In this talk, I will consider quasilinear parabolic PDEs subject to stochastic or rough perturbation and explain how various assumptions on coefficients and roughness of the noise naturally ask for different notions of solution with different regularity properties and different techniques of the proofs. On the one hand, the problems under consideration will be stochastic second order parabolic PDEs with noise smooth in space, either with a possible degeneracy in the leading order operator, where only low regularity holds true, or under the uniform ellipticity assumption, where arbitrarily high regularity can be proved under suitable assumptions on the coefficients. On the other hand, I will discuss a rough pathwise approach towards these problems based on tools from paracontrolled calculus.
 

Teaching is a learning process, with both ups and downs. In this talk, I will discuss some of the trends I have observed as I progressed from being a student to being an instructor. These include the effects of technology on Mathematics education, approaches to teaching Mathematics, etc. I will also discuss some of the lessons I have learned along the way.

The plant hormone auxin plays a central role in growth and morphogenesis. In the shoot apical meristem, the auxin flux is polarized through its interplay with PIN proteins. Concentration based mathematical models of the auxin flux permit to explain some aspects of phyllotaxis , where auxin accumulation points act as auxin sinks and correspond to primordia. Simulations show that these models can reproduce geometrically regular patterns like spirals in sunflowers or Fibonacci numbers. We propose a mathematical study of a related non-linear o.d.e. using Markov chain theory. We will next consider a concurrent model which is based on the so-called flux hypothesis, and show that it can explain the self-organization of plant vascular systems.

Counting linear extensions of a partial order (linear orders compatible with the partial order) is a classical and computationally difficult problem in enumeration and computer science. A family of partial orders that are prevalent in enumerative and algebraic combinatorics come from Young diagrams of partitions and skew partitions. Their linear extensions are called standard Young tableaux. The celebrated hook-length formula of Frame, Robinson and Thrall from 1954 gives a product formula for the number of standard Young tableaux of partition shape. No such product formula exists for skew partitions. 

In 2014, Naruse announced a formula for skew shapes as a positive sum of products of hook-lengths using ”excited diagrams” of  Ikeda-Naruse, Kreiman, Knutson-Miller-Yong in the context of equivariant cohomology. We prove Naruse’s formula algebraically and combinatorially in several different ways. Also, we show how excited diagrams give asymptotic results and product formulas for the enumeration of certain families of skew tableaux. Lastly, we give analogues of Naruse's formula in the context of equivariant K-theory.

This is joint work with Igor Pak and Greta Panova.
 

Topology began with the study of complex functions, and the interaction of topology with algebraic geometry and number theory continues to be fertile ground.  Starting with basic examples, I will explain how the function field/number field dictionary and the remarkable framework of the Weil conjectures (as established by Weil, Grothendieck, Dworkin, Deligne and many others) allows one to use counting problems to predict and discover topological properties of manifolds, and vice versa. This is joint work with Benson Farb and Melanie Matchett Wood.