# Mathematics Colloquium Spring 2017

For further information please contact Professor Junecue Suh or call the Mathematics Department at 459-2969

**Tuesday, April 4, 2017**

*No Colloquium*

**Tuesday, April 11, 2017**

**Xinwen Zhu, CalTech**

**Title: Period maps, Complex and p-adic**

Abstract: Hodge structures appear as abstractions of certain linear algebra structure on the cohomology of smooth projective algebraic varieties (more generally compact Kahler manifolds). The classifying spaces of Hodge structures are called period domains. While it is not known in general which Hodge structures come from cohomology of algebraic varieties, it is believed that those parameterized by hermitian symmetric domains (a special class of period domains) do arise in this way. Such a family is known to exist in many cases, but remains hypothetical in general, which hinders the development of arithmetic geometry related these domains.

In this talk, I will describe some recent progresses on p-adic aspects of these domains. The main tool is the p-adic non-abelian Hodge theory, which allows one to bypass the existence of the above mentioned hypothetical families.

Abstract: Hodge structures appear as abstractions of certain linear algebra structure on the cohomology of smooth projective algebraic varieties (more generally compact Kahler manifolds). The classifying spaces of Hodge structures are called period domains. While it is not known in general which Hodge structures come from cohomology of algebraic varieties, it is believed that those parameterized by hermitian symmetric domains (a special class of period domains) do arise in this way. Such a family is known to exist in many cases, but remains hypothetical in general, which hinders the development of arithmetic geometry related these domains.

In this talk, I will describe some recent progresses on p-adic aspects of these domains. The main tool is the p-adic non-abelian Hodge theory, which allows one to bypass the existence of the above mentioned hypothetical families.

**T**uesday, April 18, 2017

**Markus Linckelmann, City, University of London**

**Title: Derivations on Algebras with a View to Modular Representation Theory**

**Abstract: A derivation on an algebra A is a linear endomorphism of A which satisfies the familiar product rule. The set of derivations Der(A) on A is a vector space withsome additional structure; in particle, this space is a Lie algebra. Every element c in A gives rise to a derivation which sends any element a in A to the additive commutator ca-ac. The derivations arising in this way are called inner derivations. They form a Lie ideal in Der(A). The vector space quotient of Der(A) by this ideal is therefore again a Lie algebra. It turns out, that this Lie algebra can be interpreted as a fundamental cohomological invariant of A, namely the first Hochschild cohomology
of the algebra A.
**

**Motivated by some iconic conjectures which drive the modular representation theory of finite groups, we investigate connections between the algebra structure of A and the Lie algebra structure of its first Hochschild cohomology. What are the implications for A if its first Hochschild cohomology is a simple Lie algebra? In characteristiczero, simple Lie algebras are classified in terms of Dynkin diagrams - but in prime characteristic, the situation is far more complicated.
**

**The background motivation for this line of enquiry is the - at this point in time largely conjectural - insight that `very few' algebras arise as direct factors of finite group algebras. Formalising this insight is expected to lead to a structural interplay between finite groups and algebras. In a dream scenario, this interplay would lead to the understanding of some structural properties of finite groups, currently known only through the classification of finite simple groups. **

**Tuesday, April 25, 2017**

**Radha Kessar, City, University of London**

**Title: Local Structure and Representations of Finite Groups.**

**Abstract: Let G be a finite group and p a prime number. There are three inter-connected strands in representation theory:**

**-representations (of G) over fields of characteristic zero**

**-representations over fields of characteristic p**

**-embeddings and fusion (in G) of p-power order groups**

**I will give a guided tour of some of the main questions, conjectures and recent results in the subject using as focus the case where the relevant p-power order groups are abelian.**

**Tuesday, May 2, 2017**

**Vivek Shende, UC Berkeley**

**Title: A skeletal introduction to homological mirror symmetry **

**Abstract: I'll give an introduction to mirror symmetry emphasizing structural features of the categories of coherent sheaves and exact lagrangians. Then I'll sketch a proof at the large complex structure limit.**

**Tuesday, May 9, 2017**

**Kai-Wen Lan, University of Minnesota**

**Title: Geometric modular forms in higher dimensions**

**Abstract: Modular forms and their higher-dimensional analogues are a priori analytically defined objects which happen to have many interesting relations to other subjects (such as number theory). In this lecture, I will review how the algebraic geometry of modular curves was used for studying an important class of modular forms, and explain how the geometry of the so-called Shimura varieties can be used for an analogous theory in higher dimensions. If time permits, I will also give an overview of some interesting recent developments concerning the cohomology of Shimura varieties and other locally symmetric spaces. (There is no need to know what these objects are before you come to the talk.)**

**Tuesday, May 16, 2017**

***Special Colloquium***

**Kiran Kedlaya, UC San Diego**

**Title: Perfectoid spaces and homological conjectures
**

**Abstract: We sketch how recent work of Andre and Bhatt, together with the basic theory of perfectoid spaces, can be used to resolve some long-standing problems in commutative algebra. Among the more innocuous-looking of these is the direct summand conjecture: if R is a regular local ring, then any finite injective homomorphism from R to another ring splits in the category of R-modules.**

**Tuesday, May 23, 2017**

**Burt Totaro, UC Los Angeles**

**Title: Birational geometry and algebraic cycles**

**Abstract: A fundamental problem of algebraic geometry is to determine which algebraic varieties are rational, that is, isomorphic to projective space after removing lower-dimensional subvarieties**

from both sides. We discuss the history of the problem. Some dramatic recent progress uses a new tool in this context: the Chow group of algebraic cycles.

from both sides. We discuss the history of the problem. Some dramatic recent progress uses a new tool in this context: the Chow group of algebraic cycles.

**Tuesday, May 30, 2017**

**Ralph Abraham, UC Santa Cruz**

**Title: Math at UCSC -- The Early Days**

**Abstract: With some help from the surviving pioneer staff, I will undertake a retrospective analysis of the historical development of mathematics at UCSC through the lens of chaos theory, 1965 -- 1994. **

**Tuesday, June 6, 2017**

**Paul Yang, Princeton University**