some of the outstanding and still classical problems in algebraic combinatorics concern understanding the kronecker coefficients of the symmetric group, the multiplicities describing the decomposition of tensor products of representations into irreducibles, which are nonnegative integers lacking a positive combinatorial formula for over 75 years. recently they appeared in geometric complexity theory (gct), a program aimed to distinguish computational complexity classes (like the p vs np problem) and prove complexity theoretic bounds using algebraic geometry and representation theory.

on the combinatorial side, we \lbrack{pak-p}\rbrack will show various bounds on kronecker coefficients via character evaluations and partition enumeration, and use them to extend sylvester and stanley's theorem on the unimodality of partitions inside a rectangle and find asymptotic bounds. on the gct side, using algebraic and combinatorial methods, we \lbrack{burgisser-ikenmeyer-p, ikenmeyer-p}\rbrack show that the relevant kronecker and plethysm coefficients of the general linear group are positive, thereby disproving a mulmuley and sohoni conjecture on the existence of ``occurrence obstructions" and practically showing that the 'p vs np' problem is even more difficult. in the reverse direction, gct arguments show that rectangular kronecker coefficients are larger than plethysm coefficient in a stable range \lbrack{ikenmyer-p}\rbrack, establishing a connection between apriori unrelated and greatly mysterious multiplicities.

initial data in general relativity must satisfy certain underdetermined differential equations called the constraint equations. a natural problem is to find a parameterization of all possible initial data. a standard method for this is called the conformal method. in this talk, we'll discuss the successes and failures of this method, and future directions for research.

i will present recent joint work with arnaud brothier on actions of locally compact groups on von neumann algebras. we study algebraic properties of the associated crossed-product algebras, and prove among other things a correspondance result between certain subalgebras of this crossed-product and closed subgroups of the acting group. this generalizes results of izumi-longo-popa. i will explain our (very different) approach, and give related questions and conjectures.

one of the outstanding open problems in the study of fluids is the global regularity of smooth solutions to the three-dimensional incompressible euler equation. i will begin by introducing the incompressible euler equation as well as some classical well-posedness results. then i will discuss various attempts to understand the global regularity problem and related problems moving into recent results. one popular attempt to understand the global regularity problem is to study simplified lower dimensional models that can be satisfactorily solved. a major issue with studying simplified models is that they may have no bearing on the dynamics of the actual 3d euler equation--especially since the closer a model gets to modeling the dynamics of 3d euler the more challenging understanding the dynamics of the model is. in recent works with i. jeong, we derived a “good” model through the use of symmetry properties of the equation. in particular, we proved that if singularity formation can be established for a particular two-dimensional equation, then there is singularity formation for the full 3d euler equation for finite-energy solutions lying in a critical space where there is local well-posedness. similar results can be proven for the surface quasi-geostrophic (sqg) equation arising in atmospheric dynamics and there a one-dimensional model is derived. i will then discuss recent results on some of these models and their implications.

at the beginning of this talk, we will introduce the background of tensor network states (tns) in various areas such as quantum physics, quantum chemistry and numerical partial differential equations. famous tns includes tensor trains (tt), matrix product states (mps), projected entangled pair states (peps) and multi-scale entanglement renormalization ansatz (mera). then we will explain how to define tns by graphs and we will define tensor network ranks which can be used to measure the complexity of tns. we will see that the notion of tensor network ranks is an analogue of tensor rank and multilinear rank. we will discuss basic properties of tensor network ranks and the comparison among tensor network ranks, tensors rank and multilinear rank. if time permits, we will also discuss the dimension of tensor networks and the geometry of tns. this talk is based on papers joined with lek-heng lim.

artin's conjecture on primitive roots, which was originally formulated for multiplicative groups, has a natural analogue for elliptic curves. in this survey talk, i will discuss the analogy and focus on``new'' phenomena such as the existence of ``never-primitive'' points.

schubert curves are the spaces of solutions to certain one-dimensional schubert problems involving flags osculating the rational normal curve. the real locus of a schubert curve is known to be a natural covering space of $rp^1$, so its real geometry is fully characterized by the monodromy of the cover. it is also possible, using k-theoretic schubert calculus, to relate the real locus to the overall (complex) riemann surface.

we present a local algorithm for computing the monodromy operator in terms of jeu de taquin-like operations on certain skew young tableaux, and use it to provide purely combinatorial proofs of some of the connections to k-theory. we will also explore partial progress in this direction in the type c setting of the orthogonal grassmannian. this is joint work with jake levinson.

this is part of a series lectures studying stability of symplectic forms
on noncompact manifolds. the case of compact manifolds is well understood
thanks to seminal work of jurgen moser in the 1960s.