from number theory to knots, hecke algebras have applications within
many areas of mathematics.

in this talk we describe a pictorial calculus for computing convolution
products in affine hecke algebras over fields of characteristic zero.

convolution products of this type have been understood since the work of
iwahori and matsumoto [1965].

however, using results of parkinson, ram and schwer [2006], we can now
draw pictures illustrating the rich combinatorial nature of these products.

we describe this pictorial calculus in the example of
$\mathrm{sl}_3(\mathbb{q}_p)$. its applicability is limited to
characteristic zero.

let $w^{(n)}$ be the $n$-letter word obtained by repeating a fixed word $w$, and let $r_n$ be a random $n$-letter word over the same alphabet. we show several results about the length of the longest common subsequence (lcs) between $w^{(n)}$ and $r_n$; in particular, we show that its expectation is $\gamma_w n-o(\sqrt{n})$ for an efficiently-computable constant $\gamma_w$.\\

this is done by relating the problem to a new interacting particle system, which we dub ``frog dynamics''. in this system, the particles (`frogs') hop over one another in the order given by their labels. stripped of the labeling, the frog dynamics reduces to a variant of the pushasep.\\

in the special case when all symbols of $w$ are distinct, we obtain an explicit formula for the constant $\gamma_w$ and a closed-form expression for the stationary distribution of the associated frog dynamics.\\

froggies on a pond\\
they get scared and hop along\\
scaring others too.\\

their erratic gait\\
gives us tools to calculate\\
lcs of words.\\

(joint work with boris bukh)

the (euclidean) isoperimetric inequality says that any set has larger perimeter than a ball with the same area. the quantitative isoperimetric inequality says that the difference in perimeters is bounded from below by the square of the distance from our set e to the ``closest'' ball of the same area. in this talk, we will discuss an extension of this result to closed riemannian manifolds with analytic metrics. in particular, we show that a similar inequality holds but with the distance raised to a power that depends on the geometry. we also have examples which show that a greater power than two is sometimes necessary and that the analyticity condition is necessary. this is joint work with o. chodosh (stanford) and l. spolaor (ucsd).

grothendieck's inequality guarantees that a certain discrete optimization problem-optimizing a bilinear form
over +1, -1 inputsis equivalent up to a constant, called grothendieck's constant, to a continuous optimization
problem -- optimizing the same bilinear form over unit vectors in a hilbert space. this is important for
convex relaxation, because the former contains np-hard problems such as max-cut, whereas the latter is
a semidefinite program and can be solved in polynomial time. a world apart from convex relaxation is
algebraic computational complexity, where it is well-known that the exponent of matrix multiplication is
exactly the sharp lower bound for the (log of) tensor rank of the strassen matrix multiplication tensor.
we show that grothendieck's constant is the sharp upper bound on a tensor norm of strassen matrix
multiplication tensor. hence these two important quantities from disparate areas of theoretical computer
science strassen's exponent of matrix multiplication and grothendieck's constantare just different measures
of the same underlying tensor. this allows us to rewrite grothendieck's inequality as a norm inequality for
a 3-tensor, which in turn gives a family of natural generalized inequalities. we show that these are locally
sharp and prove that grothendieck's inequality is unique.

a fundamental question in modern cell biology is how cellular and subcellular structures are formed and maintained given their particular molecular components. how are the different shapes, sizes, and functions of cellular organelles determined, and why are specific structures formed at particular locations and stages of the life cycle of a cell? in order to address these questions, it is necessary to consider the theory of self-organizing non-equilibrium systems. we are particularly interested in identifying and analyzing novel mechanisms for pattern formation that go beyond the standard turing mechanism and diffusion-based mechanisms of protein gradient formation. in this talk we present three examples of non-classical biological pattern formation: (i) space-dependent switching diffusivities and cytoplasmic protein gradients in the c. elegans zygote (ii) transport models of cytoneme-based morphogenesis. (iii) hybrid turing mechanism for the homeostatic control of synaptogenesis in c. elegans.

free probability is a subfield of mathematics at the intersection of
operator algebras, complex analysis, probability, and combinatorics. it is
used, among other things, to study the ``$n=\infty$'' case of various $n
\times n$ random matrix models. a concept of central importance in free
probability is \textit{free independence}, the ``noncommutative analogue''
of independence (of random variables) from classical probability. the goal
of this talk is to develop a rigorous understanding of the throw-away
clause in the previous sentence with an interesting mix of analysis and
algebra. time permitting, we may also discuss why classical independence
and free independence are in a precise sense the only ``reasonable''
notions on independence.