Useful references for my students and anyone else.
Topics  References (with free links where possible even for published papers)  
ConnesKreimer Hopf algebra Renormalization Hopf algebras Renormalization in this language 
Notes from some Hopf algebra lectures Notes from Erik Panzer's lectures (1, 2, 3, 4 with extra notes) in Saskatoon Notes from MATH 821 (see lectures 9 through 14) 

Multiple zeta values Transcendentality conjectures Multiple polylogarithms 
Hoffman Algebraic aspects of MZVs Panzer notes from a KOSMOS summer university Bowman and Bradley, Multiple polylogarithms: a brief survey 

Denominator reduction Parametric Feynman integration using polylogarithms 
Brown On the periods of some Feynman integrals  
Feynman integrals in momentum space position space parametric space and how to move between them 

Combinatorial DysonScwhwinger equations Combinatorial specifications Foissy's results 
My thesis Flajolet and Sedgewick Analytic Combinatorics Foissy General DysonSchwinger equations and systems Foissy Faà di Bruno subalgebras of the Hopf algebra of planar trees from comb. DSEs 

Physics setup and language Fields, Lagrangians, path integral 
Zee QFT in a nutshell Cvitanovic Field Theory  
Important QFTs and their Feynman rules  
Graph theory highlights Robertson Seymour theorem edge, vertex, internally connectivity 
What is your favorite graph theory reference?  
Linear algebra highlights Graph Laplacian, exploded Laplacian Matrix tree theorem Schur complementation Plücker and Dodgson identities 
Start with Wikipedia and go from there By exploded Laplacian I mean the matrix M you'd find in my or Francis' work. 

Hot news! When I wrote this list: gravitational waves The previous semester: quasi polynomiality of graph isomorphism 
You should keep up on what is exciting! LIGO detection of gravitational waves Community overview of Babai's result 