# Research Interests

My research interests center around the applications of algebraic topology to geometry. For instance, I have been interested in the connections between homotopy theory, symplectic geometry and group actions. In particular, lately I've been working on applications of Lusternik-Schnirelmann category to group actions and to geometry. I've also been working on more geometric things ranging from shapes of balloons and pneumatic domes to the rational homotopy of gauge groups. In recent years, I have worked on the topology and geometry of co-Kaehler manifolds and the subject of topological complexity. Topological complexity has links to LS category, but is an even harder invariant to calculate (or estimate). The invariant TC(X) is interesting because it gives a lower bound on how many regions configuration space must be split into in order to give motion planners for robots. The following papers give some idea of the kinds of things I'm talking about:

A. Angel, H. Colman, M. Grant and J. Oprea, Morita invariance of equivariant Lusternik-Schnirelmann category and invariant topological complexity, submitted.

G. Lupton, J. Oprea and N. Scoville, A fundamental group for digital images, submitted.

G. Lupton, J. Oprea and N. Scoville, Homotopy theory in digital topology, submitted.

G. Lupton, J. Oprea and N. Scoville, Subdivision of maps of digital images, submitted.

G. Bazzoni, G. Lupton and J. Oprea, Homotopy invariants and almost non-negative curvature, submitted.

M. Farber, J. Oprea, Higher topological complexity of aspherical spaces,Topology and its Applications 258 (2019) 142-160.

M. Farber, M. Grant, G. Lupton and J. Oprea, Bredon cohomology and robot motion planning, Algebraic and Geometric Topology 19 (2019) 2023-2059.

M. Farber, M. Grant, G. Lupton and J. Oprea, An upper bound for topological complexity, Topology and its Applications 255 (2019) 109-125.

J. Carrasquel, G. Lupton and J. Oprea, Topological complexity, robotics and social choice, Oberwolfach Snapshots of Modern Mathematics, No. 5 (2018).

G. Bazzoni, G. Lupton and J. Oprea, Parallel forms, co-Kaehler manifolds and their models, Bulletin of the Belgian Mathematical Society Volume 25, Number 1 (2018), 1-11.

G. Bazzoni, J.-C. Marrero and J. Oprea, A splitting theorem for compact Vaisman manifolds, Rendiconti Seminario Matematico Univ. Pol. Torino (Workshop for Sergio Console) Vol. 74, 1 (2016), 21 - 29.

E. Macias, J. Oprea, J. Strom and D. Tanre, Height functions on Quaternionic Stiefel manifolds, J. Ramanujan Math. Soc. 32 no. 1 (2017) 1-16.

G. Bazzoni, G. Lupton and J. Oprea, Hereditary properties of co-Kaehler manifolds, Differential Geometry and its Applications 50 (2017) 126-139.

I. Mladenov and J. Oprea, On the geometry of the rotating liquid drop, Mathematics and Computers in Simulation 127, (2016) 194-202. The Maple file is rotdrop3.mw. This is a revised version of the original file that now works with Maple 15 and with Maple 2015, but not Maple 16 or 17. Because of Maple's idiotic formatting in certain versions, choose Save Target As and change the file extension to .mw before downloading. This will give you a Maple file as opposed to an xml file.

Y. Felix, J. Oprea and D. Tanre, Lie-model for Thom spaces of tangent bundles, Proc. Amer. Math. Soc. 144 (2016), 1829-1840.

M. Grant, G. Lupton and J. Oprea, A mapping theorem for topological complexity, Algebraic & Geometric Topology 15 (2015) 1643-1666.

M. Grant, G. Lupton and J. Oprea, New lower bounds for the topological complexity of aspherical spaces, Topology and its Applications 189 (2015) 78-91.

J. Oprea, Aspects of Lusternik-Schnirelmann category, Geometry, Integrability and Quantization XVI, Proceedings of the Sixteenth International Conference on Geometry, Integrability and Quantization, Varna, Bulgaria June 6-11, 2014, (2015).

