Fabian Haiden

Harvard University Department of Mathematics

Science Center

One Oxford Street

Cambridge, MA 02138

USA


email: haiden@math.harvard.edu

 

Current Teaching


Math 21b - Linear algebra and differential equations, Course head: Oliver Knill (website)


Math 118 - Dynamical Systems  (syllabus | website)




Papers and Preprints


Flat surfaces and stability structures (with L. Katzarkov, M. Kontsevich)


We identify spaces of half-translation surfaces, equivalently complex curves with quadratic differential, with spaces of stability structures on Fukaya-type categories of punctured surfaces. This is achieved by new methods involving the complete classification of objects in these categories, which are defined in an elementary way. We also introduce a number of tools to deal with surfaces of infinite area, where structures similar to those in cluster algebra appear.

This text subsumes my dissertation “Stability of 1-dimensional A-branes”.


Dynamical systems and categories (with G. Dimitrov, L. Katzarkov, M. Kontsevich)


We study questions motivated by results in the classical theory of dynamical systems in the context of triangulated and A-infinity categories. First, entropy is defined for exact endofunctors and computed in a variety of examples. In particular, the classical entropy of a pseudo-Anosov map is recovered from the induced functor on the Fukaya category. Second, the density of the set of phases of a Bridgeland stability condition is studied and a complete answer is given in the case of bounded derived categories of quivers. Certain exceptional pairs in triangulated categories, which we call Kronecker pairs, are used to construct stability conditions with density of phases. Some open questions and further directions are outlined as well.


An orbit construction of phantoms, Orlov spectra, and Knörrer periodicity (with D. Favero, L. Katzarkov)


Refined combinatorial torsion (diploma thesis)




Videos and Slides


Flat surfaces and stability in categories (video), Workshop: Wall Crossing, Quantum Integrable Systems, and TQFT


Fukaya categories of surfaces and Teichmüller theory (video), Landau-Ginzburg Theory and Fano Varieties


Quadratic differentials of exponential type and stability (slides), Conference on Homological Mirror Symmetry