Bathymetry smoothing The depth of many seas, particularly in the Adriatic Sea, has the inconvenience of varying too much for realistic oceanic modelling. The problem showed up in the computation of pressure difference in terrain following models. When the depths vary too much, artificial currents occur which compromise the stability of the model.

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M. Dutour Sikirić, I. Janeković, M. Kuzmić, A new approach to bathymetry smoothing in sigmacoordinate ocean models, Ocean Modelling 29 (2009) 128
136. 
Space fullerenes A space fullerene is a tiling of Euclidean space by fullerenes. Such tilings occur in Metallurgy, Crystallography, Soap froth, Optimization problems, etc. I enumerated small space fullerenes

LM. Dutour Sikirić, O. DelgadoFriedrichs, M. Deza, Space fullerenes: computer search for new FrankKasper structures, Acta crystallographica A 66 (2010) 602615. LM. Dutour Sikirić, M. Deza, Space fullerenes: computer search for new FrankKasper structures II, Structural Chemistry 234 (2012) 11031114. 
Coupling of circulation and wave models I coupled the circulation model ROMS with the wave model WWM. Below the effect of coupling:

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M. Dutour Sikirić, A. Roland, I. Tomažić, I. Janeković, Hindcasting the Adriatic Sea nearsurface motions with a coupled wavecurrent model, Journal of Geophysical Research  Oceans, 117 (2012) C00J36. LM. Dutour Sikirić, A. Roland, I. Janeković, I. Tomažić, M. Kuzmić, Coupling of the Regional Ocean Modelling System and Wind Wave Model, Ocean Modelling 72 (2013) 5973. 
Geometry of Numbers A lattice L is a discrete subgroup set of the Euclidean space In a lattice L an empty sphere is a sphere containing no lattice points in its interior and containing n+1 affinely independent points on its surface. A Delaunay polytope is the convex hull of the points on the surface of an empty sphere.

L M. Dutour Sikirić, A. Schuermann, F. Vallentin, Complexity and algorithms for computing Voronoi cells of lattices, Mathematics of computation 78 (2009), 17131731. 
Combinatorial types The 3dimensional lattice Delaunay polytopes have been classified by E. Fedorov in 1885 (see below). I enumerated them in dimension 6.

L M. Dutour, The sixdimensional Delaunay polytopes, European Journal of Combinatorics 25 (2004) 535548. 
Perfect Delaunay polytopes If the only deformation preserving a lattice Delaunay polytope are the isometries and homotheties, then it is called perfect. Such polytopes correspond to the extreme rays of a polyhedral cone, named the Erdahl cone. I created an infinite series of extreme Delaunay polytope, which generalize the classical Schafli polytope. Furthermore I introduced classical techniques of combinatorial optimization in this subject, which allowed us to find 2, 27 extreme Delaunay polytopes in dimension 7 and 8, respectively. 
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M. Deza, M. Dutour, The hypermetric cone on seven vertices, Experimental Mathematics 12 (2004) 433440. L M. Dutour, Infinite Serie of Extreme Delaunay polytope, European Journal of Combinatorics 26 (2005) 129132. L M. Dutour, Adjacency method for extreme Delaunay polytopes, Voronoi's Impact on Modern Science, Book 3, 94101. L M. Dutour Sikirić, K. Rybnikov, Delaunay polytopes derived from the Leech lattice, preprint. 
Ltype domains and the lattice covering problem A family of balls of equal radius such that every point belongs to at least one ball is called a covering; its density is the average number of balls to which points belongs to. See below a 2dimensional covering: 
L A. Schuermann, M. Dutour Sikirić, F. Vallentin, A generalization of Voronoi's reduction theory and its application, Duke Math. J. 142 (2008), 127164. 
Polyhedral computation with symmetry The programs lrs and cdd allow to compute the dual description of a polytope. Unfortunately, in many cases, we cannot apply them for memory and time reasons. Starting from the Adjacency Decomposition method of Christof & Reinelt I progressively build an extremely good algorithm for computing dual description of polytopes with symmetry.

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D. Bremner, M. Dutour Sikirić, A. Schuermann, Polyhedral representation conversion up to symmetries, CRM proceedings, volume 48. L M. Dutour Sikirić, A. Schuermann, F. Vallentin, Classification of eight dimensional perfect forms, Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 2132. L M. Dutour Sikirić, A. Schuermann, F. Vallentin, The contact polytope of the Leech lattice, submitted. 
Elementary polycycles and Face regular maps A (R,q)polycycle is a plane graph, whose faces, besides some disjoint holes, are igons for i in R, and whose vertices, outside of holes, are qvalent. Such polycycle is called elliptic, parabolic or hyperbolic if 1/q + 1/r  1/2 (where r is the maximum of R) is positive, zero or negative, respectively. Such polycycles can be uniquely decomposed into some simpler elementary polycycles. We classify the elementary elliptic (R,q)polycycles, i.e. elementary ({2,3,4,5},3), ({2,3},4) and ({2,3},5)polycycles. This gives a very efficient proof and enumeration technique, which we applied to the enumeration of face regular maps. A 3valent torus or spherical map with p and qgonal faces is called pRi (respectively qRj) if every pgonal (respectively qgonal) face is adjacent to exactly i pgonal (respectively j qgonal) faces. We considered the question of existence, finiteness and classification for those classes of graphs.

