Our Team

Aaron Lauda is a Professor of Mathematics at the University of Southern California, studying representation theory, low-dimensional topology, and applications in mathematics and theoretical physics. He holds a joint appointment in the Department of Physics and Astronomy and is a member of the USC Center for Quantum Information Science and Technology. He is best known for his categorification of quantum sl(2), a symmetry governing quantum link, and 3-manifold invariants connected with Chern-Simons theory. Together with Mikhail Khovanov, he extended this work to define a general theory of categorified quantum groups, helping to initiate the study of higher representation theory. His work developed applications of these new symmetries to link homology theory and traditional representation theory. More recently, his research has focused on the algebraic structure of Heegaard Floer homology, establishing a new link to the BGG category O, hypertoric geometry, and the amplituhedron. Lauda is a recipient of the Sloan Research Fellowship and is a Fellow of the American Mathematical Society and the Simons Foundation. In 2021 he was awarded a Doctorate of Science (D.Sci) from Cambridge University.

Sergei Gukov is a Professor of Mathematics and Theoretical Physics at the California Institute of Technology. He is also a member of the Max Planck Society and an external scientific member at the Institute for Mathematics in Bonn. His previous work includes various generalizations of the Volume Conjecture, the study of ramification in gauge theory approach to the geometric Langlands program, and categorification of quantum group invariants. In recent years, he has been exploring new algebraic structures in topology and in quantum field theory, working on construction of new TQFTs, and use of artificial intelligence (AI) to tackle problems in knot theory and topology.

Jørgen Ellegaard Andersen is a professor of mathematics at the Danish Institute of Advanced Study and director of Centre for Quantum Mathematics, SDU, Denmark. Andersen is a quantum topologist who specializes in the study of the Witten-Reshetikhin-Turaev Topological Quantum Field Theory from the gauge theory perspective. A very important part of these TQFT’s are the quantum representations of the mapping class groups and using the gauge theory construction of these, Andersen has proved that they are asymptotic faithfulness. In joint work with Kenji Ueno, Andersen has proved that the gauge theory construction is equivalent to the original combinatorial construction of Reshetikhin and Turaev, by showing that the modular functors which arise from conformal field theory are isomorphic to the combinatorial ones constructed from modular tensor categories. In a collaboration with Rinat Kashaev, Andersen has constructed the Teichmüller TQFT, which is based on a combinatorial quantization of Teichmüller theory. This theory is expect to be closely related to the WRT-TQFT via the program of resurgence, which Andersen is currently working on jointly with Bertrand Eynard, Maxim Kontsevich and Marcos Marino under the ERC-Synergy program “ReNewQuantum”. This program further includes the Geometric Recursion program which Andersen created joint with Borot and Orantin and which is a kind of categorification of Topological Recursion. Andersen was recently designated Knight of the Order of Dannebrog by the Danish Queen.

Anna Beliakova is a Professor at the University of Zurich. She works on low-dimensional topology, quantum groups, and Khovanov homology. In her PhD, she gave the first proof of the relationship between Turaev–Viro and Witten–Reshetikhin—Turaev (WRT) 3-manifold invariants. Later on, she unified WRT invariants of any rational homology 3-sphere into a unique power series and used it to prove the integrality of WRT invariants for all 3-manifolds at all roots of unity. Together with her collaborators, she developed a theory of traces in bicategories, invented Quantum Annular Khovanov homology, and interpreted “modified” trace in non-semisimple TQFTs as a symmetrised integral. Her most recent achievement is a construction of a new family of invariants of 4-dimensional 2-handlebodies from ribbon Hopf algebras. Beliakova is a board member of the National Centre of Competence in Research SWISSMAP.

