# Mathematics - Research

Postgraduate students in the School may read for a Ph.D. or M.Sc. degree by research. They may also pursue a one-year, full-time taught course in High-Performance Computing. There are no formal course requirements for those pursuing a degree by research, but research students are expected to participate fully in appropriate seminars. Prospective students are expected to possess a good honours degree (i.e. an upper second class at least) and to have the necessary background to pursue advanced study in their chosen field of research.

Research Programmes

The School has two broad research groups in Pure Mathematics and Theoretical Physics areas. Research in the School is being funded by IRC, SFI, ERC, H2020, the Royal Society, and the Simons Foundation.

Pure Mathematics: The main research groups concentrate on partial differential equations, operator algebras, operator theory and complex analysis, several complex variables, real and complex algebraic geometry, algebra, algorithms, numerical analysis and scientific computing as well as history of mathematics.

Partial Differential Equations

• Paschalis Karageorgis: Hyperbolic nonlinear partial differential equations, especially nonlinear wave and Schrödinger equations. Problems of existence and qualitative properties of solutions;

• John Stalker: Hyperbolic partial differential equations, especially those systems which are of particular physical interest. Mostly these are the Einstein equations of general relativity, but also the Euler equations of fluid mechanics and the equations governing nonlinear elasticity.

Complex Analysis and Geometry

• Andreea Nicoara works in several complex variables, real and complex algebraic geometry, model theory.

• Dmitri Zaitsev has interests including several complex variables (CR geometry), real and complex algebraic geometry, symplectic geometry and Lie group actions as well as connections with Partial Differential Equations and Combinatorics. His other interests include applications of Category Theory to Functional Reactive Programming and Algorithmic Complexity.

Algebra, Algebraic Geometry, and Algebraic Topology

• Sergey Mozgovoy: Moduli spaces of quiver representations, Nakajima quiver varieties, quivers with superpotentials, non-commutative crepant resolutions of 3-Calabi-Yau varieties, brane tilings and dimer models, enumerative problems in representation theory, refined Donaldson-Thomas invariants and BPS counting, wall-crossing formulas, moduli spaces of Higgs bundles, moduli spaces of vector bundles on surfaces, Hilbert schemes of surfaces and Hilbert schemes of plain curve singularities, Hall algebras of curves and quivers.

Algorithms

• Colm Ó Dúnlaing works on the theory of computation, algorithm design, computational complexity, and computational geometry.

Numerical Analysis and Scientific Computing

• Kirk Soodhalter : numerical linear algebra, ill-posed problems, Krylov subspace methods, data-driven iterative methods, high-performance computing applications.

History of Mathematics

• David Wilkins works on the history of mathematics, concentrating on the work of Hamilton and contemporaries of the 19th century.

Theoretical Physics research groups focus on String Theory and Lattice Quantum Chromodynamics.

String Theory: This is one of the most active areas of research in physics and mathematics, lying at the frontier of both. Briefly, it is an attempt to find a unified theory of fundamental interactions, including gravity.

The group’s research concentrates on mathematical aspects of string theory with special emphasis on geometric problems and methods.

• Ruth Britto: Scattering amplitudes, perturbative gauge theory and quantum chromodynamics, quantum field theory, Feynman integrals;

• Sergey Frolov: string theory, gauge theory/string theory correspondence, integrable systems;

• Tristan McLoughlin: Quantum field theory, quantum gravity, string theory, gauge/gravity correspondence;

• Jan Manschot: Quantum field theory, gravity, string theory, number theory, geometry;

• Andrei Parnachev: Conformal field theory, holography, strongly coupled quantum field theories;

• Samson Shatashvili: supersymmetric gauge theories, Donaldson and Seiberg-Witten theory, integrable systems, topological strings, string field theory.

Lattice Quantum Chromodynamics: The discretisation of QCD on a space-time lattice allows the analytically insoluble equations governing the dynamics of quarks and gluons to be simulated numerically, providing results that are of direct relevance to elementary particle physics and which shed light on theories of strongly-interacting matter. The group is a partner in the H2020 Program for European Joint Doctorates - HPC-LEAP.

• Mike Peardon: Monte Carlo techniques, algorithms for simulating quantum field theories, hadron spectroscopy and scattering;

• Sinead Ryan: QCD at zero and finite temperature, heavy quark physics, spectroscopy of strong exotic matter;

• Stefan Sint: Non-perturbative renormalisation techniques, determination of quark masses and the strong coupling constant, CKM and Standard Model phenomenology.

## Entry requirements

Applicants must normally have an excellent primary degree, and / or professional qualification, in a relevant discipline from a reputable institution. In addition, PhD applicants should have an excellent Masters degree from a reputable institution. All applicants must have a fluent command of the English language. Since the demand for places is extremely high, these minimum requirements do not guarantee admission. Preference is given to the strongest academic applicants.

## Application dates

Research Applications

No closing dates apply for application for higher degrees by research but there are only two registration periods (September and March). Applicants are advised to apply as early as possible prior to their chosen registration period as supervisory capacity may be limited.

