Cette thèse est motivée par les problèmes de constructions de surfaces algébriques réelles. Nous nous intéressons plus particulièrement au problème de construire des surfaces algébriques réelles avec un grand nombre d'anses. Ce problème est relié à la conjecture de Viro, dont un contre exemple a été construit pour la première fois par I. Itenberg en 1993. L'outil fondamental de nos constructions est le patchwork de Viro, qui peut également s'interpréter par la géométrie tropicale. En utilisant la géométrie tropicale, et plus particulièrement les modifications tropicales, nous donnons une nouvelle construction d'une famille de courbes algébriques réelles planes avec un nombre asymptotiquement maximal d'ovals pairs. Cette famille avait été construite initialement en 2006 par E. Brugallé. En utilisant la méthode générale du patchwork, nous donnons ensuite une construction d'une sextique réelle avec 45 anses, améliorant ainsi un résultat de 2001 de F. Bihan. Enfin, nous nous penchons sur l'étude des surfaces algébriques réelles dans P1xP1xP1 et nous construisons notamment une famille de surfaces algébriques réelles de tridegré (2k,2l,2) dans P1xP1xP1 avec un premier nombre de Betti asymptotiquement maximal. Cette construction utilise une généralisation de la méthode du patchwork de Viro faite par E. Shustin en 1998. / In this thesis, we focus on constructions of real algebraic surfaces. The main problem we focus on is to construct real algebraic surfaces with a big number of handles. This problem is related to Viro's conjecture. A couterexample to Viro's conjecture was constructed at the first time by I. Itenberg in 1993. The fundamental tool to our constructions is Viro's patchworking. Viro's patchworking can be reformulated in terms of tropical geometry. Using tropical geometry, and more precisely tropical modifications, we give a new construction of a family of real algebraic plane curves with asymptotically a maximal number of even ovals. This family was first constructed in 2006 by E. Brugallé. Using Viro's patchworking, we construct a real sextic with 45 handles, improving a result of F. Bihan obtained in 2001. At least, we focus on the study of real algebraic surfaces in P1xP1xP1. More precisely, we construct a family of real algebraic surfaces of tridegree (2k,2l,2) in P1xP1xP1 with asymptotically a maximal first Betti number. This construction uses a more general version of Viro's patchworking due to E. Shustin in 1998.

The Haudenosaunee Confederacy is threaded together with sken:nen, the radical practice of peacemaking. As a Kanien’keha:ka woman, I am responsible for finding ways of bringing our peace-full teachings, gifts and intellect into the future. This thesis braids together a resurgent ethic of sken:nen with Haudenosaunee knowledges, Indigenous feminisms and decolonial futurities by taking up the question: Oh nisa’taro:ten? (What is the contour of your clay?), posed in Kanien’keha to situate me in relation to the lands I come from. I am taking this ancestral question seriously by exploring the relationships that make up the ‘clay’ of my contemporary Haudenosaunee Indigeneity as it is shaped by life in an active settler colonial state. Tracing the rhythmic gestures of my grandmothers’ hands, I have created a patchworking star quilt methodology to gather fragments of my decolonial curiosities, weaving them into layered story-maps that capture constellations of my movements through settler occupied places. Through the assimilative policies of the Indian Act, quilting simultaneously became an act of survivance and resistance for my grandmothers; by picking up an intergenerational practice of patchworking as methodology, I am jumping into the ruptures of my contemporary Haundenosaunee identity, roles and responsibilities. Patchworking story-maps involves tracing genealogies of intergenerational trauma, rupturing geographies of lateral violence, overflowing either/or binary cuts of identity (non)belonging, and navigating the urbanized displacements of Indigenous peoples from lands, communities and relationships. In an effort to mobilize the knowledges and practices of sken:nen, and to ensure that my work is accessible to a wider audience, my story-maps have been shared in a digital format using Instagram to stitch moments of Indigenous presence, memory and language (back) into the fabric of cityscapes that are riddled with the logics of settler colonialism. This thesis aims to create generative spaces to explore, transform and (re)imagine futurities of peacemaking that move towards more accountable and inclusive webs of relationality rooted in fluid traditions and (star)world building. / Graduate