
Michał Adamaszek (PhD, Warwick 2012): My research is in combinatorial algebraic topology with connections to graph theory, applied topology and face enumeration. I am also interested in algorithmic and probabilistic aspects of these topics and often do computer experiments in my work.


Hiroshi Ando (PhD, Kyoto 2012): I study structures of von Neumann algebras and some Polish groups related to them. Recently I have been working on ultraproducts of type III factors and some Borel equivalence relations related to unbounded selfadjoint operators on a separable Hilbert space.


Sara E. Arklint (PhD, Copenhagen 2012): My research is concerned with classification of nonsimple (purely infinite) C*algebras. So far I've been focusing on the classification functor filtered Ktheory, and on the classification of CuntzKrieger algebras and graph algebras. 

Tobias Barthel (PhD, Harvard/Oxford 2014): I am interested in stable homotopy theory and its interactions with representation theory and algebraic geometry. In particular, much of my work is inspired by ideas from chromatic and transchromatic homotopy theory. More recently, I have also been thinking about various forms of (local) duality in algebra and topology. 

Samik Basu (PhD, Harvard 2009): I am interested in homotopy theory, particularly stable homotopy theory, and the obstruction theory of ring structures. My thesis was written about associative ring structures in generalised Thom spectra, and the computation of the Topological Hochschild Homology in certain examples. 

Alexander Berglund (PhD, Stockholm University, 2008): Algebraic topology and interactions with commutative algebra. Currently, my research interests revolve around operads, Koszul duality, and Hochschild cohomology of Ainfinity algebras, with a view towards applications in String topology. 

Christopher Cave (PhD, Southampton 2015): My research is in coarse geometry which lies in the intersection of topology, geometry and operator algebras. My current interest in studying exact and nonexact groups and their consequences. I also have a lot of interaction with geometric group theory, in particular sequence of expanders graphs and their applications to geometric group theory. 

Dustin Clausen (PhD, MIT 2013): I am interested in connections between homotopy theory and number theory. Currently I'm studying a homotopytheoretic approach to reciprocity laws. This approach is based on the equivalence between the category of finite free Zmodules, which can be thought of as numbertheoretic, and the category of tori, which can be thought of as topological. Combined with the machinery of algebraic Ktheory, this equivalence allows to use topological arguments to obtain numbertheoretic results.
Financed by Hesselholt's Niels Bohr Professorship.


Tyrone Crisp (PhD, Penn State 2012): I study group representations, using tools from operator algebra and noncommutative geometry. At the moment I am particularly interested in reductive groups over local fields and rings, and in various kinds of induction/restriction constructions which relate representations of these groups to representations of their subgroups.


Christopher Davis (PhD, MIT 2009): My research is primarily in the areas of number theory and algebraic geometry, especially padic cohomology. Through Witt vectors and the de RhamWitt complex, my research is also related to algebraic topology.
Financed by Hesselholt's Niels Bohr Professorship.


Dieter Degrijse (Phd, Leuven 2013): My research interests include geometric group theory, algebraic topology and homological finiteness conditions.


Steven Deprez (PhD, Leuven 2011): My research centers on the study of type II1 factors. There are many constructions for type II1 factors, but it is very hard to decide if two, a priori different, constructions give the same type II1 factor. For example, every ICC group G gives rise to a type II1 factor LG. But all amenable ICC groups G give the same type II1 factor LG, which we call the hyperfinite type II1 factor. For that reason, various invariants have been introduced, for example the fundamental group (which is not related to the fundamental group of a topological space) and the outer automorphism group. These invariants are usually very hard to compute, but Sorin Popa's deformation/rigidity theory allows us to compute these invariants in specific cases. Often, type II1 factors are constructed from groups or from actions of groups on probability spaces. Deformation/rigidity theory uses measurable and geometric properties of these groups and actions. For that reason I am also interested in measurable and geometric group theory. 

Elden Elmanto (PhD, Northwestern 2018): My research interesest is in algebraic geometry and its interactions with homotopy theory. This occurs mainly through MorelVeovodsky's motivic homotopy theory, the theory of motives, and algebraic Ktheory.


Dominic Enders (Phd, Westfälische WilhelmsUniversität Münster, 2013): My main research interests are operator algebras, in particular the structure theory and classification of C*algebras. More precisely, I am interested in semiprojectivity (or, more generally, perturbation questions), dimension theories for C*algebras, classification via Ktheory and the interplay between those topics.


