Research Training Area 1

Topological Combinatorics is a problem-driven area which lies on the crossroads between various facets of Geometry and Algebraic Topology. Fascinating interplay between Discrete Geometry (which provides intriguing problems), advanced Algebraic Topology (which offers a wide range of methods) and Combinatorial Geometry (which gives final answers) motivates all of our research projects. We are particularly proud of our contributions in solving the following problems: the Bárány-Larman conjecture for primes-1, the Nandakumar & Ramana Rao problem, the Vassiliev extended conjecture and Hung’s injectivity conjecture on the cohomology of an unordered configuration spaces, as well as the topological Tverberg conjecture.

Mathematical physics can be approached in many different ways. Here, it is a cross-section of traditional areas of mathematics, involving asymptotic analysis, low-dimensional geometry and geometry of moduli spaces, algebraic structures and combinatorics, which are often (at least historically) enlightened by ideas born in contemporary physics, in particular statistical physics, quantum field theory and string theory. More specific areas of active research are the enumerative geometry of complex curves, the related (quantum) geometry of moduli spaces of geometric structures (relation to RTA 2); (supersymmetric) gauge theories and conformal field theories; random matrix theory (relation to RTA 3) and free probability; the combinatorics and geometry of flat or hyperbolic surfaces (relation to RTA 4 and 5); and the deep relations and unifying perspectives behind these problems.

My research focuses on geometric variational problems and their connections to other areas of mathematics, such as differential topology. I study critical points of geometric functionals as well as parabolic evolution equations that arise from their gradient flows. Central questions include properties of critical points such as existence and uniqueness and analysis of singularities of geometric flows. I also investigate global features — such as embeddedness and symmetry of minimizers of geometric functionals or stability analysis of critical points. I am particularly fascinated by non-orientable surfaces, the Willmore functional, and geometric partial differential equations of fourth order in general.

Symplectic Structures and Hamiltonian Systems of ODE are studied through properties of holomorphic curves and their solutions spaces: bounds on their symplectic area, compactness, transversality. One obtains obstructions for Lagrangian embeddings, symplectic embeddings of subsets of the standard sysmplectic space into other such subsets, existence and action of closed periodic orbits of Hamiltonian systems.

An important question in topology is the classification of manifolds. Therefore, one might ask to what extent a manifold is determined by its homotopy type. Since invariants of spaces studied in algebraic topology typically depend only on the homotopy type, more refined invariants, such as algebraic K-theory, are needed to distinguish manifolds. Understanding the algebraic K-theory of group rings, for example by proving the Farrell-Jones conjectures, becomes an important goal and is a central theme in our group. Modern formulations freely make use of higher algebra as developed by Jacob Lurie. There is a rich interplay between (geometric) group theory, noncommutative (higher) algebra, geometry, and (stable) homotopy theory.

An important branch of Mathematical Physics is concerned with fundamental questions in string theory and quantum field theory. The latter provides the conceptual and mathematical framework for elementary particle physics. The ‘heart’ of the standard model of particle physics is the quantum field theory of gauge fields. It describes all the fundamental forces in nature – with the very special exception of gravity. Gravity is however included in the theory of strings. An exciting conjecture, the so-called AdS/CFT correspondence, proposes that certain gauge and string theories are mathematically equivalent. This offers an entirely new way to explore the fundamental interactions of matter. In recent years, remarkable progress has been made towards a quantitative understanding of this correspondence based on ideas of exact integrability, with important links to purely mathematical questions. The ultimate goal is to deepen our mathematical and physical understanding of the AdS/CFT correspondence and its applications to both particle theory as well as to general relativity.

Geometric knot theory examines connections between the topological type of a knot and the geometry of space curves representing it, thus requiring a combinations of techniques from geometry and topology. Basic results like the Fáry–Milnor theorem show that knotted curves are more complicated that unknotted curves. There are also various notions of optimal shape for a given knot type, obtained by minimizing different geometric energies.

Modern (mathematical) Gauge Theory emerged in the 1980s with the breathtaking work of Donaldson, building on analytical foundations laid by Uhlenbeck and Taubes. He discovered that a detailed understanding of moduli spaces of anti-self-dual Yang–Mills instantons (solutions to a partial differential equation from theoretical particle physics) leads to astounding results about 4–manifolds. Gauge Theory has connections with many areas of mathematics, e.g., analysis, algebraic geometry, representation theory; and, to this day, remains an essential tool in 4–manifold topology. Recent trends in gauge theory include the study of versions of anti-self-duality higher dimensions, especially in the context of Riemannian manifolds with special and exceptional holonomy, as well as knew equations in 4–dimensions, e.g., the Vafa–Witten equation and the Kapustin–Witten equation (which is expected to have connections to both knot theory and the geometric Langlands program).

Symplectic geometry emerged more than a century ago as a mathematical setting for classical mechanics. The subfield of symplectic topology – which studies the global properties of symplectic manifolds, and their implications for other fields such as algebraic geometry or mathematical physics – has exploded with activity since the introduction of Floer homology and holomorphic curve methods in the 1980s. The invariants studied in modern symplectic topology rely on an intricate mixture of techniques from geometric/functional analysis and homological algebra, often with a healthy dose of adventurous physicist’s intuition thrown in. They can be applied to study a variety of fundamental questions, including the existence and classification of symplectic or contact manifolds, the qualitative behavior of Hamiltonian dynamical systems, and the development of new invariants in low-dimensional topology. On the analytical side, many interesting questions about the technical foundations of holomorphic curve invariants remain only partially understood: these involve e.g. the interplay between symmetry and transversality conditions, the local geometric structure of moduli spaces, and the influence of this local structure on their global topology. Such questions also have relevance for neighboring fields, such as mathematical gauge theory and its application to the study of smooth manifolds.