artists, coders, and other creative professionals are never on the cutting edge of research. however, their use cases are often the most memorable and effective tools for communicating what mathematics, and applied mathematics especially, is capable of. when google shows off its neural networks, it makes your selfies look like van gogh drew them. when ibm wants to prove its advancements in nlp, it plays jeopardy. this talk aims to present a handful of examples of computational art - and to explore both its technical background and it's impact. we will discuss the ways in which mathematics impacts these projects, and the ways in which the art communicates mathematics.

a ${\rm ii}_1$ factor $m$ is called prime if it cannot be decomposed as a tensor product of ${\rm ii}_1$ subfactors. naturally, if $m$ is not prime, one asks if $m$ can be uniquely factored as a tensor product of prime subfactors. the first result in this direction is due to ozawa and popa in 2003, who gave a large class of groups $\mathcal{c}$ such that for any $\gamma_1, \dots, \gamma_n \in \mathcal{c}$, the associated von neumann algebra $l(\gamma_1) \,\overline{\otimes}\, \cdots \,\overline{\otimes}\, l(\gamma_n)$ is uniquely factored in a strong sense. this talk will consider the case where $\gamma$ is icc group that is measure equivalent to a product of non-elementary hyperbolic groups. in joint work with daniel drimbe and adrian ioana, we show that any such $\gamma$ admits a unique decomposition $\gamma = \gamma_1 \times \gamma_2 \times \cdots \times \gamma_n$ such that $l(\gamma) = l(\gamma_1) \,\overline{\otimes}\, \cdots \,\overline{\otimes}\, l(\gamma_n)$ is uniquely factored in sense of ozawa and popa. using this, we provide the first examples of prime ${\rm ii}_1$ factors arising from lattices in higher rank lie groups.

heavy-tailed distributions are ubiquitous in modern statistical analysis and machine learning problems. this talk gives a simple principle for robust high-dimensional statistical inference via an appropriate shrinkage on the data. this widens the scope of high-dimensional techniques, reducing the moment conditions from sub-exponential or sub-gaussian distributions to merely bounded second moment. as an illustration of this principle, we focus on robust estimation of the low-rank matrix from the trace regression model. it encompasses four popular problems: sparse linear models, compressed sensing, matrix completion, and multi-task regression. under only bounded $2+\delta$ moment condition, the proposed robust methodology yields an estimator that possesses the same statistical error rates as previous literature with sub-gaussian errors. we also illustrate the idea for estimation of large covariance matrix. the benefits of shrinkage are also demonstrated by financial, economic, and simulated data.

the analog of the riemann curvature tensor for noncommutative tori manifests itself in the term $a_4$ appearing in the heat kernel expansion of the laplacian of curved metrics. this talk presents a joint work with alain connes, in which we obtain an explicit formula for the $a_4$ associated with a general metric in the canonical conformal structure on noncommutative two-tori. our final formula has a complicated dependence on the modular automorphism of the state or volume form of the metric, namely in terms of several variable functions with lengthy expressions. we verify the accuracy of the functions by checking that they satisfy a family of conceptually predicted functional relations. by studying the latter abstractly we find a partial differential system which involves a natural flow and action of cyclic groups of order two, three and four, and we discover symmetries of the calculated expressions with respect to the action of these groups. at the end, i will illustrate the application of our results to certain noncommutative four-tori equipped with non-conformally flat metrics and higher dimensional modular structures.

let $p: ... \to c_2 \to c_1 \to p^1$ be a $\mathbb{z}_p$-cover of the
projective line over a finite field of characteristic $p$ which ramifies
at exactly one rational point. in this talk, we study the $p$-adic
newton slopes of l-functions associated to characters of the galois
group of $p$. it turns out that for covers $p$ such that the genus of $c_n$ is a quadratic polynomial in $p^n$ for $n$ large, the newton slopes are uniformly distributed in the interval $[0,1]$. furthermore, for a large class of such covers $p$, these slopes behave in an even more regular way. this is joint work with hui june zhu.

recently, venkatesh introduced the derived hecke algebra to explain extra endomorphisms on the cohomology of arithmetic manifolds: the crucial local construction is a derived version of the spherical hecke algebra of a reductive p-adic group. working with p-torsion coefficients, we will describe a satake homomorphism for the derived spherical hecke algebra of a p-adic group. this will allow us to understand its structure well enough to attack some global questions, which are work in progress.

the hodge theory of surfaces provides a link between enumerative geometry and modular forms, via the cohomological theta correspondence. i will present an approach to studying the gromov-witten invariants of weierstrass fibrations over $p^2$, proving part of a conjectural formula coming from topological string theory.

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.