In an ongoing work, we introduce the notion of isometric orbit equivalence for probability measure preserving actions of marked groups. This notion asks the Schreier graphings defined by the actions of the marked groups to be isomorphic. In the first part of the talk, we will prove that pmp actions of a marked group whose Cayley graph has a discrete automorphisms group are rigid up to isometric orbit equivalence. In a second time, we will explain how to construct pmp actions of the free group that are isometric orbit equivalent but not conjugate.

Efficient optimization has become one of the major concerns in data analysis. There has been a lot of focus on first-order optimization algorithms because of their low cost per iteration. In 1983, Nesterov's Accelerated Gradient method (NAG) was shown to converge in $O(1/k^2)$ to the minimum of the convex objective function $f(x)$, improving on the $O(1/k)$ convergence rate exhibited by the standard gradient descent methods, which is the phenomenon referred to as acceleration. It was shown that NAG limits to a second order ODE, as the time step goes to 0, and that the objective function $f(x(t)$) converges to its optimal value at a rate of $O(1/t^2)$ along the trajectories of this ODE. In this talk, we will discuss how the convergence of $f(x(t))$ can be accelerated in continuous time to an arbitrary convergence rate $O(1/t^p)$ in normed spaces, by considering flow maps generated by a family of time-dependent Bregman Lagrangian and Hamiltonian systems which is closed under time resca
ling. We will then discuss how this variational framework can be exploited together with the time-invariance property of the family of Bregman Lagrangians using adaptive geometric integrators to design efficient explicit algorithms for symplectic accelerated optimization. Finally, we will discuss briefly the generalization from normed spaces to Riemannian manifolds.

The Invariant Subspace Problem is one of the most famous problems in Operator Theory, and is concerned with the search of non-trivial, closed, invariant subspaces for bounded operators acting on a separable Banach space. Considerable success has been achieved over the years both for the existence of such subspaces for many classes of operators, as well as for non-existence of invariant subspaces for particular examples of operators. However, for the most important case of a separable Hilbert space, the problem is still open.
\\
\\
A natural, related question deals with the existence of invariant subspaces for perturbations of bounded operators. These types of problems have been studied for a long time, mostly in the Hilbert space setting. In this talk I will present a new approach to these ``perturbation'' questions, in the more general setting of a separable Banach space. I will focus on the recent history, presenting several new results that were obtained along the way with this new approach, and examining their connection and relevance to the Invariant Subspace Problem.

In the 1960's, Kervaire and Milnor boiled down the problem of counting smooth structure on spheres of dimension greater than 4 to the computation of stable homotopy groups of spheres and the Kervaire invariant one problem. In the following decades, the elements of Kervaire invariant one whose dimension are less or equals to 62 are shown to exist, and finally, Hill, Hopkins and Ravenel in their 2016 paper show that the Kervaire invariant one elements doesn't exist for dimension larger or equals to 254, leaving the 126-dimensional case open.
\\
\\
The $C_2$-equivariant Real bordism spectrum and its norms are crucial in HHR's solution, and computation of them is a central topic in equivariant stable homotopy theory. In this talk, I will explore two aspects of Real bordism and its norms:
\\
\\
1. How computation in Real bordism helps us to understand Lubin-Tate E-theories at p = 2. In particular, we can understand almost all actions of finite subgroups of the Morava stablizer groups on E-theories in homotopy.
\\
\\
2. The relation between Real bordism and the Segal conjecture. This relation allows us to bring new tools and perspective into this equivariant computation, and we will show how a spectral sequence based on (Real) topological Hochschild homology can help in understanding Real bordism and its norms.
\\
\\
This talk is based on joint work with Beaudry, Hill, Lawson, Meier and Shi.

One of the largest problems in Quantum Computation is how you deal with errors. Alexei Kitaev invented a method whereby we can use the discretization of a manifold to encode logical information in a subspace of the Hilbert space that corresponds to the homology of the Manifold itself. This has been vastly generalized, but we will restrict to looking at his original example of the toric code. We will go through how it can be used as an error-correcting code and several methods on how to actually compute on such a code. If time permits we can discuss recent results where expander graphs (and in general combinatorial methods) are used to construct a very good code which is then used to construct a manifold that solves a certain problem in differential topology.

There has been a growing interest in the study of nonlocal models as more general and sometimes more realistic alternatives to the conventional PDE models. We will give an introduction to nonlocal models in this talk. In particular, we will focus on the nonlocal models with a finite range of nonlocal interactions, which serve as bridges connecting the classical PDEs, nonlocal discrete models and the fractional differential equations. This talk will cover topics including nonlocal modeling, nonlocal calculus and numerical analysis for the nonlocal models.

