Profile
Profile

Prof. Dr. Catharina Stroppel

E-mail: stroppel(at)math.uni-bonn.de
Phone: +49 228 73 6838
Fax: +49 228 73 7916
Homepage: http://www.math.uni-bonn.de/people/stroppel/
Room: 4.007
Office hours: Wednesdays, 12-14
Location: Mathematics Center
Institute: Mathematical Institute
Research Areas: Former Research Area F (Leader)
Research Area F* (Leader)
Research Area C
Former Research Area E
Mathscinet-Number: 700232

Academic Career

1991 - 1998

Diploma and school teacher degree Mathematics / Theology, Freiburg

1994 - 2000

Scientific Assistant, University of Freiburg

1998 - 2001

PhD in Mathematics (supervisor: Prof. W. Soergel), University of Freiburg

2000 - 2001

Teaching Assistant, University of Freiburg

2001 - 2003

Research Associate in Pure Mathematics, University of Leicester, England, UK

2003 - 2004

Associate Professor (CAALT Postdoc), University of Aarhus, Denmark

2004 - 2005

Research Associate, University of Glasgow, Scotland, UK

2005 - 2007

Lecturer, University of Glasgow, Scotland, UK

2007 - 2008

Reader, University of Glasgow, Scotland, UK

2007 - 2008

Von-Neumann Fellow, Institute of Advanced Study, Princeton, NJ, USA

2008 - 2010

Professor (W2), University of Bonn

Since 2010

Professor (W3), University of Bonn

Research Profile

I am interested in geometric and combinatorial aspects of representation theory in particular in connection with topology and category theory.

Research Projects and Activities

DFG Priority Program SPP 1388 “Representation theory”
Member

DFG Research Training Group GRK 1150 “Homotopy and Cohomology”
Member

DFG Collaborative Research Center SFB/TR 45 “Periods, moduli spaces and arithmetic of algebraic varieties”
Member

Contribution to Research Areas

Former Research Area E
One of my research interests is the interaction of geometry with representation theory and combinatorics. Schubert calculus is a classical representation theoretic tool to describe cohomology and representation rings in terms of symmetric functions. I am interested in modern aspects of this idea, of understanding integral structures, fusion algebras, canonical bases, and their connections to (topological) field theories. One of my recent contributions here is a combinatorial description of the affine {<br>rm sl}(n) Verlinde algebra with an explicit connection to quantum cohomology of Grassmannians [1]. In [2], we connect the cohomology theory of flag varieties and Springer fibers with the representation theory of Lie algebras and coherent sheaves related to nilpotent orbits in the Lie algebra {<br>rm gl}(n). Together with [3] this establishes a first concrete relationship between the geometric versions of Khovanov homology, the original algebraic-topological version and the Lie theoretic construction.
Former Research Area F
Representations of the symmetric group, more general Coxeter groups and their related Hecke algebras are an important topic in representation theory. They can be constructed geometrically in terms of convolution algebras of functions or sheaves on flag varieties or related varieties. Khovanov's categorification of the Jones polynomial and the important role played by Hecke algebras in knot theory leads to the question if these algebras, their representations and their representation categories can be categorified. This produces new interesting knot invariants, but also a more detailed description of the involved categories.
I categorified the complete Reshetikhin-Turaev quantum sl(2)-invariant for tangles and obtained a representation theoretic version of Khovanov homology [4], [5]. In this context interesting braid group actions on derived categories play an important role. I used categorification techniques to establish unknown equivalences of categories.
In this way several problems about decomposition numbers or Jordan-Hoelder multiplicity formulas could be solved or refined, as for instance for generalized Verma modules in [6].
Research Area C
Fusion rings and categorification questions are of interest for mathematicians and physicists. In particular allows categorification the interpretation of inverted quantum numbers and formal power series in q as as Euler characteristics of infinite complexes of graded vector spaces. We used this to categorify parts of the Reshethikin-Turaev-Viro invariants for 3-manifolds, [7], [8], [9]. Fusion rings arising from quantum groups at roots of unities were studied from an integrable systems point of view in [1], from an algebraic point of view in [10] and where used to study the famous Brauer centralizer algebras in [11] , [12]. One of the first successful categorifications was the famous Khovanov homology of links. It categorifies the Jones polynomial and lifts to an invariant of cobordisms of tangles up to signs. We addressed these sign issues in two papers describing a slightly twisted version of Khovanov homology which is functorial, see [13], [14].
Research Area F*
One of my research interests is the interaction of geometry with representation theory and combinatorics. I studied in particular categories of representations of Lie superalgebras [15], [16] with its connections to the geometry of perverse sheaves on Grassmannians [17], Springer fibers [18] and its connections to algebras arising in classical invariant theory [19].