Mikhail Khovanov is a Professor of Mathematics at Columbia University. He is best known for Khovanov homology, which is a homology theory for link embedded in 3-space which has the Jones polynomial as its own characteristic. His other notable work includes constructing SL(3) link homology, discovering Khovanov-Rozansky homology (joint with Lev Rozansky) which categorifies quantum SL(N) link invariants, and joint work with Aaron Lauda on categorified quantum groups. Khovanov’s recent research centers around using foams and the universal construction of topological theories to investigate link homology and topology in low dimensions.

Peter Kronheimer is a Professor at Harvard University. His early work centered on hyper-Kähler geometry, and in particular the construction of gravitational instantons, but he is now known principally for his contributions to low-dimensional topology through applications of gauge theory, an area he has pursued in collaboration with Tomasz Mrowka. Results of this collaboration include the first proof of Milnor’s conjecture on the unknotting number of algebraic knots, the Thom conjecture on the genus-minimizing property of algebraic curves, a proof of Bing’s Property P conjecture concerning surgery on knots, and the proof that Khovanov homology detects the unknot. Current research projects include a conjectural program to apply gauge theory to a new proof of the 4-color theorem and establishing bounds on the degrees of rational curves lying on surfaces of general type. He is a recipient of the Whitehead Prize of the London Mathematical Society, the Förderpreis from the Mathematisches Forschungsinstitut Oberwolfach, and the Veblen and Doob prizes from the American Mathematical Society. He was elected a Fellow of the Royal Society in 1997 and was a plenary speaker at the International Congress of Mathematicians in 2018.

Ciprian Manolescu is a professor at Stanford University. He works in low dimensional topology and gauge theory. His research is centered on constructing new versions of Floer homology and applying them to questions in topology. With collaborators, he showed that many Floer-theoretic invariants are algorithmically computable. He also developed a new variant of Seiberg-Witten Floer homology, which he used to disprove the Triangulation Conjecture in high dimensions. His most recent work is on Khovanov homology and its applications to four-dimensional topology. He is a recipient of the European Mathematical Society Prize, the E. H. Moore Prize from the American Mathematical Society, and was an invited speaker at the International Congress of Mathematicians in 2018. He is also a Simons Investigator.

Cris Negron is an assistant professor at the University of Southern California. His work concerns interactions between representation theory, tensor categories, and mathematical physics. In joint work with Terry Gannon he established the “logarithmic Kazhdan-Lusztig correspondence” in type A_1, which relates representations of quantum groups at roots of unity to non-rational chiral conformal field theories. His recent projects focus on mathematical constructions of topological quantum field theories with infinite-dimensional state spaces, and non-semisimple categories of line operators. In studying such TQFTs one finds an interesting intertwining of geometric representation theory and tensor triangular geometry.

Tomasz (Tom) Mrowka is a Professor of Mathematics at the Massachusetts Institute of Technology. He studies the partial differential equations of high energy physics including the Yang-Mills and Seiberg-Witten equations. As first discovered by Simon Donaldson, these differential equations have many applications to the differential topology of manifolds in dimensions three and four. With Bob Gompf, Mrowka discovered the first class of irreducible four dimensional manifolds that did not admit complex structures. Highlights of a long and on going collaboration with Peter Kronheimer are a proof of the Thom conjecture about the genus minimizing property of complex curves in the projective plane and the closely related discovery four dimensional smooth manifolds have analogues of the canonical class of complex surface, the resolution of Bing’s property P conjecture, and that Khovanov homology detects the unknot. Recently they discovered a connection between the invariants of knotted graphs coming from gauge theory and the four color map theorem and hope to develop these ideas into a computer free proof of four color map theorem during the course of this collaboration.

Mrowka is a member of the National Academy of Science and the American Academy of Arts. With Kronheimer, he is was awarded the Veblen and Doob Prizes of American Mathematical Society, and was a Simons Foundation, Sloan and Guggenheim fellow.

