we will start by reviewing recent developments in random walks on homogeneous spaces. in a second part, we will discuss the notion of a $h$-expanding probability measure on a connected semisimple lie group $h$. as we shall see, for a $h$-expanding $\mu$ with $h < g$, on the one hand, one can obtain a description of $\mu$-stationary probability measures on the homogeneous space $g / \lambda$ ($g$ lie group, $\lambda$ lattice) using the measure classification results of eskin-lindenstrauss, and on the other hand, the recurrence techniques of benoist-quint and eskin-mirzakhani-mohammadi can be adapted to this setting. with some further work, these allow us to deduce equidistribution and orbit closure description results simultaneously for a class of subgroups which contains zariski-dense subgroups and further epimorphic subgroups of $h$. if time allows, we will see how, utilizing an idea of simmons-weiss, these also allow us to deduce birkhoff genericity of a class of fractal measures with respect to certain diagonal flows, which, in turn, has applications in diophantine approximation problems. \\
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joint work with roland prohaska and ronggang shi.

we consider the problem of approximating high dimensional functions using shallow neural networks, and more generally by sparse linear combinations of elements of a dictionary. we begin by introducing natural spaces of functions which can be efficiently approximated in this way. then, we derive the metric entropy of the unit balls in these spaces, which allows us to calculate optimal approximation rates for approximation by shallow neural networks. this gives a precise measure of how large this class of functions is and how well shallow neural networks overcome the curse of dimensionality. finally, we describe an algorithm which can be used to solve high-dimensional pdes using this space of functions.

suppose l is a link in s\^{}3, we show that $\pi_1(s^3-l)$ admits an irreducible meridian-traceless representation in su(2) if and only if l is not the unknot, the hopf link, or a connected sum of hopf links. as a corollary, $\pi_1(s^3-l)$ admits a (not necessarily meridian-traceless) irreducible representation in su(2) if and only if l is neither the unknot nor the hopf link. this result generalizes a theorem of kronheimer and mrowka to the case of links. the proof is based on singular instanton floer theory and an observation about finite simple graphs.
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this is joint work with yi xie.

we present a direct unified analytic proof of the classical boundary harnack principle for solutions to linear uniformly elliptic equations in either divergence or non-divergence form. the proof extends also to (appropriate) h\"older domains. the strategy also applies to the parabolic context. applications of the bhp to free boundary problems are discussed."

the growth of random surfaces has attracted a lot of attention in probability theory in the last ten years, especially in the context of the kardar-parisi-zhang (kpz) equation. most of the available results are for exactly solvable one-dimensional models. in this talk i will present some recent results for models that are not exactly solvable. in particular, i will talk about the universality of deterministic kpz growth in arbitrary dimensions, and if time permits, a necessary and sufficient condition for superconcentration in a class of growing random surfaces.

from polynomial to deep neural network approximation in high dimensions approximating multi-dimensional functions from pointwise samples is a ubiquitous task in data science and scientific computing. this task is made intrinsically difficult by the presence of four contemporary challenges: (i) the target function is usually defined over a high- or infinite-dimensional domain; (ii) generating samples can be very expensive; (iii) samples are corrupted by unknown sources of errors; (iv) the target function might take values in a function space. in this talk, we will show how these challenges can be substantially overcome thanks to the ``blessings" of sparsity and monte carlo sampling.
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first, we will consider the case of sparse polynomial approximation via compressed sensing. focusing on the case where the target function is smooth (e.g., holomorphic), but possibly highly anisotropic, we will show how to obtain sample complexity bounds only mildly affected by the curse of dimensionality, near-optimal accuracy guarantees, stability to unknown errors corrupting the data, and rigorous convergence rates of algebraic and exponential type. then, we will illustrate how the mathematical toolkit of sparse polynomial approximation via compressed sensing can be employed to obtain a practical existence theorem for deep neural network (dnn) approximation of high-dimensional hilbert-valued functions. this result shows not only the existence of a dnn with desirable approximation properties, but also how to compute it via a suitable training procedure in order to achieve best-in-class performance guarantees. we will conclude by discussing open research questions.
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the presentation is based on joint work with ben adcock, casie bao, nick dexter, sebastian moraga, and clayton g. webster.

a classical transcendence result of schneider on the modular
$j$-invariant states that, for $\tau \in \mathbb{h}$, both $\tau$ and
$j(\tau)$ are in $\overline{\mathbb{q}}$ if and only if $\tau$ is
contained in an imaginary quadratic extension of $\mathbb{q}$. the space
$\mathbb{h}$ has a natural interpretation as a parameter space for
$\mathbb{q}$-hodge structures (or, in this case, elliptic curves), and
from this perspective the imaginary quadratic points are distinguished
as corresponding to objects with maximal symmetry. this result has been
generalized by cohen and shiga-wolfart to more general moduli of hodge
structures (corresponding to abelian-type shimura varieties), and by
ullmo-yafaev to higher dimensional loci of extra symmetry (special
subvarieties), where bialgebraicity is intimately connected with the
pila-zannier approach to the andre-oort conjecture.
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in this talk, i will discuss work in progress with christian klevdal on
an analogous bialgebraicity characterization of special subvarieties in
scholze's local shimura varieties and more general diamond moduli of
$p$-adic hodge structures.

this talk is about optimizing polynomial functions under constraints. a general method is to apply the moment-sos hierarchy of semidefinite programming relaxations. the convergence is based on various positivstellensatz. closely related polynomial optimization is convex algebraic geometry.
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it concerns geometric properties of convex semialgebraic sets through semidefinite programming. we are going to review basic results for these topics.