G. Bazzoni and J. Oprea, On the structure of coKahler manifolds , Geometriae Dedicata, 170, Issue 1, (2014) 71-85.

J. Oprea, Applications of Lusternik-Schnirelmann category and its generalizations, J. Geom. Symm. in Physics 36 (2014) 59-97.

M. Grant, G. Lupton and J. Oprea, Spaces of topological complexity one, Homology, Homotopy and Applications 15(2) (2013) 73-81.

J. Oprea and J. Strom, On Fox's m-dimensional category and theorems of Bochner type ,Topology and its Applications, 159, Issue 5, (2012) 1448-1461.

J. Oprea and J. Strom, Mixing categories, Proc. Amer. Math. Soc. 139, Number 9, (2011) 3383-3392.

J. Oprea and J. Strom, Lusternik-Schnirelmann category, complements of skeleta and a theorem of Dranishnikov, Algebraic & Geometric Topology 10 (2010) 1165-1186.

J. Oprea and J. Strom, On the realizability of Gottlieb groups, Contemporary Mathematics 519 (2010) 181-188.

I. Mladenov and J. Oprea, Balloons, domes and geometry, J. Geom. Symm. in Physics 15 (2009) 53-88.

Y. Felix and J. Oprea, Rational homotopy of gauge groups, Proc. Amer. Math. Soc. 137 (2009) 1519-1527.

J. Oprea, The Propagation of Non-Lefschetz Type, the Gottlieb Group and Related Questions, Journal of Fixed Point Theory and Applications 3 no. 1 (2008) 63-77.

Y. Felix, J. Oprea and D. Tanre, Algebraic Models in Geometry, Oxford Graduate Texts in Mathematics 17, Oxford University Press (2008).

M. Hadzhilazova, I. Mladenov and J. Oprea, Unduloids and their geometry, Archivum Mathimaticum (BRNO) Tomus 43 no. 5 (2007) 417-429.

I. Mladenov and J. Oprea, On some deformations of the Mylar balloon, Proceedings of the XV International Workshop on Geometry and Physics, Puerto De La Cruz, Tenerife, Canary Islands, Spain, Sept. 11-16, 2006, Real Sociedad Matematica Espanola, Volumen 11 (2007) 310-315.

I. Mladenov and J. Oprea, The Mylar balloon: New viewpoints and generalizations, Proc. Eighth International Conference on Geometry, Integrability and Quantization June 9-14, 2006, Varna, Bulgaria, SOFTEX (2007) 246-263.

J. Oprea and D. Tanre, Flat circle bundles, pullbacks and the circle made discrete , IJMMS 2005:21 (2005) 3487-3495. DOI: 10.1155/IJMMS.2005.3487. Click on the link for a reprint (up to 50).

C. Allday and J. Oprea, A c-symplectic free $$S^1$$-manifold with contractible orbits and $${\rm cat} = \frac{1}{2} {\rm dim}$$, Proc. Amer. Math. Soc. 134 no. 2 (2006) 599-604.

I. Mladenov and J. Oprea, The Mylar balloon revisited, The American Mathematical Monthly 110 no. 9 (2003) 761-784.

I. Mladenov and J. Oprea, Unduloids and their closed geodesics, Proc. Fourth International Conference on Geometry, Integrability and Quantization June 6-15, 2002, Varna, Bulgaria, Coral Press (2003) 206-234.

O. Cornea, G. Lupton, J. Oprea and D. Tanre Lusternik-Schnirelmann Category, American Mathematical Society Mathematical Surveys and Monographs vol. 103 (2003).

J. Oprea and Y. Rudyak, Detecting elements and Lusternik--Schnirelmann category of 3-manifolds,Contemp. Math. 316 (2003) 181-191.

J. Oprea, Bochner-type theorems for the Gottlieb group and injective toral actions, Contemp. Math. 316 (2003) 175-180.

J. Oprea, Category bounds for nonnegative Ricci curvature manifolds with infinite fundamental group, Proceedings of the American Math. Society 130 no.3 (2002) 833-839.

