**TIME:**Tuesday 4:00 - 5:30 pm.**ROOM:**628.**ORGANIZERS:**Taylor Brysiewicz, Timo de Wolff, Kevin Kordek, and Ola Sobieska.**Email:**tbrysiewicz(at)math.tamu.edu, dewolff(at)math.tamu.edu, kordek(at)math.tamu.edu, sobieska(at)math.tamu.edu

**GOALS:**- The goal of this seminar is to provide graduate students with an introduction to two topics: Category Theory and De Rham Cohomology.
- This seminar is designed to be relaxed and to move at a moderate pace.

**STRUCTURE:**- The first 6 weeks will be dedicated to learning about Category Theory while the last six weeks will be dedicated to De Rham Cohomology.
- Each week, a student volunteer will prepare a 50 minute lecture covering the suggested reading for the previous week.
- The remaining 40 minutes will be left open for discussion. This could include questions about the material, exercises, and/or proofs.
- The last meeting for each subject will be facilitated by a faculty member and will focus on a more advanced aspect of the topic and/or modern research in that area.

**REFERENCES:**- For the Category Theory portion, we plan to use "Categories for the Working Mathematician" by Mac Lane.
- For the De Rham Cohomology part of the seminar, we will use Lee's "Introduction to Smooth Manifolds" and "Differential Forms in Algebraic Topology" by Bott and Tu.

**CATEGORY THEORY SCHEDULE**Date Facilitator Reading Assignment Topic 8/30/16 N/A N/A Organizational meeting 9/6/16 Pablo Sanchez Ocal CWM I. 1-4 Definition of category, functor, natural transformations, etc 9/13/16 Konrad Wrobel CWM I. 5-8 Monics, epis, zeros, foundations, large categories, hom-sets, etc 9/20/16 Alex Ruys de Perez CWM II. 1-4 Duality, contravariance/opposites, products of categories, functor categories, etc 9/27/16 Adam Deaton CWM II. 5-8 The category of all categories, comma categories, graphs/free categories, quotient categories, etc 10/4/16 Christian Williams CWM III. 1-4 Universal arrow, Yoneda lemma, coproducts/colimits, products and limits, etc 10/11/16 Dr. Eric Rowell N/A Rigid, balanced, monoidal categories

**DE RHAM COHOMOLOGY SCHEDULE**Date Facilitator Reading Assignment Topic 10/18/16 Nida Obatake TBA Differential Forms and Manifolds. Integration. Stoke's Theorem. 10/25/16 Dustin McPhate TBA The De Rham Complex and De Rham Cohomology. Examples and formal properties. 11/1/16 Fulvio Gesmundo BT I.2, I.4, I.5. L 17 (Theorem 17.20 on), 18. Examples and Tools. Mayer Vietoris, Kunneth Theorem, De Rham Theorem, etc. 11/8/16 David Buzinski BT I.5 Poincare Duality for Compact Orientable Manifolds. Examples. 11/15/16 Kash Bari BT I.6 More Poincare Duality (more geometric viewpoint) 11/22/16 Dr. Paulo Lima-Filho N/A Characteristic Classes for Real and Complex Vector Bundles