For parameterized systems, one standard problem is to determine the set of parameters which "best" fits given data.  Two examples of this will be summarized in this talk, both of which can be solved using homotopies.  The first is variational inference in which one searches in a parameterized family of probability distributions for a probability distribution that best fits the given data.  The second is synthesizing a linkage whose coupler curve best approximates the given data.  This talk is joint work with Emma Cobian, Fang Liu, and Daniele Schiavazzi (variational inference) and Aravind Baskar and Mark Plecnik (approximate synthesis).

 

What is the most singular possible (reduced) hypersurface in positive characteristic? One answer to this question comes from finding a lower bound on an invariant called the F-pure threshold of a polynomial in terms of its degree. In this talk, I'll introduce the F-pure threshold and then discuss a class of hypersurfaces which obtain a minimal F-pure threshold.  We’ll further investigate some of their surprising algebraic and geometric properties including their connection to surfaces with many lines.  This talk is based on joint work with Zhibek Kadyrsizova, Jennifer Kenkel, Jyoti Singh, Karen E. Smith, Adela Vraciu, and Emily E. Witt.

In this talk, we’ll present some recent work on traveling waves in water that carry vortices in their bulk. We show that for any supercritical Froude number (non-dimensionalized wave speed), there exists a continuous one-parameter family of solitary waves with a submerged point vortex in equilibrium. This family bifurcates from an irrotational laminar flow, and, at least for large Froude numbers, it extends up to the development of a surface singularity. These are the first rigorously constructed gravity wave-borne point vortices without surface tension, and notably our formulation allows the free surface to be overhanging. Through a separate numerical study, we find strong evidence that many of the waves do indeed have an overturned air—water interfaces. Finally, we prove that generically one can perform a desingularization procedure to obtain a solitary wave with a submerged hollow vortex. Physically, these can be thought of as traveling waves carrying spinning bubbles of air in their bulk.

We will also discuss some work in progress on the existence of imploding vortex configurations that experience finite-time self-similar collapse.

This is joint work with Ming Chen, Kristoffer Varholm, and Miles Wheeler.
 

Density estimation for Gaussian mixture models is a classical problem in statistics that has applications in a variety of disciplines. Two solution techniques are commonly used for this problem: the method of moments and maximum likelihood estimation. This talk will discuss both methods by focusing on the underlying geometry of each problem.

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

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.