Virtual

Symmetry protected topological (SPT) phases have recently attracted a lot of
attention from physicists and mathematicians as a topological classification
scheme for gapped ground states. In this talk I will briefly introduce the
operator algebraic approach to SPT phases in the infinite-volume limit. In
particular, I will focus on the quasifree (free-fermionic) setting, where we

can adapt tools from algebraic quantum field theory to describe phases of
gapped ground states using K-homology and the coarse index.

Let X be a closed Riemannian manifold, and represent the algebra C(X) of continuous functions on X on the Hilbert space L^2(X) by multiplication.  Inspired by the heat kernel proof of the Atiyah-Singer index theorem, I'll explain how to describe K-homology (i.e. the dual theory to Atiyah-Hirzebruch K-theory) in terms of parametrized families of operators on L^2(X) that get more and more 'local' in X as time tends to infinity.

I'll then switch perspectives from C(X) -- the prototypical example of a commutative C*-algebra -- to noncommutative C*-algebras coming from discrete groups, and explain how the underlying large-scale geometry of the groups can give rise to approximate 'decompositions' of the C*-algebras.  I'll then explain how to use these decompositions and localization in the sense above to compute K-homology, and the connection to some conjectures in topology, geometry, and C*-algebra theory.

Over the complex numbers, the Bomolgorov-Tian-Todorev theorem asserts that Calabi-Yau varieties have unobstructed deformations, so any n^{th} order deformation extends to higher order.  We prove an analogue of this statement for the nicest kind of Calabi-Yau varieties in characteristic p, namely ordinary ones, using derived algebraic geometry. In fact, we produce canonical lifts to characteristic zero, thereby generalising results of Serre-Tate, Deligne-Nygaard, Ward, and Achinger-Zdanowic. This is joint work with Taelman.

Let $A$ be an $n$ by $n$ matrix or a bounded linear operator on a complex Hilbert space $(H, \langle \cdot , \cdot \rangle , \| \cdot \|)$. A closed set $\Omega \subset \mathbb{C}$ is a $K$-spectral set for $A$ if the spectrum of $A$ is contained in $\Omega$ and if, for all rational functions $f$ bounded in $\Omega$, the following inequality holds:
\[\| f(A) \| \leq K \| f \|_{\Omega} ,\]
where $\| \cdot \|$ on the left denotes the norm in $H$ and $\| \cdot \|_{\Omega}$ on the right denotes the $\infty$-norm on $\Omega$. A simple way to obtain a $K$ value for a given set $\Omega$ is to use the Cauchy integral formula and replace the norm of the integral by the integral of the resolvent norm:
\[f(A) = \frac{1}{2 \pi i} \int_{\partial \Omega} ( \zeta I - A )^{-1}
f( \zeta )\,d \zeta \Rightarrow
\| f(A) \| \leq \frac{1}{2 \pi} \left( \int_{\partial \Omega}
\| ( \zeta I - A )^{-1} \|~| d \zeta | \right) \| f \|_{\Omega} .\]
Thus one can always take
\[K = \frac{1}{2 \pi} \int_{\partial \Omega} \| ( \zeta I - A )^{-1} \| | d \zeta | .\]
In M. Crouzeix and A. Greenbaum, Spectral sets: numerical range and beyond, SIAM J. Matrix Anal. Appl., 40 (2019), pp. 1087-1101, different bounds on $K$ were derived.  I will show how these compare to that from the Cauchy integral formula for a variety of applications.  In case $A$ is a matrix and $\Omega$ is simply connected, we can numerically compute what we believe to be the optimal value for $K$ (and, at least, is a lower bound on $K$).  I will show how these values compare with the proven bounds as well.