John D. Foley (PhD, UCSD 2012): My research combines algebraic topology and geometric group theory to study infinite (or infinite dimensional) groups. I am especially interested in the possible extension of homotopy Lie groups as developed in the theory of pcompact and plocal compact groups to homotopy KacMoody groups. More broadly, I am interested in how ntypes generalize groups as higher symmetries and how homotopy can be used to study small models combinatorially.


Olivier Gabriel (PhD, Paris 7 2011): My research interests center on operator algebras and noncommutative geometry. I work with spectral triples, quantum groups, Ktheory and cyclic cohomology. I am especially interested in ergodic actions on C*algebras. 

Giovanni Gandini (PhD, Southampton 2011): I work in geometric group theory, often using tools from group cohomology. I am particularly interested in (homological) finiteness properties of groups. Recently I also took an interest in the isomorphism conjectures.


Matthew Gelvin (PhD, MIT 2010): I work in the intersection of algebraic topology and finite group theory, studying fusion systems and their classifying spaces. My graduate thesis describes group actions from a fusion and plocal finite grouptheoretic perspective; I hope to continue in this vein by exploring the representation theory of fusion system and their connections with modular representation theory. 

Heiko Gimperlein (PhD, Hannover 2010): My research centers around geometric analysis on noncompact or singular manifolds and includes applications to representation theory and numerical analysis. Particular topics are pseudodifferential operators on singular spaces, Ktheory of the corresponding operator algebras, propagation of singularities for the wave equation, analytic methods for Lie groups and their representations as well as theoretical numerical analysis of nonconvex variational problem. 

Magnus Goffeng (Phd, University of Gothenburg 2015): My research evolves around the more analytic aspects of noncommutative geometry. It is mainly concerned with problems from index theory and spectral theory.


Márton Hablicsek (PhD, University of WisconsinMadison 2014): My research is in algebraic geometry with focus on Derived Algebraic Geometry and Combinatorial Algebraic Geometry. 

Rune Haugseng (Phd, Massachusetts Institute of Technology 2013): I am interested in algebraic topology and higher category theory, and their connections with other areas such as (derived) geometry and (topological) quantum field theory. 

Markus Hausmann (Phd, University of Bonn, 2016): My research is in algebraic topology, in particular different forms of equivariant homotopy theory. At the moment I am studying global equivariant spectra and try to apply them to problems in equivariant and nonequivariant topology. 

Ellen Henke (PhD, Birmingham 2010): My research focuses on studying saturated fusion systems. These are categories satisfying important features of fusion in finite groups. They have been studied by algebraic topologists, modular representation theorists and recently also by local group theorists. Many concepts and results from finite group theory have been translated into the language of fusion systems. An important long term goal appears to be the classification of all simple saturated fusion systems, at least for the prime 2. Achieving this goal will certainly require a huge amount of work, and with my research I hope to contribute to that. The main project I am working on at the moment aims to classify certain fusion systems that I call minimal. Minimal fusion systems can be seen as analogs of Thompson's Ngroups whose classification set a pattern for the classification of finite simple groups. Moreover, they appear to play a significant role, since every nonsolvable fusion system has a section which is minimal. 

Richard Hepworth (PhD, Edinburgh 2005): I work in the field of Algebraic Topology, and in particular in String Topology. This was introduced almost 10 years ago by Chas and Sullivan, and it shows how to extract interesting algebra from a manifold by thinking about the loops, or strings, in that manifold. I recently proved a result that completely describes the algebraic quantities when the manifold is a Lie group, and I am currently hoping to extend this in several directions.


Gijsbert Heuts (PhD, Harvard 2015): I am interested in studying unstable homotopy theory using methods from Goodwillie calculus and chromatic homotopy theory. I also think about higher category theory and its applications to these topics. 

Renee Hoekzema (PhD, Oxford, 2018): My mathematics research concerns algebraic and geometric topology, particularly in the study of manifolds, cobordism categories and topological quantum field theories. I am also interested in mathematical (palaeo)biological questions as well as mathematical physics, particularly relativity and quantum gravity. 

Rune Johansen (PhD, Copenhagen 2011): My research focuses on the interplay between symbolic dynamics and operator algebras, and I use computer programs to do experimental investigations of such mathematical structures in the search for input to formulate conjectures, theorems, and proofs. Currently, I am primarily working on experimental investigations of automorphisms of graph algebras and isomorphisms of Leavitt path algebras. I am also interested in the flow equivalence classification of sofic shifts with a special focus on renewal systems and betashifts. Additionally, I am involved in a project concerning colouring problems for LEGObuildings. 

Søren Knudby (PhD, Copenhagen 2014): My main interest is the study of von Neumann algebras and relations to group theory. In particular, I study approximation properties for groups and von Neumann algebras to see how they relate and complement each other. A classic example is that of amenability which for a group may be formulated as an approximation property and is reflected as hyperfiniteness of the group von Neumann algebra. 