Selected Publications

[1] Christian Korff, Catharina Stroppel
The {\widehat{\germsl}(n)_k}-WZNW fusion ring: a combinatorial construction and a realisation as quotient of quantum cohomology
Adv. Math. , 225: (1): 200--268
2010
DOI: 10.1016/j.aim.2010.02.021
[2] Catharina Stroppel, Ben Webster
2-block Springer fibers: convolution algebras and coherent sheaves
Comment. Math. Helv. , 87: (2): 477--520
2012
DOI: 10.4171/CMH/261
[3] Catharina Stroppel
Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology
Compos. Math. , 145: (4): 954--992
2009
DOI: 10.1112/S0010437X09004035
[4] Catharina Stroppel
Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors
Duke Math. J. , 126: (3): 547--596
2005
DOI: 10.1215/S0012-7094-04-12634-X
[5] Igor Frenkel, Mikhail Khovanov, Catharina Stroppel
A categorification of finite-dimensional irreducible representations of quantum {\germsl_2} and their tensor products
Selecta Math. (N.S.) , 12: (3-4): 379--431
2006
DOI: 10.1007/s00029-007-0031-y
[6] Volodymyr Mazorchuk, Catharina Stroppel
Categorification of (induced) cell modules and the rough structure of generalised Verma modules
Adv. Math. , 219: (4): 1363--1426
2008
DOI: 10.1016/j.aim.2008.06.019
[7] Igor Frenkel, Catharina Stroppel, Joshua Sussan
Categorifying fractional Euler characteristics, Jones-Wenzl projectors and 3j-symbols
Quantum Topol. , 3: (2): 181--253
2012
DOI: 10.4171/QT/28
[8] Pramod N. Achar, Catharina Stroppel
Completions of Grothendieck groups
Bull. Lond. Math. Soc. , 45: (1): 200--212
2013
DOI: 10.1112/blms/bds079
[9] Catharina Stroppel, Joshua Sussan
Categorified Jones-Wenzl projectors: a comparison
Perspectives in representation theory
of Contemp. Math. : 333--351
Publisher: Amer. Math. Soc., Providence, RI
2014
DOI: 10.1090/conm/610/12194
[10] Henning Haahr Andersen, Catharina Stroppel
Fusion rings for quantum groups
Algebr. Represent. Theory , 17: (6): 1869--1888
2014
[11] H. Haahr Andersen, C. Stroppel, D. Tubbenhauer
Cellular structures using \textbfU\_q-tilting modules
to appear in Pacific Journal of Math
2015
[12] H. Haahr Andersen, C. Stroppel, D. Tubbenhauer
Semisimplicity of Hecke and (walled) Brauer algebras
to appear in the Journal of the Australian Mathematical Society
2015
[13] M. Ehrig, C. Stroppel, D. Tubbenhauer
The Blanchet-Khovanov algebras
to appear in Perpectives in Categorification in Algebra, Geometry and Physics
2015
[14] M. Ehrig, C. Stroppel, D. Tubbenhauer
Generic \mathfrakgl\_2-foams, web and arc algebras
ArXiv e-prints
2016
[15] Antonio Sartori, Catharina Stroppel
Categorification of tensor product representations of {\germsl_k} and category {\CalO}
J. Algebra
,
428: : 256--291
2015
DOI: 10.1016/j.jalgebra.2014.12.043
[16] M. Ehrig, C. Stroppel
On the category of finite-dimensional representations of \operatornameOSp(r|2n): Part I
Representation Theory-Current Trends and Perspectives
of EMS Series of Congress Reports : 109--170
2017
[17] Michael Ehrig, Catharina Stroppel
Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians
Selecta Math. (N.S.) , 22: (3): 1455--1536
2016
DOI: 10.1007/s00029-015-0215-9
[18] Michael Ehrig, Catharina Stroppel
2-row Springer fibres and Khovanov diagram algebras for type D
Canad. J. Math. , 68: (6): 1285--1333
2016
DOI: 10.4153/CJM-2015-051-4
[19] Michael Ehrig, Catharina Stroppel
Koszul gradings on Brauer algebras
Int. Math. Res. Not. IMRN (13): 3970--4011
2016
DOI: 10.1093/imrn/rnv267
[20] Jonathan Brundan, Catharina Stroppel
Highest weight categories arising from Khovanov's diagram algebra IV: the general linear supergroup
J. Eur. Math. Soc. (JEMS)
, 14: (2): 373--419
2012
DOI: 10.4171/JEMS/306
[21] Volodymyr Mazorchuk, Catharina Stroppel
Projective-injective modules, Serre functors and symmetric algebras
J. Reine Angew. Math. , 616: : 131--165
2008
DOI: 10.1515/CRELLE.2008.020

