||Geometry of Orbifolds
Orbifolds are topological spaces which are largely similar to
manifolds outside of some singular points, whose neighborhoods
can be modeled by quotients of Euclidean space by finite
isometry groups. Such spaces abound in mathematics and
physics, e.g., moduli spaces of curves and in string theory.
This talk will sketch the construction of these mildly singular spaces
(which includes carrying around a heap of extra data).
||A Brief Glimpse of Grothendieck Topologies
A Grothendieck topology is a topology that one can put on a
category C which generalizes and gives explicit axioms for the
notion of an open cover. In this talk I will attempt to
motivate the definition and use of Grothendieck topologies
past the standard "Grothendieck thought about it so you should
too", and show some of the consequences of using such a
categorical construction. In particular we will define the
Étale Topology and briefly discuss how this provides us with
a robust cohomology theory.
||Geometry of Orbifolds II: Metrics, Connections and
In this talk we'll take a pedestrian approach to
(defining and) constructing differentiable maps between orbifolds. This will
lead us to the notion of ``orbibundles'' and automatically
provide us with standard geometric gadgets such as metrics and
connections. We will also make an attempt at using these
orbibundles to produce objects solving universal problems,
e.g., in moduli theory.
||Chern classes from the point of view of differential
There are many approaches one can take in defining Chern
classes of vector bundles. In this talk we introduce these
invariants from the point of view of differential geometry and
we will discuss a limited amount of applications due to time
constraints. Students with an understanding of differential
geometry at the level of MA562 should be able to follow the
talk for the most part.
When working with algebraic group actions on varieties we would like to have a notion of quotient which makes sense as
an algebraic variety. In 1965 Mumford developed a method for constructing such an object for certain types of
"well-behaved" actions. I will present some basic definitions and make an introduction as to what to expect from such a
||The Cremona Group
The Cremona group is the group of birational automorphisms of projective n-space and is an interesting albeit immensely
mysterious object in algebraic geometry. In this talk we will review what a rational map between varieties as well as
some examples, a la blow-ups, etc. We will then go on to talk about the Cremona group and some of the known results for
it including, time permitting, the structure of it for n=2. This talk is intended to be fairly basic and although some
algebraic geometry will be discussed, a deep mastery of it is certainly not requisite.
||Geometry of Fourier Integral Operators
Fourier integral operators (FIOs) are a useful tool in analysis of PDEs with deep geometrical significance. Here we will
explain their relation to Lagrangian manifolds (and the Lagrangians of physics) and symplectic geometry. If time permits
we will also talk about thedifficulties involved in composing FIOs.
||The Chern Class of Line Bundles
In this talk we introduce the Chern class of a line bundle and discuss basic applications. Familiarity with basic
notions of complex geometry will be advantageous but good understanding of Riemannian geometry should be enough to
follow the talk for the most part.
||Delign-Mumford-Knudson Compactification of M_0,n
M_0,n (n>=3) is the moduli space of genus zero curve with n marked points over complex numbers. It can be thought of as
the configuration space of n points on a sphere up to automorphisms that respect marked points. We will describe
Deligne-Mumford-Knudsen compactification of this space, and explain about the topological and algebraic operad structure
that it exhibits. This leads to the notion of (formal) Frobenius manifold. Quantum cohomology is a manifestation of this
||Operads: Definitions and Examples
Officially defined by Peter May in the 1970s, operads have become important tools in many areas of mathematics,
including topology,algebra, and geometry. In this talk I will try to give an intuitive feeling for what an operad is
(although the rigorous definition will be presented), along with some hopefully interesting examples. Only basic
knowledge of topology and algebra will be needed, and no knowledge ofcategory theory will be required (however
familiarity with the definition of a symmetric monoidal category will be helpful).
||Residual Finiteness and Topology
In this talk I will introduce residual finiteness for groups and its interpretation on the profinite topology. I will
relate this concept with covering spaces from which we can demonstrate various examples of groups are residually finite,
such as surface groups and free groups. If time permits, I will discuss asymptotic group behavior and residual