# Geometry, Representations and Some Physics.

## The GRaSP seminar series at Purdue

##### Schedule
 Date Speaker Title/Abstract 01/25 Artur Jackson 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). 01/31 Nick Miller 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. 02/07 Artur Jackson Geometry of Orbifolds II: Metrics, Connections and Universality 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. 02/15 Tamás Darvas Chern classes from the point of view of differential geometry 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. TBA Andrés Figueroa Geometric Quotients 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 quotient. 3/22 Nick Miller 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. 3/29 Andrew Homan 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. 04/02 Tamás Darvas 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. 04/12 Byeongho Lee 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 structure. 4/19 Jason Lucas 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). 04/26 Mark Pengitore 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 finiteness.
##### Notes:
The GRaSP seminar aims to be a useful seminar series whose target audience is graduate students and intends to focus on topics (broadly) related to geometry, representation theory, and mathematical physics. This semester will largely be devoted to covering background material in both geometry and physics which is required to get the seminar off the ground before trecking into deeper topics.

Loose topics list:
• Complex geometry
• 3- and 4-manifolds
• Knot theory
• Dynamical systems
• Group theory
• Representation theory
• Geometric Langlands
• Conformal field theory (CFT)
• Geometric invariant theory (GIT)
• Resolution of singularities
• String theory
• Quantum field theory
• Geometric quantization
• Quantum groups
• Higher categories
• D-branes
• K-theory
• Yang-Mills
• Gauge theory
• Geometric Analysis
• General Relativity
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