The purpose of this article is to define and study the specialization map in the context of Scholze's category of diamonds and to prove some basic results on its behavior. Our specialization map generalizes the classical specialization map that appears in the theory of formal schemes. Afterwards, as an example of interest, we study the specialization map for p-adic Beilinson-Drinfeld Grassmanians and moduli spaces of mixed-characteristic shtukas associated to reductive groups over Z p . Finally, as an application, we describe the geometric connected components of moduli spaces of mixed-characteristic shtukas at hyperspecial level in terms of the connected components of affine Deligne Lusztig varieties. This generalizes a similar theorem of Chen, Kisin, and Viehmann that describes the connected components of unramified Rapoport-Zink spaces in terms of connected components of affine Deligne Lusztig varieties. Hansen for many very helpful conversation that cleared misunderstandings of the author, for providing a reference and a proof to proposition 1.4.2, for kindly reading an early draft of this article and for pointing out a serious mistake of an early draft. To Eugen Hellmann for a very helpful conversation that lead to a big simplification in the proof of theorem 2.3.14. To João Lourenço for some correspondances that encouraged the author to pursue theorem 4 in this generality. To the author's PhD advisor Sug Woo Shin for giving him the freedom to pursue his interests, for masterfully guiding his effort in the right direction, for providing very helpful feedback at every stage of this project and for his generous constant support.The author would also like to thank Johanes Anschűtz, Alexander Bertoloni, Dong Gyu Lim, Zixin Jiang and Alex Youcis for helpful conversations and correspondances.

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Given two convex polytopes, the join, the cartesian product and the direct sum of them are well understood. In this paper we extend these three kinds of products to abstract polytopes and introduce a new product, called the topological product, which also arises in a natural way. We show that these products have unique prime factorization theorems. We use this to compute the automorphism group of a product in terms of the automorphism groups of the factors and show that (non trivial) products are almost never regular or two-orbit polytopes. We finish the paper by studying the monodromy group of a product, show that such a group is always an extension of a symmetric group, and give some examples in which this extension splits. arXiv:1603.03585v1 [math.CO]• If P is the prism (or the bipyramid) over a 3-polytope having the property that all its vertex figures (faces) are isomorphic to an n-gon, with n not congruent to 0 modulo 9;• If = and each Q i has rank 2.By [2], we already know that the monodromy group of a pyramid over an n-gon is an extension of S 4 by (C m ) 4 , where m = p gcd (3,p) , and that such extension splits whenever n is not congruent to 0 modulo 9. Our techniques to show Theorem B can also be used to show this result. Abstract polytopesAbstract polytopes generalise the (face-lattice) of classical polytopes as combinatorial structures. In this chapter we review the basic theory of abstract polytopes and refer the reader to [11] for a detail exposition of the subject.An (abstract) n-polytope (or (abstract) polytope of rank n) P is a partially ordered set whose elements are called faces that satisfies the following properties. It contains a minimum face F −1 and a maximum

We compute the connected components of arbitrary parahoric level affine Deligne-Lusztig varieties for quasisplit reductive groups, by relating them to the connected components of infinite level moduli spaces of p-adic shtukas, where we use v-sheaf-theoretic techniques such as the specialization map of kimberlites.As an application, we deduce new CM lifting results on integral models of Shimura varieties at arbitrary parahoric levels. We also prove various results beyond the quasi-split case, by studying the cohomological dimensions of Newton strata inside the B + dR -affine Grassmannian. Contents 1. Introduction 1 Acknowledgements 10 2. Preliminaries and background 10 3. Generic Mumford-Tate groups 28 4. Proof of main theorems 31 5. Beyond the quasisplit case. 40 References 45

Antiprisms of polygons are classical convex vertex-transitive polyhedra. In this paper, for any given (abstract) polytope, we define its antiprism. We then find the automorphism group of the antiprism of P in terms of the extended group of P (the groups of automorphisms and dualities) as well as some transitivity properties. We also give a relation between some products of abstract polytopes and their antiprisms.