Veige joined the faculty of the University of Florida as an assistant professor of chemistry (inorganic chemistry) in 2004. In 2010, Veige received the Alfred P. Sloan fellowship award, the only researcher to be so honored in Florida in 2010. He was promoted to an associate professor in 2011. He is currently the director of the Center for Catalysis in the Department of Chemistry at the University of Florida.
His research focuses on the design, synthesis, isolation, and characterization of novel inorganic molecules for application in the production of fertilizers, polymers, and pharmaceuticals. His research has included the preparation of chiral catalysts, synthesis of nitriles via N-atom transfer to acid chlorides, chromium catalyzed aerobic oxidation, an alkene isomerization catalyst, a highly active alkene polymerization catalyst, and a highly active alkyne polymerization catalyst.