Modern machine learning methods (i.e. neural networks) have been very successful in tasks such as image classification and speech recognition, but have been shown to be extremely brittle to small, adversarially-chosen perturbations of their inputs. This is a critical issue in many deep learning applications (e.g. object detection, robotic perception, ranking and recommendation, etc.). In this talk, I will provide an overview of the problem of adversarial robustness, formally introduce some general principles (what we know and what we don't know about this phenomenon), and discuss heuristic solutions (methods that appear to work in practice) and recent certification techniques (how do we provably - and efficiently - guarantee robustness?).

Lojasiewicz inequalities are a popular tool for studying the stability of geometric structures. For mean curvature flow, Schulze used Simon's reduction to the classical Lojasiewicz inequality to study compact tangent flows. For round cylinders, Colding and Minicozzi instead used a direct method to prove Lojasiewicz inequalities. We'll discuss similarly explicit Lojasiewicz inequalities and applications for other shrinking cylinders and Clifford shrinkers.

We provide a new general upper bound for the minimal L2-worst-case recovery error in the framework of RKHS, where only n function samples are allowed. This quantity can be bounded in terms of the singular numbers of the compact embedding into the space of square integrable functions. It turns out that in many relevant situations this quantity is asymptotically only worse by square root of log(n) compared to the singular numbers. The algorithm which realizes this behavior is a weighted least squares algorithm based on a specific set of sampling nodes which works for the whole class of functions simultaneously. These points are constructed out of a random draw with respect to distribution tailored to the spectral properties of the reproducing kernel (importance sampling) in combination with a sub-sampling procedure coming from the celebrated proof of Weaver's conjecture, which was shown to be equivalent to the Kadison-Singer problem. For the above multivariate setting, it is still a fundamental open problem whether sampling algorithms are as powerful as algorithms allowing general linear information like Fourier or wavelet coefficients. However, the gap is now rather small. As a consequence, we may study well-known scenarios where it was widely believed that sparse grid sampling recovery methods perform optimally. It turns out that this is not the case for dimensions d greater than 2.
\\
\\
This is joint work with N. Nagel and M. Schaefer from TU Chemnitz.

Given an elliptic curve E, Kolyvagin used CM points on
modular curves to construct a system of classes valued in the Galois
cohomology of the torsion points of E. Under the conjecture that not
all of these classes vanish, he gave a description for the Selmer group
of E. This talk will report on recent work proving new cases of
Kolyvagin's conjecture. The methods follow in the footsteps of Wei
Zhang, who used congruences between modular forms to prove Kolyvagin's
conjecture under some technical hypotheses. We remove many of these
hypotheses by considering congruences modulo higher powers of p. The
talk will explain the difficulties associated with higher congruences of
modular forms and how they can be overcome. I will also provide an
introduction to the conjecture and its consequences, including a `converse theorem': algebraic rank one implies analytic rank one.

Our work was motivated by the analysis projects using the linked US SEER-Medicare database to study mortality in men of age 65 years or older who were diagnosed with prostate cancer. Such data sets contain up to 100,000 human subjects and over 20,000 claim codes. For studying mortality the number of deaths are the ``effective'' sample size, so here we are in the situation of p is greater than n which is referred to as having high-dimensional predictors. In addition, a patient might die of cancer, or of other causes such as heart disease etc. These are referred to as competing risks. How to best perform prediction which inevitably involves variable selection for this type of complex survival data had not been previously investigated. Interest may also lie in comparing treatments such as radical prostatectomy versus conservative treatment. In this case the data were obviously not randomized with regard to the treatment assignments, and confounding most likely exists, possibly even beyond the commonly captured clinical variables in the SEER database. We will showcase research work done by our former PhD students from the UCSD Math Dept to account for such unobserved confounding, as well as efforts to make use of the high dimensional claims codes which have been shown to contain rich information about the patients survival.

Numerical algebraic geometry studies methods to approach problems in algebraic geometry numerically. Especially, finding roots of systems of equations using theory in algebraic geometry involves symbolic algorithm which requires expensive computations. However, numerical techniques often provides faster methods to tackle these problems. We establish numerical techniques to approximate roots of systems of equations and ways to certify its correctness.
\\
\\
As techniques for approximating roots of systems of equations, homotopy continuation method will be introduced. Since numerical approaches rely on heuristic method, we study how to certify numerical roots of systems of equations. Krawczyk method from interval arithmetic and Smale's alpha theory will be used as main paradigms for certification. Furthermore, as an approach for multiple roots, we establish the local separation bound of a multiple root. For a regular quadratic multiple zero, we give their local separation bound and study how to certify an approximation of such multiple roots.

This talk, the first on the third paper https://arxiv.org/abs/2009.03243 of a series, is partly a continuation of talks given in the fall.
See: http://www.math.ucsd.edu/\~{}benchow/cc-seminar.html
\\
\\
Some review will be given to make the talks more self-contained.