Vera Roshchina, RMIT University will be lecturing at Summer School 2016, she chatted to us about her research, reflected on her career and gave advice to budding mathematicians.
What are the most interesting “big questions” in your field?
There are several open problems related to conic optimisation, for example, what is left of the Hirsch Conjecture, the 9th problem of Stephen Smale and the generalised Lax conjecture.
The first problem I mentioned, the Hirsch conjecture, stated that in an n-facet polytope in d-dimensional Euclidean space every two vertices are connected by an edge path with length no more than n – d. This conjecture was proved to be wrong by Francisco Santos in 2010, but an important question remains regarding the upper bound on the length of the shortest path, in particular whether a polynomial bound exists.
The ninth problem of Stephen Smale is to find a strongly polynomial algorithm to decide the consistency of a system of linear inequalities. That is, given a real matrix A of size m×n and an m dimensional real vector b, we are looking for an algorithm that decides whether there exists an x such that Ax ≥ b, and does so in a polynomial number of arithmetic operations (as a function of n and m).
The generalised Lax conjecture asks whether a rather general kind of convex cones called hyperbolicity cones can be represented as an intersection of a linear subspace with the cone of positive semidefinite matrices.
What kind of problems are you interested in broadly in the field? (maths as a whole)
I like to think about open problems geometrically, so I tend to focus on problems that are finite dimensional and are essentially continuous. At the same time I find discrete mathematics truly fascinating, mostly because a lot of problems are so very easy to understand, and some proofs can be explained even to a relatively uninitiated person, whilst solving such problems is prohibitively difficult. So I enjoy learning about combinatorics and thinking about some open questions, but in terms of contribution I do not hold my hopes very high!
What are your favourite applications of your work?
Even though I work in optimisation, I am more interested in studying the geometry of a general class of problems rather than focussing on practical applications. What I find truly amazing though is the recent contribution of conic optimisation to solve theoretical problems. In particular, recent work by Frank Vallentin and his team on bounds for packing problems is fascinating, and of course the resolution of Kepler’s conjecture by Thomas Hales which heavily utilised numerical computation by means of linear programming is worth mentioning.
What are some other areas of maths that are particularly interesting to you?
Some of the most exciting research directions that I successfully explored came from reading random articles on the web or attending talks in seemingly unrelated areas of maths. This happened with some of the research I did on invisibility in dynamical billiards and that is also how I got interested in studying facial structure of convex cones, which is my main research interest at the moment. My preferences are defined by my limitations rather than anything else: my background in nonsmooth optimisation defines the range of ideas that I can truly appreciate.
Why did you become a mathematician?
Doing a PhD in maths was an easy choice: who wouldn’t want to move to a new country in their early twenties, being financed for three years of having fun? Unfortunately the job market is too competitive, and getting a stable position nowadays is more like winning a lottery, so I can say that maybe I was a lucky winner, and I would like to also encourage young people to keep their options open.
Do you have any advice for future mathematicians?
No, not really. It may be much more fun to figure things out by yourself!
Biggest mathematical/statistical regret?
I wish I had more time to study mathematics deeply when I was an undergraduate student.
Biggest mathematical/statistical success?
The end product of any academic research is communication of new ideas, and you also need someone to listen to what you have to say and to respond. So when this kind of communication happens—maybe you solve an open problem or introduce a new idea, then someone else takes notice and develops this further—I think that’s very rewarding.
