Mathematics Colloquium Spring 2017

Tuesdays 4:00 p.m.
McHenry Building Room 4130
Refreshments served at 3:30 in the Tea Room (4161)
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.

Tuesday, 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.

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. 

Transcript of talk is available HERE

Tuesday, June 6, 2017

Paul Yang, Princeton University

I will discuss several basic geometric inequalities for CR geometry, including isoperitmetric inequalities. Motivated by concepts in conformal geometry, some results have natural approaches.