# From WSU mathematics graduate to university professor

Professor Tony Crachiola and mathematics alum (B.A. in mathematics and French, 1998; M.A. in mathematics in 1999; Ph.D. in mathematics, 2004, all from WSU), has been a professor at Saginaw Valley State University since 2005 and spent six years as chair of SVSU's Department of Mathematical Sciences. He became associate dean of SVSU's College of Science, Engineering and Technology in 2019.

Professor Crachiola's research is in affine algebraic geometry, specifically in the calculation and application of the AK invariant of rings. The AK invariant of a ring R is the intersection of the kernels of the locally nilpotent derivations on R.

That is, the AK invariant of a k-algebra R is the set of all elements x of R such that, if d: R -> R is a k-linear homomorphism satisfying the Leibniz rule such that, for each r in R, some iterate of d sends r to zero, then d(x) = 0. The AK invariant was first introduced by WSU's Professor Makar-Limanov (who was Tony Crachiola's doctoral thesis advisor), in order to make progress on the generalized Zariski cancellation conjecture, i.e., the conjecture that, if A,B are k-algebras which are integral domains and A and B are isomorphic as k-algebras after tensoring each with a polynomial k-algebra on n generators, then A and B were already isomorphic as k-algebras, before tensoring up with the polynomial algebra.

We recently reached out to Professor Crachiola to ask him a bit about his accomplishments and advice for WSU students.

## What kind of advice would you give to current mathematics students at WSU?

There is so much to say, but I'll focus on two pieces of advice here. First, the more you invest in your studies, the more you will benefit. This means you should seek and seize opportunities beyond your class work. Read about math outside of your classes and try to solve problems in your spare time. Talk with your professors and get to know your classmates. Participate in math competitions and clubs and attend math talks. Look for job opportunities as a tutor or grader and look for ways to do research. Second, develop and maintain a growth mindset.

Too many math students suffer from imposter syndrome, thinking they don't belong. Math doesn't come easily to most people. Your math professors spend all their time thinking about problems they don't know how to solve. It's hard to remember this because then they come to class and show you all the problems they do know how to solve! It takes time to digest new ideas and problems. If you jump to the internet or solutions manual, you are depriving yourself of the opportunity to grow.

Try to read your textbooks actively, with paper and pencil to fill in any missing steps. Keep detailed notes and recopy them after class to organize and improve them. Ask questions in class and during office hours.

## How did your degrees in mathematics help you on your career path?

This is pretty straightforward in my case. I was a math major in college and then went on to pursue graduate studies in math. After completing my Ph.D., I began my professional academic career as a visiting assistant professor at Loyola University New Orleans and then went on to become a professor at Saginaw Valley State University.

So my math degrees were directly relevant to my career path. That said, I have more recently moved into an administrative role and I think my math background has been an asset there. Mathematical thinking is by its nature transferable to just about any situation that involves problem-solving, planning or analysis.

## Is there anything you wish you would have known at the start of your education in mathematics at WSU?

There are probably many things I wish I had known, but sometimes you have to go through an experience before you can really know them. I enjoyed math in high school, but like many math majors, I didn't really see what math is until I got to abstract algebra. The transition from calculus to upper-level courses can be challenging. The focus changes abruptly from asking "how" to "why." Along these lines, I also didn't know as a freshman that math involves writing.

You don't really have a solution to a problem if you can't communicate it to others, so you have to express yourself clearly and precisely. I also wish I had known how social math can be. I was slow to understand this, coming from an educational background where the most important thing was to keep your eyes on your own paper at all times. With maybe a few exceptions, math is done best as a collaborative experience and you will always understand your work better after talking with other people.

## Were there some hardships you experienced at WSU that you had to overcome? What did you do to overcome them?

I was very fortunate to have a strong support system as a student and I avoided any serious tragedies by chance. We don't have to let our hardships define us, but we don't often get a straight-line path to our goals. No one knows this better than our current students. I am in awe of the strength and determination they have shown to persevere through all the disruptions caused by this pandemic.

I also understand it hasn't come without a cost. For what it is worth, when you are facing hardship it is so important to reach out and tell someone what you need. This gets back to my earlier advice, to talk with your professors and get to know your classmates. You can help each other through difficult situations. But if you are feeling alone without support, your campus has people who can help.

Learn more about Professor Crachiola.