in this paper, we consider polynomial optimization with correlative sparsity. we construct correlative sparse lagrange multiplier expressions (cs-lmes) and propose cs-lme reformulations for polynomial optimization problems using the karush-kuhn-tucker optimality conditions. correlative sparse sum-of-squares (cs-sos) relaxations are applied to solve the cs-lme reformulation. we show that the cs-lme reformulation inherits the original correlative sparsity pattern, and the cs-sos relaxation provides sharper lower bounds when applied to the cs-lme reformulation, compared with when it is applied to  the original problem. moreover, the convergence of our approach is guaranteed under mild conditions. in numerical experiments, our new approach usually finds the global optimal value (up to a negligible error) with a low relaxation order, for cases where directly solving the problem fails to get an accurate approximation. also, by properly exploiting the correlative sparsity, our cs-lme approach requires less computational time than the original lme approach to reach the same accuracy level.

in this survey-style talk i discuss the (old) idea of studying turbulence in stochastically-forced fluid equations. i will discuss definitions of chaos, anomalous dissipation, and various other predictions by physicists that can be phrased as mathematically precise conjectures in this context. then, i will discuss some recent work by my collaborators and i on various aspects, namely (1) a straightforward characterization of anomalous dissipation that implies the classical kolmogorov 4/5 law for 3d nse (joint with michele coti zelati, sam punshon-smith, and franziska weber); (2) the study of "lagrangian chaos" and exponential mixing of scalars and how it leads to a proof of anomalous dissipation and of the power spectrum predicted by batchelor in 1959 for the simpler problem of batchelor-regime passive scalar turbulence (joint with alex blumenthal and sam punshon-smith); (3) the more recent proof of "eulerian chaos" for galerkin truncations of the navier-stokes equations (joint with alex blumenthal and sam punshon-smith).

in 1995, kuperberg introduced a collection of web bases, which combinatorially encode $sl_2$ and $sl_3$ invariant tensors. by schur-weyl duality, these bases are also bases for the specht modules corresponding to partitions $(k,k) \vdash 2k$ and $(k,k,k) \vdash 3k$ respectively, and have nicer symmetry properties than the standard polytabloid basis. in 2017, rhoades introduced a basis for the specht module corresponding to partition $(k,k,1^{n-2k}) \vdash n$ indexed by noncrossing set partitions and with similarly nice symmetry properties. in this talk, we will explore these bases and the connections between them, and discuss how these connections might be used to create a similar basis for the specht module corresponding to $(k,k,k,1^{n-3k}) \vdash n$.

the foundation of equivariant stable homotopy theory is laid by lewis-may-steinberger in the 80's, while people's understanding of the computational aspect of the subject is very limited even until today. the reason is that the equivariant homotopy groups are $ro(g)$-graded, and even the coefficient rings of eilenberg-maclane spectra involve complicated combinatorics of cell structures. in this talk i'll illustrate the advantages of tate squares in doing $ro(g)$-graded computations. several eilenberg-maclane spectra of particular interest will be discussed: the eilenberg-maclane spectra associated with the constant mackey functors $\mathbb{z}$, $\mathbb{f}_2$, and the burnside ring. time permitting, i'll also talk about some structures of the homotopy of $hm$, for $m$ a general $c_{2^n}$-mackey functor.

we'll give a quick introduction to the field of random matrix theory and give partial proof of the famed semicircle law for wigner matrices. to do so, we will incorporate tools from combinatorics, probability, and topology to understand the genus expansion of a random matrix. no background in probability is required, though it may be helpful.

the unstable foliation, that changes the horizontal components of period coordinates, plays an important role in the study of translation surfaces, including their deformation theory, and in the understanding of horocycle invariant measures.

in this talk, we show that measures of large dimension equidistribute in affine invariant manifolds and give an effective rate. an analogous result in the setting of homogeneous dynamics is crucially used in the effective equidistribution results of lindenstrauss-mohammadi and lindenstrauss--mohammadi--wang. background knowledge on translation surfaces and homogenous dynamics will be explained.

mean curvature flow is the negative gradient flow of the area functional, and it has attracted a lot of interest in the past few years. in this talk, we will discuss a pde-based, gauge theoretic, construction of codimension-two mean curvature flows based on the yang-mills-higgs functionals, a natural family of energies associated to sections and metric connections of hermitian line bundles. the underlying idea is to approximate the flow by the solution of a parabolic system of equations and study the corresponding singular limit of these solutions as the scaling parameter goes to zero. this is based on joint work with a. pigati and d. stern.