J. Oprea and J. Walsh, Quotient maps, group actions and Lusternik-Schnirelmann category, Topology and its Applications 117 no. 3 (2002) 285-305.

J. Oprea and A. Tralle (ed.), Homotopy and Geometry, Volume 45 in the Proceedings of the Stefan Banach Center, Warsaw, Poland 1998.

J. Oprea, Homotopy Theory and Circle Actions on Symplectic Manifolds, Workshop on Homotopy and Geometry, Volume 45 Proceedings of the Stefan Banach Center, Warsaw, 1998.

Y. Rudyak and J. Oprea, On the Lusternik-Schnirelmann category of symplectic manifolds and the Arnold conjecture, Math. Zeitschrift, 230 no. 4 (1999) 673-678.

A. Tralle and J. Oprea, Symplectic Manifolds with No Kähler Structure, volume 1661 in Springer Lecture Notes in Mathematics 1997.

J. Oprea and A. Tralle, Koszul-Sullivan models and the cohomology of certain solvmanifolds, The Annals of Global Analysis and Geometry, 15 (1997) 347-360.

R. Geoghegan, A. Nicas and J. Oprea, Higher Lefschetz traces and spherical Euler characteristics, Trans. AMS, 348 (1996) 2039-2062.

J. Oprea, Symplectic geometry and homotopy theory, Proceedings of the Daewoo Workshop in Pure Mathematics held at Chungju City, Korea 1995, Proceedings of Workshops in Pure Mathematics 15 Part III, Korean Academic Council (1996) 1-41.

G. Lupton and J. Oprea, Cohomologically symplectic spaces: toral actions and the Gottlieb group, Trans. AMS, 347 (1995) 261-288.

G. Lupton and J. Oprea, Symplectic manifolds and formality, J. Pure and Appl. Alg., 91 (1994) 193-207.

C. McCord and J. Oprea, Rational Ljusternik-Schnirelmann category and the Arnol'd conjecture for nilmanifolds, Topology 32 (1993) 701-717.

J. Oprea and J. Pak, Principal bundles over tori and maps which induce the identity on homotopy, Topology and its Applications 52 (1993) 11-22.

J. Oprea, The category of nilmanifolds, L'Enseignement Mathematique, vol. 38 no. 1-2 (1992) 27-40.

J. Oprea, Finite group actions on spheres and the Gottlieb group, J. Korean Math. Soc., vol. 28 no.1 (1991) 65-78.

J. Oprea, A homotopical Conner-Raymond theorem and a question of Gottlieb, Canadian Bulletin of Mathematics, Vol. 33(2) (1990) 219-229.

Along with Peter Bubenik, I organized a NSF-CBMS Workshop on Algebraic Topology in Applied Mathematics. The main speaker is Robert Ghrist of the University of Pennsylvania. The workshop was held at Cleveland State University August 3-7, 2009. For more information, see CBMS Workshop 2009.

Along with Yves Felix and Daniel Tanre, I lectured on Algebraic Models in Geometry at a special Summer School in Louvain-La-Neuve in June 2007. See Summer School on Algebraic Models.

Along with Octav Cornea and Daniel Tanre of Lille and Greg Lupton of Cleveland State, I organized an AMS-IMS-SIAM Summer Research Conference, Lusternik-Schnirelmann Category in the New Millennium, which was held at Mt. Holyoke in Summer 2001.

In June 1997, I co-organized a conference with Aleksy Tralle, Yves Felix, D. Lehmann and P. Zvengrowski at the Banach Center in Warsaw on Homotopy and Geometry. Of course, the real organizer was Tralle! This conference showed how homotopy theory and geometry can intertwine to reveal unexpected results and relationships. The proceedings of this conference will be published in 1998 as a volume in the Publications of the Banach Center.

My interests in homotopy theory itself are represented by the contents of Lecture Notes Volume 29, Gottlieb Groups, Group Actions, Fixed Points and Rational Homotopy available from The Global Analysis Research Center located at Seoul National University, Seoul, Korea.