Peter Ozsvath is a professor of mathematics at Princeton University. He works in low-dimensional topology and its interaction with symplectic geometry. In collaboration with Zoltan Szabo, he defined Heegaard Floer homology, an invariant for three-manifolds, four-manifolds, and knots. Recently, his research has focused on building up computational and conceptual tools for understanding these invariants, in addition to finding new applications to low-dimensional topology. He is a recipient of the Veblen Prize from the American Mathematical Society, and he is a member of the National Academy of Sciences.

Lisa Piccirillo is an assistant professor at MIT. She works in low dimensional topology, primarily on the study of 4-manifolds. Her work uses tools from knot concordance, 3-manifold topology, symplectic topology, and Khovanov homology; she is particularly interested in the differences between smooth and topological phenomena in 4-manifolds. Her work includes a new method for obstructing knot sliceness which was applied to prove that the Conway knot is not slice. Her development of the study of knot traces has led to progress in the study of exotic 4-manifolds and surfaces, and in the study of geometrically simply connected 4-manifolds. She is a recipient of a Clay Research Fellowship and the 2021 Maryam Mirzakhani New Frontiers Prize.

Raphael Rouquier is a Professor of Mathematics at the University of California, Los Angeles. He works in representation theory with categorical, homological, geometrical and topological aspects. In collaboration with Chuang, he initiated higher representation theory to solve a conjecture of Broue on modular representations of finite groups. Together with Bonnafe and Dat, he constructed a Jordan decomposition for modular representations. With Kashiwara, he introduced a new type of quantization in geometric representation theory beyond cotangent bundles. With Shan, Varagnolo and Vasserot, he proved a conjecture on characters of cyclotomic Cherednik algebras. He developed a general setting of 2-representation theory and, in collaboration with Andrew Manion, recast a part of bordered Heegaard-Floer theory in that setting. He constructed a categorical version of braid groups that has applications to representation theory and knot invariants. He introduced the notions of perverse equivalences and of Hecke algebras for complex reflection groups which have widespread applications in representation theory. He has received the Peccot Prize from the College de France, the Whitehead Prize from the London Mathematical Society, the Adams Prize from the University of Cambridge and the Elie Cartan Prize from the French Academy of Science. He is also a Simons Investigator.

Lev Rozansky is a professor at the University of North Carolina at Chapel Hill. His interests lie at the intersection of low-dimensional topology, algebraic geometry and string theory. In a joint work with E. Witten he introduced 3d B-twisted sigma-models whose targets are holomorphic symplectic varieties. Together with A. Kapustin and N. Saulina he described the 2-category of boundary conditions of these models. Its objects are fibrations over Lagrangian subvarieties and their morphisms are matrix factorizations. Together with M. Khovanov he constructed a matrix factorization based categorification of the sl(n) Reshetikhin-Turaev link invariant and a categorification of the HOMFLY-PT polynomial. Together with A. Oblomkov he constructed a categorification of the HOMFLY-PT polynomial which is based on the 3d B-model whose target is the Hilbert scheme of points in a complex plane. This construction interprets the link homology as quantum vibrations of a stack of D2 branes in a IIA string theory. Currently he explores the properties and other applications of this string-motivated approach to link homology and categorification of Lie and Hecke algebras.

Zoltan Szabo is a Professor of Mathematics at Princeton University. Some of his earlier work centers on the study of exotic smooth structures on 4-manifolds and the minimal genus problem for symplectic four-manifolds. In collaboration with Peter Ozsvath he has studied surgery problems for Seiberg-Witten invariants that led to the proof of the Symplectic Thom Conjecture. In another collaboration with Peter Ozsvath he defined Heegaard Floer homology; an invariant for closed oriented three-dimensional manifolds. This theory has applications in knot theory and on surgery problems for three-dimensional manifolds. His more recent research is centered on further developing knot Floer homology and Heegaard Floer homology, studying topological applications and finding new computational methods for these invariants. He is a honorary member of the Hungarian Academy of Sciences, and a recipient of the Veblen Prize from the American Mathematical Society.