In exceptional circumstances it may be possible to register retrospectively. Applicants wishing to be considered for retrospective admission should contact the Graduate Studies Office by emailing research.admissions@tcd.ie

## Enrolment dates

Next Intake: March 2024 / September 2024

## Research

School Description:

Postgraduate study in the School of Mathematics offers students a range of subjects in pure mathematics, theoretical physics, and interdisciplinary subjects such as bioinformatics. The School is small and the setting is informal which encourages close contact with staff, postdoctoral fellows, visiting scholars and fellow postgraduate students. The workshops and guests of the School's Hamilton Mathematics Institute (www.hamilton.tcd.ie) in addition to its joint seminars with the School of Theoretical Physics of the Dublin Institute for Advanced Studies and TCD's three neighbouring universities provide a stimulating intellectual backdrop to a student's stay at TCD.

Postgraduate students in the School may read for a Ph.D. or M.Sc. degree by research. They may also pursue a one-year, full-time taught course in High-Performance Computing. There are no formal course requirements for those pursuing a degree by research, but research students are expected to participate fully in appropriate seminars. Prospective students are expected to possess a good honours degree (i.e. an upper second class at least) and to have the necessary background to pursue advanced study in their chosen field of research.

Research Programmes

The School has two broad research groups in Pure Mathematics and Theoretical Physics areas. Research in the School is being funded by IRC, SFI, ERC, H2020, the Royal Society, and the Simons Foundation.

Pure Mathematics: The main research groups concentrate on partial differential equations, operator algebras, operator theory and complex analysis, several complex variables, real and complex algebraic geometry, algebra, algorithms, numerical analysis and scientific computing as well as history of mathematics.

Partial Differential Equations

•Paschalis Karageorgis: Hyperbolic nonlinear partial differential equations, especially nonlinear wave and Schrödinger equations. Problems of existence and qualitative properties of solutions;

•John Stalker: Hyperbolic partial differential equations, especially those systems which are of particular physical interest. Mostly these are the Einstein equations of general relativity, but also the Euler equations of fluid mechanics and the equations governing nonlinear elasticity.

Functional analysis

•Richard M. Timoney: Operator spaces, complex analysis.

Complex Analysis and Geometry

•Andreea Nicoara works in several complex variables, real and complex algebraic geometry, model theory.

•Dmitri Zaitsev has interests including several complex variables (CR geometry), real and complex algebraic geometry, symplectic geometry and Lie group actions as well as connections with Partial Differential Equations and Combinatorics. His other interests include applications of Category Theory to Functional Reactive Programming and Algorithmic Complexity

Algebra, Algebraic Geometry, and Algebraic Topology

•Vladimir Dotsenko works on homological and homotopical algebra, combinatorics, representation theory, Gröbner bases.

•Sergey Mozgovoy: Moduli spaces of quiver representations, Nakajima quiver varieties, quivers with superpotentials, non-commutative crepant resolutions of 3-Calabi-Yau varieties, brane tilings and dimer models, enumerative problems in representation theory, refined Donaldson-Thomas invariants and BPS counting, wall-crossing formulas, moduli spaces of Higgs bundles, moduli spaces of vector bundles on surfaces, Hilbert schemes of surfaces and Hilbert schemes of plain curve singularities, Hall algebras of curves and quivers.

Algorithms

•Colm Ó Dúnlaing works on the theory of computation, algorithm design, computational complexity, and computational geometry.

Numerical Analysis and Scientific Computing

•Kirk Soodhalter : numerical linear algebra, ill-posed problems, Krylov subspace methods, data-driven iterative methods, high-performance computing applications.

History of Mathematics

•David Wilkins works on the history of mathematics, concentrating on the work of Hamilton and contemporaries of the 19th century.

Theoretical Physics research groups focus on String Theory and Lattice Quantum Chromodynamics.

String Theory: This is one of the most active areas of research in physics and mathematics, lying at the frontier of both. Briefly, it is an attempt to find a unified theory of fundamental interactions, including gravity.

The group's research concentrates on mathematical aspects of string theory with special emphasis on geometric problems and methods.

•Ruth Britto: Scattering amplitudes, perturbative gauge theory and quantum chromodynamics, quantum field theory, Feynman integrals;

•Sergey Frolov: string theory, gauge theory/string theory correspondence, integrable systems;

•Tristan McLoughlin: Quantum field theory, quantum gravity, string theory, gauge/gravity correspondence;

•Jan Manschot: Quantum field theory, gravity, string theory, number theory, geometry;

•Andrei Parnachev: Conformal field theory, holography, strongly coupled quantum field theories;

•Samson Shatashvili: supersymmetric gauge theories, Donaldson and Seiberg-Witten theory, integrable systems, topological strings, string field theory;

•Dmytro Volin: Integrable systems, gauge/gravity correspondence, representation theory, quantum spectral curve.

Lattice Quantum Chromodynamics: The discretisation of QCD on a space-time lattice allows the analytically insoluble equations governing the dynamics of quarks and gluons to be simulated numerically, providing results that are of direct relevance to elementary particle physics and which shed light on theories of strongly-interacting matter. The group is a partner in the H2020 Program for European Joint Doctorates - HPC-LEAP.

•Marina Krstic Marinkovic: Standard Model phenomenology, muon g-2, heavy quark physics, Monte Carlo techniques, high performance computing;

•Mike Peardon: Monte Carlo techniques, algorithms for simulating quantum field theories, hadron spectroscopy and scattering;

•Alberto Ramos: Non-perturbative effects in quantum field theories, lattice field theory and its applications to understand the phenomenology of the strong interactions;

•Sinead Ryan: QCD at zero and finite temperature, heavy quark physics, spectroscopy of strong exotic matter;

•Stefan Sint: Non-perturbative renormalisation techniques, determination of quark masses and the strong coupling constant, CKM and Standard Model phenomenology.