Job Kuit (PhD, Utrecht 2011): My research centers around integral geometry and harmonic analysis on symmetric spaces. I am particularly interested in horospherical transforms. Together with Erik van den Ban (Utrecht University) and Henrik Schlichtkrull (University of Copenhagen) I am currently working towards a notion of cusp form for reductive symmetric spaces. We hope to get a better understanding of the Plancherel decomposition for these spaces by using cusp forms. 

Alexander Kupers (PhD, Stanford 2016): My research is in applications of algebraic topology to algebra and geometry. In particular, I am interested in the homology groups and homotopy groups of spaces of automorphisms of various objects. 

Anssi Lahtinen (PhD, Stanford 2010): I am an algebraic topologist with interests in string topology, parametrized homotopy theory and field theories. I am currently especially interested in the string topology of classifying spaces, which is a field studying the homology groups of the free loop space of the classifying space of a compact Lie group. As discovered by Chataur and Menichi, these homology groups enjoy a rich algebraic structure, as they are the value of the circle in a Homological Conformal Field Theory, a field theory where the operations are parametrized by homology classes of spaces of cobordisms. Jointly with Richard Hepworth, I am working on a novel construction of these field theories that will improve on Chataur and Menichi's original construction in a number of ways. 

Cyril Levy (PhD, ENSLyon 2009): I am interested in the construction of spectral triples for Dirac operators on manifolds with boundary and applications to the spectral action of ChamseddineConnes. We recently obtained, with Bruno Iochum and Dmitri Vassilevich, a regular spectral triple for a rather general geometric setting which includes the skewsymmetric torsion and the chiral bag conditions on the boundary. It turns out that theta=0 is a critical point of the associated spectral action in any dimension and at all orders of the expansion. The first four leading terms of the action have been computed for vanishing chiral parameter (theta=0), and torsiondependent terms have been identified. 

Frank H. Lutz (PhD, Berlin 1999): My main field of research is experimental topology. I have been involved in various projects on constructing extremal or otherwise interesting triangulations of manifolds. With Jesper Møller I work on colorings of simplicial complexes and manifolds. Together with Bruno Benedetti and Karim Adiprasito I have initiated and is developing random discrete Morse theory. Mimi Tsuruga and FL recently combinatorialized the AkbulutKirby handle body description of the CappellShaneson spheres. The resulting triangulations are used, in collaboration with Konstantin Mischaikow and Vidit Nanda, as nontrivial examples for testing the homology software CHomP. With Menachem Lazar, Robert MacPherson, Jeremy Mason and David Srolovitz I work on the topological microstructure analysis of metals and steel. 

Ehud Meir (PhD, Technion 2010): I work in cohomology of groups and in Hopf algebras, as well as in some related areas. My Phd dealt with some questions relating the cohomology of a group to that of a finite index subgroup. I later studied some questions which relates finite dimensional Hopf algebras to properties of the field over which they are defined. For example, if H is a finite dimension semisimple Hopf algebras (over a field F of characteristic zero), we can ask what are the possible simple quotients of H (I have proved that up to Brauer equivalence we get everything), or if H is already defined over the ring of integers of F (in a joint work with Juan Cuadra, we have proved that it is not always the case). I have also studied fusion categories, which can be viewed as a generalization of the representation categories of Hopf algebras.


Johan Öinert (PhD, Lund 2009): My research is concerned with the ideal structure of noncommutative rings, in particular graded rings such as generalized crossed products and skew group rings. I am also interested in the interplay between topological dynamical systems and various types of algebraic constructions such as C*crossed product algebras. 

Irakli Patchkoria (PhD, Bonn 2013): My research interests mainly lie in stable homotopy theory and homological algebra. I am especially interested in equivariant stable homotopy theory.


Dan Petersen (PhD, KTH Stockholm 2013): I am interested in the topology and cohomology of algebraic varieties, particularly moduli spaces.


Maria RamirezSolano (PhD, Copenhagen 2013): I work in the research area of operator algebras and tilings. I am currently working with Uffe Haagerup on the experimental side of the investigation of the Thompson group F. Part of my PhD thesis involved computer experiments while investigating a nonstandard hierarchical tiling. 

Oscar RandalWilliams (DPhil, Oxford 2009): I am interested in studying "moduli spaces of manifolds", both in the sense of diffeomorphism groups of manifolds, which are very interesting yet opaque, and in the sense of the spaces representing cobordism theory, which are far more computable. The MadsenWeiss theorem identifies spaces of these two flavours in dimension 2: a large part of my research, joint with Søren Galatius, is towards generalising this result to higherdimensional manifolds. The other aspect of my research is focused on applying results centered around the MadsenWeiss theorem to answer cohomological questions about spaces related to moduli spaces of curves, for example to study the universal Picard variety, or moduli spaces of rspin curves. 