Publication List

Awards

1998

Ferdinand-von-Lindeman Prize for the best diploma thesis at the faculty, University of Freiburg

2007

Whitehead Prize, London Mathematical Society

2007

Von-Neumann Award, Institute of Advanced Study

Selected Invited Lectures

2009

Lecture series on representation theory and combinatorics, Beijing, China

2009

Lecture series on structures on Lie representation theory, Bremen

2009

Summer school on link homology, Paris, France

2010

Oporto Meeting on Geometry, Topology and Physics, Faro, Portugal

2010

Lectures on categorification, Aarhus, Denmark

2010

International Congress of Mathematicians, invited speaker, Hyderabad, India

2011

Lecture series on Lie superalgebras, Cargese, France

2012

Lecture series on Springer fibers, Northeastern University, Boston, MA, USA

2012

Lecture series on categorification, Luminy, France

2013

Lecture series on categorified invariants of manifolds, MPI, Bonn

2013

Summer school on Category O, Freiburg

2014

Lecture series on Khovanov algebras (Program Math. Structures and Computations), Lyon, France

2014

Lecture series on categorification (Program on Algebraic Lie Theory), Glasgow, Scotland, UK

2016

Lecture series on representation theory of Lie superalgebras and categorification, Workshop, Bonn

Offers

2007

University of Wisconsin-Madison, WI, USA

2009

University of Vienna, Austria

2010

University of Chicago, IL, USA

2013

University of Glasgow, Scotland, UK

Selected PhD students

Hoel Queffelec (2013): “Sur la catégorification des invariants quantiques sln : étude algébrique et diagrammatique”,
now Chargé de recherche CNRS, Institut Montpelliérain Alexander Grothendieck, University of Montpellier, France

Antonio Sartori (2014): “Categorification of tensor powers of the vector representation of Uq(gl(1|1))”,
now Research Assistant, University of Freiburg

Hanno Becker (2015): “Homotopy-Theoretic Studies of Khovanov-Rozansky Homology”

Joanna Meinel (2016): “Affine nilTemperley-Lieb Algebras and Generalized Weyl Algebras”,
now Postdoc, University of Stuttgart

Supervised Theses

  • Master theses: 7, currently 5
  • Diplom theses: 8
  • PhD theses: 6, currently 4
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