.

We study a model for lipid-bilayer membrane vesicles exhibiting phase separation, incorporating a phase field together with membrane fluidity and bending elasticity. We prove the existence of a plethora of equilibria in the large, corresponding to symmetry-breaking solutions of the Euler-Lagrange equations. We also numerically compute a special class of such solutions, namely those possessing icosahedral symmetry. We overcome several difficulties along the way. Due to inherent surface fluidity combined with finite curvature elasticity, neither the Eulerian (spatial) nor the Lagrangian (material) description of the model lends itself well to analysis. This is resolved via a singularity-free radial-map description, which effectively eliminates the grossly under-determined mid-plane deformation. We then use well known group-theoretic selection techniques combined with global bifurcation methods to obtain our results.

Various concepts of weak solution have been suggested for the fundamental equations of fluid dynamics over the last few decades. However, such weak solutions may be non-unique, or at least their uniqueness is unknown. Nevertheless, a conditional notion of uniqueness, the so-called weak-strong uniqueness, can be established in various situations. We present some recent results, both positive and negative, on weak-strong uniqueness in the realm of incompressible and compressible fluid dynamics. Applications to the convergence of numerical schemes will be indicated.

Computing the solutions $u$ of an equation $f(u, \lambda) = 0$ as the parameter $\lambda$ is varied is a central task in applied mathematics and engineering. In this talk I will present a new algorithm, deflated continuation, for this task.

Deflated continuation has three main advantages. First, it is capable of computing disconnected bifurcation diagrams; previous algorithms only aimed to compute that part of the bifurcation diagram continuously connected to the initial data. Second, its implementation is extremely simple: it only requires a minor modification to any existing Newton-based solver. Third, it can scale to very large discretisations if a good preconditioner is available.

Among other problems, we will apply this to a famous singularly perturbed ODE, Carrier's problem. The computations reveal a striking and beautiful bifurcation diagram, with an infinite sequence of alternating pitchfork and fold bifurcations as the singular perturbation parameter tends to zero. The analysis yields a novel and complete taxonomy of the solutions to the problem, and demonstrates that a claim of Bender & Orszag (1999) is incorrect. We will also use the algorithm to calculate distinct local minimisers of a topology optimisation problem via the combination of deflated continuation and a semismooth Newton method.