# Welcome to the Hodge Institute

We are the algebra, geometry, number theory and topology research groups in the School of Mathematics at the University of Edinburgh. Together we have 15 permanent faculty, and a large number of PhD students and post-docs. We run several weekly seminars, together with many working groups and the SMSTC graduate training programme. Edinburgh has a rich mathematical history, which has been turned into a mathematical map of Edinburgh by Sophie Powell-Hall.

In general, our work traverses algebraic geometry, algebraic number theory, algebraic topology, category theory, noncommutative algebra, representation theory. Specific research interests include:

- Algebraic Geometry related to neighbouring fields: Combinatorics, Commutative Algebra, Gauge Theory, Mathematical Physics, Homotopy theory, Symplectic Geometry
- Algebraic Surgery Theory
- Birational Geometry, Positivity properties of divisors
- Category Theory and its Applications
- Derived Categories and Moduli Spaces
- Derived Algebraic Geometry
- Geometry of Algebraic Numbers
- Homological Commutative Algebra
- Knots, Links and Braids
- Noncommutative Algebraic Geometry
- Noncommutative Ring Theory
- Representation Theory related to: Combinatorics, Lie theory, Mathematical Physics
- Topology of Manifolds
- Toric and tropical geometry

From the abstract for Sir Michael Atiyah's lecture on Clerk Maxwell's Influence on Mathematics at the meeting "Inspiring Brilliance: Celebrating Maxwell's Genius and Legacy" at the Royal Society of Edinburgh, November 9, 2015 : *Maxwell’s Equations describe the behaviour of electro-magnetic waves in a way that combines geometry and analysis in perfect harmony. This led William Hodge, from the same Edinburgh–Cambridge stable as Maxwell, to extend the mathematical formalism to higher dimensions, with profound consequences. In due course, this fed into physical theories of the late 20th Century, which in turn, but unexpectedly, stimulated whole new and exciting developments in mathematics, now studied at the Hodge Institute in Edinburgh. For both physics and mathematics, Maxwell is the pioneering hero.*

Michael Atiyah has received too many prizes to list here! Agata Smoktunowicz has won a European Mathematical Society Prize, the Whitehead Prize of the London Mathematical Society, the Whittaker Prize of the Edinburgh Mathematical Society and the Wacław Sierpiński Prize of the Polish Academy of Sciences, and also gave an invited lecture at the 2006 International Congress of Mathematicians. She was awarded an ERC advanced investigator grant in 2013. Andrew Ranicki won the London Mathematical Society Junior Whitehead Prize in 1983 and the LMS Senior Berwick Prize in 1994. Iain Gordon was awarded the LMS Berwick Prize in 2005, was an EPSRC Leadership Fellow 2008-2013, and was an invited speaker at the 2010 International Congress of Mathematicians. Jon Pridham held an EPSRC 5-year Fellowship, ending in 2016. In 2013, Arend Bayer won an ERC starting grant and Michael Wemyss was awarded a 5-year Early Career Fellowship by EPSRC. In 2015, David Jordan won an ERC starting grant and Arend Bayer won the Cambridge University Adams Prize. In 2016, Arend Bayer was awarded the Whitehead Prize of the London Mathematical Society and the Whittaker Prize of the Edinburgh Mathematical Society. Nick Sheridan and Ben Davison both hold a Royal Society University Research Fellowship, and Ben Davison also holds and ERC starting grant.

In 2018, a group at the Hodge Institute, together with researchers in Glasgow and Sheffield, was awarded the EPSRC programme grant Enhancing Representation Theory, Noncommutative Algebra and Geometry.