Leonel Robert (PhD, Toronto 2006): My field of research is the classification and structure of C*algebras. In its broadest formulation, the classification problem asks "What is the *right* invariant that classifies the* right* class of C*algebras?" Although there may not be a unique answer to this question, various structural properties of the C*algebra arise naturally when looking for the right class: Zstability, finite nuclear dimension, the dichotomy of purely infinite vs stably finite, approximation by ``nice" subalgebras, etc. On the side of the classifying invariants, the most standard ones are Ktheory (filtered, with coefficients), the cone of traces, and more recently, the Cuntz semigroup. 

Simon Rose (PhD, University of British Columbia 2012) I study the enumerative geometry of hyperelliptic curves and surfaces, and in particular their connections with number theory and physics. Lately I have taken an interest in tropical geometry and how it fits into this picture.


Beren Sanders (PhD, UCLA 2014) My primary research interests lie in the theory and applications of triangulated categories, especially tensor triangular geometry and examples arising in stable homotopy theory, modular representation theory, algebraic geometry, and noncommutative topology. Other interests include equivariant homotopy theory, motivic homotopy theory, higher category theory, and the representation theory of groups and associative algebras.


David Schrittesser (PhD, Vienna 2010): My research is in set theory, where I have worked on forcing axioms, large cardinals, and descriptive set theory.
I work on problems in descriptive set theory, but also, for example, on forcing and some questions connected to the automorphism group of the measure algebra. I also hope to be able to apply descriptive set theoretic methods in other fields, especially operator algebra.


Farbod Shokrieh: I am interested in nonArchimedean analytic and tropical geometry, combinatorics, algebraic and arithmetic geometry. 

Tatiana Shulman (PhD, Moscow 2006): I am working in Operator Algebras. Since each commutative C*algebra (operator algebra) is the algebra of all continuous functions on some (locally) compact space, one can try to generalize topological notions to the category of all (not necessary commutative) C*algebras. In particular there is a theory of noncommutative absolute retracts and absolute neighborhood retracts which I am very ineterested in. 

David Sprehn (PhD, University of Washington 2015): I work in group cohomology. I am studying the modular cohomology of the general linear groups over finite fields, and of other finite groups of Lie type. More generally, I am interested in group actions and characteristic classes.


Wolfgang Steimle (PhD, Münster 2010): My research focuses on the interplay between the topology of manifolds and algebraic Ktheory. More specifically, I am interested in the fibering problem ("Is a given map between manifolds homotopic to a fiber bundle projection?"), in metrics of positive scalar curvature ("Are there exotic families of such metrics on a given manifold?"), and on the BökstedtMadsen map that relates to cobordism category to Waldhausen's Ktheory.
Financed by Hesselholt's Niels Bohr Professorship.


Gábor Szabó (PhD, University of Münster 2015): My research focuses on the fine structure of simple C*algebras and the classification of group actions on these objects. I am also interested in the structure of crossed products, in particular regarding the interplay between C*algebras and topological dynamics. 

Otgonbayar Uuye (PhD, Penn State 2008): I study Ktheory and KKtheory of operator algebras and index theoretic problems in NCG using cyclic cohomology.I did my PhD with Nigel Higson at Penn State on the local index theorem in noncommutative geometry; I studied index characters of Khomology classes over noncommutative spaces, in particular multiplicativity properties, meromorphic continuation of regularized traces, asymptotic expansions of heat kernels etc. 

Stefan Wagner (PhD, Darmstadt 2011): My research is mainly concerned with translating the geometric ideas and concepts of the theory of fibre bundles to Noncommutative Geometry. In particular, I am interested in the noncommutative geometry of principal bundles (and hence as well in noncommutative generalizations of coverings). I am also interested in finding "geometric" invariants for noncommutative spaces  currently I am working on a noncommutative version of the fundamental group for C*algebras. 

Guozhen Wang (PhD MIT 2015): My research interest is homotopy theory, especially those computational aspects. My current interest is computing stable and unstable homotopy groups of spheres, using techniques from chromatic theory and Goodwillie calculus.


Sinan Yalin (PhD Lille 1 2013): My research focuses on interactions between homotopy theory of algebraic structures and various topics in topology, geometry and mathematical physics. For this, I use methods coming from homotopy theory, homotopical algebra, higher category theory and derived geometry.