(joint with  Michel Crouzeix and Natalie Wellen)
 

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I will discuss new work on practically popular algorithms, including the kernel polynomial method (KPM) and moment matching method, for approximating the spectral density (eigenvalue distribution) of an n x n symmetric matrix A. We will see that natural variants of these algorithms achieve strong worst-case approximation guarantees: they can approximate any spectral density to epsilon accuracy in the Wasserstein-1 distance with roughly O(1/epsilon) matrix-vector multiplications with A. Moreover, we will show that the methods are robust to *in accuracy* in these matrix-vector multiplications, which allows them to be combined with any approximation multiplication algorithm. As an application, we develop a randomized sublinear time algorithm for approximating the spectral density of a normalized graph adjacency or Laplacian matrices. The talk will cover the main tools used in our work, which include random importance sampling methods and stability results for computing orthogonal polynomials via three-term recurrence relations.

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Solving large-scale Bayesian inverse problems governed by complex models suffers from the twin difficulties of the high dimensionality of the uncertain parameters and computationally expensive forward models. In this talk, we focus on 1. reducing the computational cost when solving these problems (via joint parameter and state dimension reduction) and 2. accounting for the error due to using a reduced order forward model (via Bayesian Approximation Error (BAE)).  To reduce the parameter dimension, we exploit the underlying problem structure (e.g., local sensitivity of the data to parameters, the smoothing properties of the forward model, the fact that the data contain limited information about the (infinite-dimensional) parameter field, and the covariance structure of the prior) and identify a likelihood-informed parameter subspace that shows where the change from prior to posterior is most significant. For the state dimension reduction, we employ a proper orthogonal decomposition (POD) combined with the discrete empirical interpolation method (DEIM) to approximate the nonlinear term in the forward model. We illustrate our approach with a model ice sheet inverse problem governed by the nonlinear Stokes equation for which the basal sliding coefficient field (a parameter that appears in a Robin boundary condition at the base of the geometry) is inferred from the surface ice flow velocity. The results show the potential to make the exploration of the full posterior distribution of the parameter or subsequent predictions more tractable.

This is joint work with Ki-Tae Kim (UC Merced), Benjamin Peherstorfer (NYU) and Tiangang Cui (Monash University).

Function approximation, as a goal in itself or as an ingredient in scientific computing, typically relies on having a basis. However, in many cases of interest an obvious basis is not known or is not easily found. Even if it is, alternative representations may exist with much fewer degrees of freedom, perhaps by mimicking certain features of the solution into the “basis functions" such as known singularities or phases of oscillation. Unfortunately, such expert knowledge typically doesn’t match well with the mathematical properties of a basis: it leads instead to representations which are either incomplete or overcomplete. In turn, this makes a problem potentially unsolvable or ill-conditioned. We intend to show that overcomplete representations, in spite of inherent ill-conditioning, often work wonderfully well in numerical practice. We explore a theoretical foundation for this phenomenon, use it to devise ground rules for practitioners, and illustrate how the theory and its ramifications manifest themselves in a number of applications.

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Courserequirements: Basicmathematicalanalysis.

Examination and grading: The exam will consist in the presentation of some previously as- signed article or book chapter (of course the student must show a good knowledge of those issues taught during the course which are connected with the presentation.).

SSD: MAT/05 Mathematical Analysis
Aim: to make students aware of smooth and non-smooth controllability results and of some

applications in various fields of Mathematics and of technology as well.

Course contents:

Vector fields are basic ingredients in many classical issues of Mathematical Analysis and its applications, including Dynamical Systems, Control Theory, and PDE’s. Loosely speaking, controllability is the study of the points that can be reached from a given initial point through concatenations of trajectories of vector fields belonging to a given family. Classical results will be stated and proved, using coordinates but also underlying possible chart-independent interpretation. We will also discuss the non smooth case, including some issues which involve Lie brackets of nonsmooth vector vector fields, a subject of relatively recent interest.

Bibliography: Lecture notes written by the teacher.

Aimed: An introduction to the nonlinear theory of hyperbolic PDEs, as well as its close connections with the other areas of mathematics and wide range of applications in the sciences.

Aimed: An introduction to the nonlinear theory of hyperbolic PDEs, as well as its close connections with the other areas of mathematics and wide range of applications in the sciences.