# Department Colloquium Summer 2019

## Talks in Summer Semester 2019

At the Colloquium of the Department of Mathematics international researchers report on their work. The colloquium will be held on Wednesdays in "Hörsaal 3" (MI 00.06.011). We invite all interested parties.

**These are the lecture dates in the summer semester 2019:**

22 May: Jean-David Fermanian, John v. Neumann Guest Professor (14:30)

12 June: Robert Calderbank (14:30) + Ngoc Tran (16:00)

3 July: Rayan Saab (14:30) + Melina Freitag (16:00)

17 July: Karlheinz Brandenburg (14:30) + Claus Scheiderer (16:00)

### Topics at the Department Colloquium

#### Audio and Acoustics Signal Processing: The Quest for High Fidelity continues

17 July,14:30 - 15:30

**Prof. Karlheinz Brandenburg (TU Ilmenau)**

The dream of high fidelity has lasted since more than 100 years. In the last decades, signal processing has contributed many new solutions and a vast amount of additional knowledge to this field. These include

- simple solutions like matrix multichannel systems,
- audio coding which changed the world of music distribution and listening habits
- active noise control
- active modification of room acoustics
- search and recommendation technologies to find your favourite music
- and many more …

The talk will cover topics of audio signal processing technologies in current use and new research as well as new future applications like personalized auditory realities (PARty).

#### Spectrahedral Shadows

17 July,16:00 - 17:00

**Prof. Claus Scheiderer (Universität Konstanz)**

A spectrahedron is the solution set of a linear matrix inequality (LMI), or equivalently, an affine-linear section of the cone of positive semidefinite matrices of some size. The linear images of spectrahedra are called spectrahedral shadows or projected spectrahedra. They are the feasible sets of semidefinite programming (SDP) and are convex semi-algebraic sets. We will discuss examples and constructions of spectrahedral shadows, and will also present examples of sets that fail to be a spectrahedral shadow. For optimizing over a concretely given spectrahedral shadow, the matrix size in the lifted LMI has strong influence on performance parameters. We will discuss some recent results on how large these matrix sizes have to be chosen.

#### Binary data embeddings: Theory and Applications

03 July, 14:30 - 15:30

**Prof. Rayan Saab (University of California)**

High dimensional and large data sets often impose a significant burden on applications both in terms of computational complexity and storage cost.

To reduce these burdens, it is often necessary to obtain lower dimensional, preferably binarized, representations of such data that simultaneously preserve important geometric properties, or even permit accurate reconstruction. Thinking of a set $X \subset \mathbb{R}^N$ as either a high-dimensional data set, or even as a class of signals (such as natural images), in this talk we present fast methods to replace points from $X$ with points in a lower-dimensional cube $\{\pm 1\}^m$. That is, we embed $X$ into the binary cube, and we endow the binary cube with a function (a pseudo-metric) that preserves Euclidean distances in the original space. We discuss applications of these ideas to compressed sensing, which deals with efficient data acquisition, as well as on-going work related to machine learning.

#### A low-rank approach to the solution of weak constraint variational data assimilation problems

03 July,16:00 - 17:00

**Dr. Melina Freitag (University of Bath)**

Large-scale variational data assimilation problems commonly arise in many applications like numerical weather prediction and oceanography. We give a brief introduction to common data assimilation methods. More specifically, weak constraint four-dimensional variational data assimilation is an important method for incorporating observations into a model. The system arising within the minimisation process can be formulated as a saddle point problem. In this talk we present a low-rank approach which exploits the structure of this system using techniques and theory from solving matrix equations. Numerical experiments with the linear advection-diffusion equation, and the nonlinear Lorenz-95 model demonstrate the effectiveness of a low-rank Krylov subspace solver.

#### Recent Developments in Quantum Error Correcting Codes

12 June, 14:30 - 15:30

**Prof. Robert Calderbank (Duke University)**

Quantum computers promise to be more capable of solving certain problems than any classical computer. Quantum error correction is fundamental to moving these devices out of the lab and to their becoming generally programmable. In this talk, we will focus on stabilizer codes, which have played a central role in quantum information theory for more than 20 years, and on the family of CSS codes in particular.

There is a natural hierarchy of unitary gates, starting with the Pauli gates at the first level, the Clifford gates at the second level, and then higher-level gates. Universal quantum computation requires that we augment gates from the Clifford group with non-Clifford operators such as the T-gate, which is a diagonal gate corresponding to p/8 rotation from the 3rd level of the hierarchy. Elements in the kth level of this hierarchy act by conjugation on Pauli matrices to produce a result in the (k-1)th level. The structure of diagonal operators in this hierarchy is of particular interest.

We introduce a new family of diagonal operators from the kth level of the Clifford hierarchy defined by quadratic forms over the ring of integers modulo 2k. This family is rich enough to capture all 1- and 2-local and certain higher locality diagonal gates in the Clifford hierarchy. Since 1- and 2-local gates are natural operations to realize in the lab, the quadratic forms illuminate a world of logical operations that might be easier to implement. We provide explicit algebraic descriptions of their action on Pauli matrices, establishing a natural recursion to diagonal unitaries from lower levels.

Our recursion makes it possible to characterize stabilizer codes preserved by tensor products of T and T-1 gates. It unifies the many code constructions known to support T gates, and leads to several new codes and code families. These include a Reed-Muller CSS family that contains a [[64,15,4]] code, where the logical operation realized by physical transversal T appears to be an order 2 diagonal gate in the 15th level of the Clifford hierarchy.

#### Tell the truth and never lose with tropical geometry

12 June, 16:00 - 17:00

**Prof. Ngoc Tran (University of Texas)**

Mechanism design is a branch of game theory and economics, aims at finding mechanisms for collective decision making, such that the outcome maximizes social welfare, while assuming that the participants (be it corporations, businesses or individuals) are selfish. The most basic class of mechanisms are incentive compatible (IC): one where the best strategy for individual participants is to be truthful and not cheat. That is, a cheater cannot win over a truth-teller. Do such mechanisms exist? How to optimize over such class? It turns out that these questions can be answered with tropical convex geometry. We review some known results, and list many open problems of interest to both fields. We do not assume background in either tropical geometry or mechanism desisgn.

Based on joint work with Robert Crowell and Bo Lin.

#### An overview of the goodness-of-fit test problem for copulas

22 May, 14:30-15:30

**Prof. Dr. Jean-David Fermanian, ENSAE Paris / John von Neumann guest professor**

In dependence modeling, a typical problem is related to the selection of "the best", or at least "a convenient" dependence structure, i.e. a copula. We will review the main “omnibus procedures” for goodness-of-fit testing for copulas: tests based on the empirical copula process, on probability integral transformations, on Kendall’s dependence function, etc, and some corresponding reductions of dimension techniques. The problems of finding asymptotic distribution-free test statistics and the calculation of reliable p-values will be discussed. Some particular cases, like convenient tests for time-dependent copulas, for Archimedean or extreme-value copulas, will be tackled too.

Further information can be found on the overview page of the Colloquium of the Department of Mathematics.