Algebraic geometry is an old subject, going back to the ancient Greeks who studied the geometry of ellipses, parabolas and hyperbolas using the language of conic sections. In the 16th century, Descartes rephrased everything in terms of coordinates. The conic sections of the Greeks became solutions to quadratic polynomial equations. Finally, in the 1960s the current framework of algebraic geometry was introduced by Grothendieck, with the advent of scheme theory. An important question in the context of conic sections is their classification: how many different types are there, and how can one be related, or "deformed", into another? It is possible to consider this problem in the three settings introduced above, giving equivalent answers. But the high level of abstraction in the last setting allows one to really explain what is specific to the situation of conic sections, and what is true more generally. My research proposal concerns these classification and deformation problems in (non-commutative) algebraic geometry: Hochschild cohomology describes previously unknown ways of deforming objects in algebraic geometry, making them non-commutative. My goal is to study and obtain exciting and unexpected connections: Can we understand symmetries of deformations? Can we translate noncommutativity back to commutativity? How can we relate geometric objects in new ways? How are deformations similar or different?