for a tracial $*$-probability space $(\mathcal{a}, \varphi)$, c\'{e}bron, dahlqvist, and male constructed an enveloping traffic space $(\mathcal{g}(\mathcal{a}), \tau_\varphi)$ that extends the trace $\varphi$. the cdm construction provides a universal object that allows one to appeal to the traffic probability framework in generic situations, prioritizing an understanding of its structure. we show that $(\mathcal{g}(\mathcal{a}), \tau_\varphi)$ comes equipped with a canonical free product structure, regardless of the choice of $*$-probability space $(\mathcal{a}, \varphi)$. if $(\mathcal{a}, \varphi)$ is itself a free product, then we show how this additional structure lifts into $(\mathcal{g}(\mathcal{a}), \tau_\varphi)$. here, we find a duality between classical independence and free independence. we apply our results to study the asymptotics of large (possibly dependent) random matrices, generalizing and providing a unifying framework for results of bryc, dembo, and jiang and of mingo and popa. this is joint work with camille male.

a real symmetric matrix (resp., tensor) is said to be copositive if the associated quadratic (resp., homogeneous) form is greater than or equal to zero over the nonnegative orthant. the problem of detecting their copositivity is np-hard. this paper proposes a complete semidefinite relaxation algorithm for detecting the copositivity of a matrix or tensor. if it is copositive, the algorithm can get a certificate for the copositivity. if it is not, the algorithm can get a point that refutes the copositivity. we show that the detection can be done by solving a finite number of semidefinite relaxations, for all matrices and tensors.

in this talk i will describe a family of events arising in two related probability models, one having to do with uniformly random ``fully packed loops'' (a family of combinatorial objects which are in bijection with alternating sign matrices), and another appearing in connection with a natural random walk on noncrossing matchings. the connection between the two models is highly nonobvious and was conjectured by physicistsrazumov and stroganov in 2001, and given a beautiful proof in 2010 by cantini and sportiello. another intriguing phenomenon is that the probabilities of the events in question, known as ``connectivity events'', appear to be rational functions of a size parameter n (with the simplest such formula being $3(n^2-1)/2(4n^2+1))$, but this is only conjectured in all but a few cases. the attempts to prove such formulas by myself and others have led to interesting algebraic results on a family of multivariate polynomials known as ``wheel polynomials'', and to a family of conjectural constant term identities that is of independent interest and poses an interesting challenge to algebraic combinatorialists.

a local system on the riemann sphere minus finitely many
points is defined to be ``rigid'' if it is determined by the conjugacy
classes of its monodromy operators along the missing points. katz proves
a convenient cohomological criterion characterizing irreducible rigid
local systems, which is based on an analysis of the moduli of local
systems on the punctured riemann sphere. in this talk, we will discuss
this story, and then proceed towards an analogous story in the
arithmetic setting, where, in place of local systems on the punctured
riemann sphere, we consider overconvergent isocrystals on the punctured
projective line over a field of positive characteristic.

the moving sofa problem is a well-known open problem in geometry. posed by leo moser in 1966, it asks for the planar shape of largest area that can be moved around a right-angled corner in a two-dimensional hallway of width 1. although deceptively easy to state, it turns out to be highly nontrivial to analyze, and has a rich structure that is intriguing to amateurs and experts alike. in this talk i will survey both old and new results about the problem, including a new moving sofa shape with an interesting algebraic structure that i discovered in 2016, and new bounds on the area of a moving sofa i derived more recently in joint work with yoav kallus. i will conclude with a discussion of how the heavily experimental and computer-assisted nature of the recent results offers broader lessons for aspiring research mathematicians.

the essential skeleton is an invariant of a degeneration that
appears in both berkovich geometry and minimal model theory. i will show
that for degenerations with a semistable model, the essential skeleton of a
product of degenerations is the product of their skeleta.

as an application, we are able to describe the homeomorphism type of some
degenerations of hyperkähler varieties, both for a hilbert scheme of
degenerating k3 surfaces and for a kummer variety associated to a
degeneration of an abelian surface. this is joint work with enrica mazzon.