In 1987, Margulis solved an old conjecture of Oppenheim which states that for a nondegenerate, indefinite and irrational quadratic form $Q$ in $n \geq 3$ variables, $Q(\mathbb{Z}^n)$ is dense in $\mathbb{R}$. Following this, Dani and Margulis proved the simultaneous density at integer points for a pair consisting of quadratic and linear form in $3$ variables when certain conditions are satisfied. We prove an analogue of this for the case of an inhomogeneous quadratic form and a linear form.
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This is based on joint work with Anish Ghosh.

A parallel cut-cell algorithm is described to solve the free boundary problem of the Grad-Shafranov equation.
The algorithm reformulates the free-boundary problem in an irregular bounded domain and its important aspects include a
searching algorithm for the magnetic axis and separatrix, a surface integral along the irregular boundary to determine the
boundary values, an approach to optimize the coil current based on a targeting plasma shape, Picard iterations with Aitken's
acceleration for the resulting nonlinear problem and a Cartesian grid embedded boundary method to handle the complex
geometry. The algorithm is implemented in parallel using a standard domain-decomposition approach and a good parallel
scaling is observed. Numerical results verify the accuracy and efficiency of the free-boundary Grad-Shafranov solver.

An $n$-tangle is a proper embedding of the disjoint union of n arcs into a $3$-ball, in such a way that the endpoints are mapped to $2n$ marked points on the ball’s boundary. In 1967, Conway developed the theory of a special class of $2$-tangles, called ``rational tangles," leading to important results on the classification of knots. Rational tangles themselves have an elegant classification which relates to continued fraction expansions of rational numbers. We explore this connection in the context of a fun activity, which was developed by Conway in order to demonstrate some aspects of the theory.

This talk establishes a concentration phenomenon on the empirical eigenvalue distribution (EED) of the principal submatrix in a random hermitian matrix whose distribution is invariant under unitary conjugacy. More precisely, if the EED of the whole matrix converges to some deterministic probability measure 𝔪, then its difference from the EED of its principal submatrix, after a rescaling, concentrates at the Rayleigh measure (in general, a Schwartz distribution) associated with 𝔪 by the Markov-Krein correspondence. For the proof, we use the moment method with Weingarten calculus and free probability. At some stage of calculations, the proof requires a relation between the moments of the Rayleigh measure and free cumulants of 𝔪. This formula is more or less known, but we provide a different proof by observing a combinatorial structure of non-crossing partitions.
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This is a joint work with Katsunori Fujie.

This is joint work with Will Wylie. The goal is to classify, if possible, the homogeneous geometric solitons. Here a geometric soliton is the soliton for a geometric flow. The Ricci flow is the most prominent example of such a flow, but there are man others where the Ricci tensor is replaced with some other tensor that depends in a natural way on the Riemannian structure. We will also consider some more general problems showing that our techniques can be used for other geometric problems.

We develop a new empirical Bayesian inference algorithm for solving a linear inverse problem given multiple measurement vectors (MMV) of under-sampled and noisy observable data. Specifically, by exploiting the joint sparsity across the multiple measurements in the sparse domain of the underlying signal or image, we construct a new support informed sparsity promoting prior. Several applications can be modeled using this framework. Our numerical experiments demonstrate that using this new prior not only improves accuracy of the recovery, but also reduces the uncertainty in the posterior when compared to standard sparsity producing priors.
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This is joint work with Theresa Scarnati formerly of the Air Force Research Lab Wright Patterson and now working at Qualis Corporation in Huntsville, AL, and Jack Zhang, recent bachelor degree recipient at Dartmouth College and now enrolled at University of Minnesota’s PhD program in mathematics.

We give a non-asymptotic bound on the spectral norm of a $d\times d$ matrix $X$ with centered jointly Gaussian entries in terms of the covariance matrix of the entries. In some cases, this estimate is sharp and removes the $\sqrt{\log d}$ factor in the noncommutative Khintchine inequality.
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Joint work with Afonso Bandeira

Simpson conjectured that for a reductive group $G$, rigid
$G$-local systems on a smooth projective complex variety are integral. I
will discuss a proof of integrality for cohomologically rigid $G$-local
systems. This generalizes and is inspired by work of Esnault and
Groechenig for $GL_n$. Surprisingly, the main tools used in the proof
(for general $G$ and $GL_n$) are the work of L. Lafforgue on the
Langlands program for curves over function fields, and work of Drinfeld
on companions of $\ell$-adic sheaves. The major differences between
general $G$ and $GL_n$ are first to make sense of companions for
$G$-local systems, and second to show that the monodromy group of a
rigid G-local system is semisimple.
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All work is joint with Stefan Patrikis.