the morrey conjecture pertains to the properties of quasi-convexity and rank-one convexity in functions, where the former implies the latter, but the converse relationship is not yet established. while sverak has proven the conjecture in three dimensions, it remains unresolved in the two-dimensional case. analyzing these properties analytically is a formidable task, particularly for vector-valued functions. consequently, to investigate the validity of the morrey conjecture, we conducted numerical simulations using a set of example functions by dacorogna and marcellini. based on our results, the morrey conjecture appears to hold true for these functions.

when steven strogatz wrote a 15-part series on the elements of math for the new york times, to his surprise — and his editor's — each piece climbed the most emailed list and elicited hundreds of appreciative comments. in this talk steve will describe his adventures in bringing math to the masses, and will reflect on what works… and what doesn’t.

steven strogatz is the jacob gould schurman professor of applied mathematics at cornell university. he works on nonlinear dynamics with applications to physics, biology, and the social sciences. his latest book, infinite powers, was a new york times bestseller and was shortlisted for the 2019 royal society science book prize.

presented by the uc san diego research communications program and supported by a grant from the gordon and betty moore foundation.

the classical coinvariant ring $r_n$ is obtained from the polynomial ring $\mathbb{c}[x_1, \dots, x_n]$ by quotienting by the ideal $i_n$ generated by symmetric polynomials with vanishing constant terms. the {\em superspace coinvariant ring} $sr_n$ is obtained analogously, but starting with the ring $\omega_n$ of regular differential forms on $n$-space. we describe  the bigraded hilbert series of $sr_n$ in terms of ordered set partitions and give an `operator theorem' which describes the harmonic space attached to $sr_n$. this proves conjectures of n. bergeron, li, machacek, sulzgruber, swanson, wallach, and zabrocki. this talk is based on joint work with andy wilson.

the boardman-vogt tensor product of operads encodes the notion of interchanging algebraic structures. a classic result of dunn tells us that the tensor product of two little cube operads is equivalent to a little cube operad with the dimensions added together. as models for $\mathbb{e}_k$-operads, this reflects a defining property of these operads.
in this talk, we will explore some equivariant generalizations to dunn’s additivity. along the way, we will play with little star-shaped operads, question if we really need group representations for equivariant operads, and learn to love (and hate) the tensor product.

we study a robust approximation method for solving a class of chance constrained optimization problems. the constraints are assumed to be  polynomial in the random vector. a semidefinite relaxation algorithm is proposed for solving this kind of problem. its asymptotic and finite convergence are proven under some mild assumptions.

we consider a finite volume homogeneous space endowed with a random walk whose driving measure is zariski-dense. in the case where jumps have finite exponential moment, eskin-margulis and benoist-quint established recurrence properties for such a walk. i will explain how their results can be extended to walks with finite first moment. the key is to make sense of the following claim: "the walk in a cusp goes down faster that some iid markov chain on r with negative mean". joint work with n. de saxcé.

imagine passing a signal $v$ through a noisy channel, where $v \in \mathbb{c}^n$ is a deterministic unit vector. we assume that the recipient observes a corrupted version of the signal in the form of $\theta vv^* + m$, where $\theta \in \mathbb{r}$ is the strength of the signal and $m$ is a random hermitian $n \times n$ matrix representing the noise. we consider two questions:

1. (detection) is it possible for the recipient to conclude that a signal has been passed based on the observation?
2. (recovery) if so, is it possible for the recipient to recover the signal

for rotationally invariant noise, benaych-georges and nadakuditi answered the detection question in terms of the outlying eigenvalues and the recovery question in terms of the corresponding eigenvectors. their proof crucially relies on the fact that the eigenvectors of a rotationally invariant ensemble are haar distributed (in particular, delocalized). 

we consider a general class of noise that includes non mean-field models such as random band matrices in regimes where the eigenvectors are known to be localized. in contrast to the usual approach to outliers via the resolvent, our analysis relies on moment method calculations for general vector states and a seemingly innocuous isotropic global law.

on a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics. we prove that this result may be localized to compact subdomains in an arbitrary riemannian manifold with boundary, as motivated by an attempt to generalize the riemannian penrose inequality in dimension 8. this result is a generalization of corvino's result about localized scalar curvature deformations; however, the existence part needs to be handled delicately since the problem is non-variational. for non-generic cases, we give a classification theorem for domains in space forms and schwarzschild manifolds, and show the connection with positive mass theorems.