Part 2 of Stephen’s talk.

Motivated by the multi-reference alignment (MRA) problem and questions in equivariant neural networks we study the problem of recovering the generic orbit in a representation of a finite group from invariant polynomials of degree at most 3. We prove that in many cases of interest these low degree invariants are sufficient to recover the orbit of a generic vector. 

Semester seminar calendar: https://sites.google.com/view/mathdatamizzou/home

We define the basic machinery of multiplicities and graded families and state some recent theorems on multiplicities (which we will finish proving in part 2).

In a 2014 paper, Cutkosky proved a volume equals multiplicity formula for the multiplicity of an m_R-primary ideal. We will discuss a generalization of this result to the epsilon multiplicity.

The problem is to limit the motion of a robot so that if it is commanded to work outside of its workspace, then the robot experiences a graceful degradation of performance depending upon the extent of the workspace violation.  This is demonstrated with Stewart tables, which provide six degrees of freedom. This research is used by NASA in creating a simulator for a lunar terrain vehicle. This is real applied mathematics, showcasing ODEs, the Lie group SE(3), and its Lie algebra se(3).

I will give an overview on results about the higher ramification theory of finite Galois extensions, mainly, but not only, those of prime degree, for arbitrary valuations. I willtalk about ramification groups, ramification ideals, ramification jumps, norms and traces of these ideals, Kähler differentials and their annihilators, and Dedekind differents. Several of these objects are used for the classification of defects of Galois defect extensions of prime degree. This in turn is used to prove results about the class of deeply ramified fields, which also contains the perfectoid fields.

The Brascamp–Lieb inequalities are an important family of inequalities in analysis that subsumes several inequalities significant in their own right, including Hölder’s inequality, Young’s inequality, and the Loomis–Whitney inequality. Several variants and extensions of these inequalities have been developed, some of which have proved to be very useful in contemporary harmonic analysis. In this talk, we will discuss the formulation of generalized Brascamp–Lieb inequalities for algebraic objects known as quivers, building on the recent work of Chindris and Derksen on the representation theory of quivers. No prior knowledge of Brascamp–Lieb inequalities or quivers will be assumed.

In 1961, Grunbaum asked whether the centroid $c(K)$ of a convex body $K$ is the centroid of at least $n + 1$ different $(n − 1)$-dimensional sections of $K$ through $c(K)$. A few years later, Loewner asked to find the minimum number of hyperplane sections of $K$ passing through $c(K)$ whose centroid is the same as $c(K)$.

We give an answer to these questions for $n \geq 5$. In particular, we construct a convex body which has only one section whose centroid coincides with the centroid of the body. Joint work with S. Myroshnychenko and V. Yaskin.

An origin-symmetric convex body $K$ in $\mathbb{R}^n$ is said to be in the John position if the maximal volume ellipsoid contained in it is the Euclidean ball. One of the most celebrated theorems in geometric functional analysis is John's theorem, which says that if a convex body $K$ is in John position, then there are a collection of points $u_1,\dots u_m \in S^{n-1} \cap \partial K$ and positive scalars $c_1,\dots, c_m$ for which $I_n = \sum_{j=1}^m c_j u_j \otimes u_j$, where $I_n$ is the identity operator in $\mathbb{R}^n$.  This theorem has numerous consequences such as an estimate for the Banach-Mazur distance to Euclidean space  and the reverse isoperimetric inequality due to Ball. 

Recently, much attention has been given to translating notions from convex geometry and geometric functional analysis to the world of log-concave functions.  In particular, over the last 10 years, extensions of these celebrated theorems (in some forms) have been translated to the world of log-concave functions.

In this talk, we focus on a complementary set of positions, called "maximal-intersection positions," originally introduced by Artstein-Avidan and Katzin in 2016. This collection of position includes the John (and its dual the Lowner position) as particular cases.  The extension of this notion to the world of functions is seen through the following extremal problem for the convolution of a pair of functions: given a pair of integrable log-concave functions $f$ and $g$, find

$\sup_{(T,b) \in SL_n(\mathbb{R}) \times \mathbb{R^n}} \int f(x) g(T^{-1}x-T^{-1}b)dx.$

This is based on a joint work with Steven Hoehner. 

We consider an elliptic operator, double divergence form operator L*, which is the formal adjoint of the elliptic operator in non-divergence from L. An important example of a double divergence form equation is the stationary Kolmogorov equation for invariant measures of a diffusion process. We are concerned with the regularity of weak solutions of L*u=0.

We will also discuss some applications and parabolic counterparts.