We consider a class of parametric regression problems where the signal is observed through random nonlinear functions with a difference of convex (DC) form. This model describes a broad subset of nonlinear regression problems that includes familiar special cases such as phase retrieval/quadratic regression and blind deconvolution/bilinear regression. Given the DC decomposition of the observation functions as well as an approximate solution, we formulate a convex program as an estimator that operates in the natural space of the signal. Our approach is computationally superior to the methods based on semidefinite/sum-of-squares relaxation---tailored for polynomial observation functions---and can compete with the non-convex methods studied in special regression problems. Furthermore, under mild moment assumptions, we derive the sample complexity of the proposed convex estimator using a PAC-Bayesian argument. We instantiate our results with bilinear regression with Gaussian factors and provide a method for constructing the required initial approximate solution.

In random constraint satisfaction problems, first and second moment methods are used to yield upper and lower bounds for the threshold of edge density for existence of solutions. In random hypergraph 2-coloring, Achlioptas and Moore obtain a lower bound using a standard second moment method based on the Paley-Zygmund inequality. Coja-Oghlan and Zdeborova later use an enhanced second moment method involving the Hamming geometry of the set of colorings to improve the lower bound up to a so-called condensation transition.

We adapt their methods and setup to a subshift of finite type over a sofic group and show results analogous to the aforementioned, exploiting the combination of phenomena that occur at densities between the standard and enhanced second moment thresholds to conclude that there exists an interesting example of a topological dynamical system having two different positive sofic entropies relative to two different sofic approximations. This is joint work with Dylan Airey and Lewis Bowen.

Quantum computing can be effected by a sequence of projective measurements. Such strategies are called, ``measurement only''. The key geometric concept which ensures that system information does not leak to the environment during measurement is ``equiangularity''. I will explain what the Octonions are and how they provide the largest families of equiangular subspaces of a Hilbert space.

Morse theory can be generalized to the study of maps to $R^k$. I will discuss where this leads, focusing on ambient Morse functions of embedded 3-manifolds. The relevance to the question of finite generation of the Goeritz groups $G_g$ will be explained.

In our fast paced world, it's important to take things slow every once in a while. In this talk, we'll take a leisurely stroll through some of the research I've done recently on slow Fibonacci walks, in which we try and generate numbers in a ``Fibonacci-like way'' as slowly as possible. This talk is part of the graduate student seminar ``Food for Thought,'' and in particular (1) no prior knowledge of anything is assumed, and more importantly (2) snacks will be provided.

Web geometry deals with foliations in general position. In the planar case and the complex setting, a $d$-web is given by the generic family of integral curves of an analytic or an algebraic differential equation \textit{F(x,y,y')=0} with y'-degree $d$. Invariants of these configurations as abelian relations (related to Abel's addition theorem), Lie symmetries or Godbillon-Vey sequences are investigated. This viewpoint enlarges the qualitative study of differential equations and their moduli. In the nonsingular case and through the singularities, Cartan-Spencer and meromorphic connections methods will be used. Basic examples will be given from different domains including classic algebraic geometry and WDVV-equations. Standard results and open problems will be mentioned. Illustration of the interplay between differential and algebraic geometry, new results will be presented.