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M. Deza, M. Dutour, M. Shtogrin, Elementary elliptic (R,q)polycycles, Analysis of Complex Networks, From Biology to Linguistics, edited by M. Dehmer, F. EmmertStreib, WileyBlackwell, Weinheim 2009, 351376. L M. Deza, M. Dutour Sikirić, Geometry of Chemical Graphs, Cambridge University Press, Series: Encyclopedia of Mathematics and its Applications (No. 119) L M. Deza, M. Dutour, M. Shtogrin, Elliptic polycycles with holes, Russian Math. Surveys. 60 (2005) 349351 
Zigzags and central circuits We have considered the structure of central circuit (circuit of edges, such that no two consecutive edges belong to the same face) for 4valent plane graph and zigzags (circuit of edges, such that any two, but no three, consecutive edges belong to the same face) for 3valent plane graphs. See below two parts of such objects:
The GoldbergCoxeter construction, which takes a 3 or 4valent plane graph and a pair (k,l) of integers for creating a new 3 or 4valent plane maps was considered in this context: We computed the zigzag or central circuit structure of the Goldberg construction by creating a new formalism of (k,l)product and the introduction of a finite index subgroup of SL2(Z). 
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M. Dutour, M. Deza, GoldbergCoxeter construction for 3 or 4valent plane graphs, Electronic Journal of Combinatorics 11 (2004) R20. L M. Deza, M. Dutour, Zigzag Structure of Simple Twofaced Polyhedra, Combin. Probab. Comput. 14 (2005), 3157 L M. Deza, M. Dutour, P. Fowler, Zigzags, Railroads, and Knots in Fullerenes, Journal of Chemical Information and Computer Sciences 44 (2004) 12821293. L M. Deza, M. Dutour, M. Shtogrin, 4valent plane graphs with 2, 3 and 4gonal faces, "Advances in Algebra and Related Topics" (in memory of B.H. Neumann; Proceedings of ICM Satellite Conference on Algebra and Combinatorics, Hong Kong 2002), World Scientific Publ. Co. (2003) 7397. 
Simplicial complexes All simplicial ncomplexes, whose (n2)dimensional faces are contained in 3 or 4 simplices have been classified in terms of partitions of {1,..., n+1}. See below the 2dimensional case:

L M. Deza, M. Dutour Sikirić, M. Shtogrin, On simplicial and cubical complexes with short links, Israel J. Math. 144 (2004), 109124. 
Wythoff polytopes The Wythoff construction takes a nonnecessarily convex ndimensional polytope P, a nonempty subset S of {0,...,n} and returns another ndimensional polytope W(P,S). Particular cases are duality, medial polytope, clique clomplex, ... We determined a conjecturally complete list of polytopes W(P,S) that are L1embeddable for P a regular polytope. The Wythoff construction was also used to compute the third homology group of the Mathieu group M24. 
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M. Deza, M. Dutour Sikirić, S. Shpectorov, Hypercube Embedding of Wythoffians, Ars Math. Contemp. 1 (2008), 99111 L M. Dutour, G. Ellis, Wythoff polytopes and lowdimensional homology of Mathieu groups, preprint. 
Cube packings and tilings A cube tiling (respectively packing) is a 4periodic tiling (respectively packing) of R^n by translates of cubes [1,1]^n. Much of this research came from the existence in dimension 3 of a nonextensible cube packing with 4 translation classes of cubes: 
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M. Dutour Sikirić and Y. Itoh, Continuous random cube packings in cube and torus, European Journal of combinatorics, to appear. L M. Dutour Sikirić, Y. Itoh, A. Poyarkov, Cube packings, second moment and holes, European J. Combin. 28 (2007), 715725. 
Ginzburg Landau model My PhD thesis was on the bulk GinzburgLandau model of superconductivity. This is a partial differential equation model depending on a structural parameter k and the exterior field Hext. The internal states of the model are described by an interior magnetic field Hint and a complex valued wave function phi. By using the Bochner Kodaira Nakano formula of complex hermitian geometry, I established the classical form of the phase diagram with the pure state region (phi=1 all electron supraconductor), the normal region (phi=0 no electron supraconductor) and the mixed region where both kinds of electron coexist: 
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M. Dutour, Phase diagram for Abrikosov lattice, Journal of Mathematical Physics 42 (2001) 49154926. 