# Academics

## Professor András Juhász

### Tutorial Fellow and Professor of Mathematics

### Royal Society Research Fellow

### Welcome

I am a pure mathematician working in geometric topology. In Keble, I teach undergraduates various subjects in pure mathematics. I hold a joint appointment with the Mathematical Institute, where I focus mostly on research and PhD supervision.

### Research Interests

I am mainly interested in the study of manifolds. An n-dimensional manifold is a spaces that locally looks like the standard n-dimensional coordinate space (whose points can be parametrized by n real numbers). Even though locally every n-manifold looks the same, they can have highly nontrivial global structure. Two manifolds are equivalent from a topological point of view if one can be deformed into the other without tearing or puncturing it. For example, the surface of a doughnut and the surface of a coffee mug are equivalent two-manifolds. In particular, topologists are not interested in the geometry of the object, such as how it is curved, but instead focus on the global shape.

The central question of manifold topology is classification. For example, how do we rigorously distinguish the surface of a sphere from the surface of a doughnut? One has a hole, the other does not. The only connected one-manifolds are the real line and the circle. Two-manifolds are called surfaces. The closed orientable (two-sided) ones are surfaces of doughnuts with a number of holes, while the closed non-orientable ones can be obtained by joining a number of Klein bottles together with tubes. Interestingly, the classification of manifolds is simpler in some sense in dimensions five and higher than in dimensions three and four. Hence, most of the attention of manifold topology is focused on low dimensions. This is also my main research interest. Low-dimensional topology is closely linked with knot theory, which studies how one can embed the circle in three-space.

Manifolds arise as configuration spaces of mechanical systems. Our universe is a three-manifold, and space-time is a four-manifold. According to some versions of string-theory, we live in an 11- or 13-dimensional universe. Knot theory is helpful in the study of the DNA and certain proteins.

### Recent Publications

- 'Cobordisms of sutured manifolds and the functoriality of link Floer homology' in
*Adv. Math.*, 299 (2016), 940-1038 - with I. Altman and S. Friedl, 'Sutured Floer homology, fibrations, and taut depth one foliations' in
*Transactions of the AMS*, 368 (2016), 6363-6389 - with M. Marengon, 'Concordance maps in knot Floer homology' in
*Geom. Topol.*(2016) - with S. Kang, 'Algebraic torsion for contact manifolds with convex boundary' in
*arXiv:1601.05602*(2016) - with M. Marengon, 'Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT' in
*math.GT/1503.00665*(2015) - 'A survey of Heegaard Floer homology' in
*New Ideas in Low Dimensional Topology*, ed. L. Kauffman and V. Manturov (World Scientific, 2014) - 'Defining and classifying TQFTs via surgery' in
*math.GT/1312.0823*(2014) - with M. Hedden and S. Sarkar, 'On sutured Floer homology and the equivalence of Seifert surfaces' in
*Algebr. Geom. Topol.*, 13 (2013), 505-548 - with T. Kálmán and J. Rasmussen, 'Sutured Floer homology and hypergraphs' in
*Math. Res. Lett.*, 19 (2013), 1309-1328 - with D. Thurston, 'Naturality and mapping class groups in Heegaard Floer homology' in
*arXiv:1112.2632*(2012) - with S. Friedl and J. Rasmussen, 'The decategorification of sutured Floer homology' in
*J. Topol.*, 4 (2011), 431-478 - 'The sutured Floer homology polytope' in
*Geom. Topol.*, 14 (2010), 1303-1354 - 'Floer homology and surface decompositions' in
*Geom. Topol.*, 12 (2008), 299-350 - 'Knot Floer homology and Seifert surfaces' in
*Algebr. Geom. Topol.*, 8 (2008), 603-608 - 'Holomorphic discs and suture manifolds' in
*Algebr. Geom. Topol.*, 6 (2006), 1429-1457 - 'Regular homotopy casses of singular maps' in
*Proc. London. Math. Soc.*, 90 (2005), 738-762 - 'A geometric classification of immersions of 3-manifolds into 5-space' in
*manuscripta math.*, 117 (2005), 65-83 - 'Regular homotopy classes of locally generic mappings' in
*Topology Appl.*, 138 (2004), 45-59

### Academic Biography

- ERC Starting Grant (2016 - 2021)
- Royal Society Research Fellow and Associate Professor, Mathematical Institute, Oxford (2013 - present)
- Tutorial Fellow in Mathematics, Keble College, Oxford (2013 - present)
- Royal Society Research Fellow and Senior Lecturer, Imperial College London (2012 - 2013)
- Royal Society Research Fellow, Cambridge (2011 - 2012)
- Herchel Smith Postdoctoral Research Fellow, DPMMS, Cambridge (2008 - 2011)
- Junior Research Fellow, King's College Cambridge (2008 - 2012)

#### College Contact Details

Keble College

Oxford

OX1 3PG

UK

Telephone: 01865 272736

Fax: 01865 272705

Email: juhasza@maths.ox.ac.uk

#### Faculty/Dept. Information

Mathematical Institute

Woodstock Road

Oxford

OX2 6GG

United Kingdom

Website:

http://www.maths.ox.ac.uk/people/profiles/andras.juhasz