Fachbereich Mathematik und Informatik FB 10

Private Homepage	https://yifeizhao.com
Topics in Mathematics Münster	T1: K-Groups and cohomology T2: Moduli spaces in arithmetic and geometry T7: Field theory and randomness
Current Projects	• CRC 1442 - A05: Moduli spaces of local shtukas in mixed characteristic We study the geometry and cohomology of moduli spaces of local G-shtukas, a class of moduli spaces that plays a central role in the geometrisation of Langlands correspondences. More precisely, we are interested in the geometry of the image of the period maps, want to investigate étale sheaves on the moduli spaces and aim at the local Langlands correspondence for covering groups via a metaplectic geometric Satake equivalence. online • CRC 1442 - D03: Integrability We investigate blobbed topological recursion for the general Kontsevich matrix model, as well as the behaviour of Baker–Akhiezer spinor kernels for deformations of the spectral curve and for the quartic Kontsevich model. We study relations between spin structures and square roots of Strebel differentials, respectively between topological recursion and free probability. We examine factorisation super-line bundles on infinite-dimensional Grassmannians and motivic characteristic classes for intersection cohomology sheaves of Schubert varieties. online • EXC 2044 - A1: Arithmetic, geometry and representations The Langlands programme relates representations of (the adele valued points of) reductive groups G over Q - so-called automorphic representations - with certain representations of the absolute Galois group of Q. This programme includes the study of these objects over general global fields (finite extension of Q or Fp (t)) and local fields as well. In its local form the classical programme onlyconsidered l-adic Galois representations of p-adic fields for unequal primes l neq p. In order to allow for a p-adic variation of the objects, it is absolutely crucial to extend it to the case l = p. In the global situation, the automorphic representations in question can often be realised in (or studied via) the cohomology of a tower of Shimura varieties (or related moduli spaces) attached to the group G. We will focus on the following directions within this programme: The p-adic and mod p Langlands programme asks for an extension of such a correspondence involving certain continuous representations with p-adic respectively mod p coefficients. Broadening the perspective to p-adic automorphic forms should, for example, enable us to capture all Galois representations, not just those having a particular Hodge theoretic behaviour at primes dividing p. This extended programme requires the introduction of derived categories. We will study differential graded Hecke algebras and their derived categories on the reductive group side. On the Galois side, we hope to use derived versions of the moduli spaces of p-adic Galois representations introduced by Emerton and Gee. The geometric Langlands programme is a categorification of the Langlands programme. We plan to unify the different approaches using motivic methods. In another direction, we study the geometry and arithmetic of moduli stacks of global G-shtukas over function fields. Their cohomology has been the crucial tool to establish large parts of the local and global Langlands programme over function fields. Variants of G-shtukas are also used to construct and investigate families of p-adic Galois representations. Cohomology theories are a universal tool pervading large parts of algebraic and arithmetic geometry. We will develop and study cohomology theories, especially in mixed characteristic, that generalise and unify étale cohomology, crystalline cohomology and de Rham cohomology as well as Hochschild cohomology in the non-commutative setting. Developing (topological) cyclic homology in new contexts is an important aim. A main goal is to construct a cohomology theory that can serve the same purposes for arithmetic schemes as the l-adic or crystalline cohomology with their Frobenius actions for varieties over finite fields. Ideas from algebraic geometry, algebraic topology, operator algebras and analysis blend in these investigations. online
E-Mail	yifei dot zhao at uni-muenster dot de
Phone	+49 251 83-35172
Room	100.008
Secretary	Sekretariat Dierkes Frau Gabi Dierkes Telefon +49 251 83-33730 Zimmer 414
Address	Dr. Yifei Zhao Mathematisches Institut Fachbereich Mathematik und Informatik der Universität Münster Einsteinstrasse 62 48149 Münster Deutschland
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