Brendan Guilfoyle

I am a mathematician and my main area of research is geometry. Over the past decade I have spent most of my efforts investigating the geometry of spaces of geodesics and their applications.

This work includes the proof, in collaboration with Wilhelm Klingenberg, of an 100 year-old conjecture in classical surface theory called the Caratheodory Conjecture. The complete proof has now appeared:

B. Guilfoyle and W. Klingenberg, Higher codimensional mean curvature flow of compact spacelike submanifolds, Trans. Amer. Math. Soc. 372.9 (2019) 6263-6281. CLICK HERE ,
B. Guilfoyle and W. Klingenberg, Fredholm-regularity of holomorphic discs in plane bundles over compact surfaces, Ann. Fac. Sci. Toulouse Math. Serie 6, 29.3 (2020) 565-576. CLICK HERE,
B. Guilfoyle and W. Klingenberg, Proof of the Toponogov Conjecture on complete surfaces, J. Gokova Geom. Topol. GGT 17 (2024) 1-50. (Link to follow).

More details of this can be found here.

Congratulations to my doctoral supervisor Karen Uhlenbeck (pictured on the left at my dissertation defence in 1997) who has recently been awarded the 2019 Abel Prize in Mathematics - the first woman to receive the award.

I recently dedicated the paper On isolated umbilic points to Karen for all of her support down the years. The paper shows that an arbitrarily small perturbation of the Euclidean metric does not have to satisfy the Caratheodory Conjecture (or Hamburger's umbilic index bound).

Here's a short video explainer:

Recent Research
I work on a number of related problems in geometry, analysis and topology. I have just completed a survey paper on my work over the last decade or two:

From CT Scans to 4-Manifold Topology via Neutral Geometry

and here's a recent talk I gave on some of the results in SISSA, Trieste, Italy:

Below is a list of some recent papers and preprints.

Parabolic evolution with boundary to the Bishop family of holomorphic discs (with Wilhelm Klingenberg),
Properties and transformations of Weingarten surfaces (with Morgan Robson),
A conformal mean value theorem for solutions of the ultrahyperbolic equation (with Guillem Cobos),
Roots of polynomials and umbilics of surfaces (with Wilhelm Klingenberg),
On the convergence of non-integer linear Hopf flow (with Morgan Robson),
A uniqueness theorem for incompressible fluid flows with straight streamlines,
Umbilic points on the finite and infinite parts of certain algebraic surfaces and A note on umbilic points at infinity(with Adriana Ortiz-Rodriguez),
Almost paracomplex structures on 4-manifolds (with Nikos Georgiou),
A new geometric structure on tangent bundles (with Nikos Georgiou),
Evolving to non-round Weingarten spheres: integer linear Hopf flows (with Wilhelm Klingenberg),
An extension of Asgeirsson’s mean value theorem for solutions of the ultra-hyperbolic equation in dimension four (with Guillem Cobos),
The causal topology of neutral 4-manifolds with null boundary (with Nikos Georgiou).

4-manifold Topology and the Poincaré Conjectures
Over the years I have studied 4-manifold topology and the 4-dimensional Poincaré Conjectures in both the topological and smooth categories - the former is officially deemed proven by Michael Freedman in the 1980's, the latter is officially open.
The accepted proof of the topological 4-dimensional Poincaré Conjecture is but one small consequence of a far bigger topological classification of 4-manifolds which follows from Freedman's "Disk Theorem".
The original published proof of the Disk Theorem contains numerous errors and the subsequent monograph by Freedman and Quinn did not address the central issues.
In a series of mathoverflow qestions I have probed this topic

Recently I gave a lecture on the motivation and implications of these questions: