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Women in Probability 2019MI 00.07.014 (Boltzmannstr. 3, 85748 Garching)

Sprecherinnen: Luisa Andreis (Weierstraß-Institut), Gioia Carinci (Delft University of Technology), Hanna Döring (Universität Osnabrück), Lisa Hartung (Johannes Gutenberg-Universität Mainz), Cecile Mailler (University of Bath), Eveliina Peltola (University of Geneva), Elena Pulvirenti (Universität Bonn), Ecaterina Sava-Huss (Graz University of Technology)

Local Limit Theorems And Applications2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

In the talk I shall discuss a lattice local limit theorem for Gibbs-Markov processes and sketch applications to the Poincare exponent of some Fuchsian group and local times for continued fractions and fractal Gaussian noise.

Geometrizing rates of convergence under local differential privacyBC1 2.01.10 (Parkring 11, 85748 Garching)

One of the many new challenges for data analysis in the information age is the increasing concern of privacy protection. A particularly fruitful approach to data protection that has recently received a lot of attention, is the notion of `local differential privacy’. The idea is that each data providing individual releases only a randomly perturbed version of its original data, where the randomization mechanism is required to satisfy a precise privacy definition. In this talk, we discuss the impact of a local differential privacy guarantee on the quality of statistical estimation. In this setup, the objective is not only to come up with an optimal estimation procedure that efficiently recovers information from the privatized observations, but also to devise a privatization mechanism that best facilitates subsequent estimation while respecting the required privacy provisions. In the general context of estimating linear functionals of the unknown true data generating distribution, we characterize the minimax rate of private estimation in terms of a certain modulus of continuity of the functional to be estimated and provide a construction of minimax rate optimal privatization mechanisms. Our analysis also allows for a quantification of the price of local differential privacy in terms of loss of statistical accuracy. This price appears to be highly problem dependent.

Approximability of eigenfunctions of electronic Schrödinger equation by tensor trainsMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

The N-body ground-state eigenfunction of an electronic Schrödinger equation is usually discretized in the Fock space of antisymmetrized tensors of one-particle functions. In this basis, the eigenfunction is a tensor which can be represented in a tensor train format, i.e. as a product of rank-3 tensors. Recently, the tensor train format (also known as matrix product states in quantum chemistry) were found to numerically give accurate approximations of the ground state of N-body Schrödinger equation. In this talk, some results on the approximability by tensor trains of the ground-state eigenfunction of an N-body Schrödinger equation will be presented. A precise characterization of the singular values of matrix reshapes of a single Slater determinant will be given, supported by numerical tests. Our results suggest a new scheme to label the one-particle function basis and improve the accuracy of the tensor train approximation. It is a joint work with G. Friesecke.

Finite Shadows of Infinite Groups: From Profinite Triviality to Profinite RigidityA027 (Theresienstraße 39, 80333 Mathematisches Institut, LMU)

There are many situations in geometry and group theory where it is natural, convenient or necessary to explore infinite groups via their finite-dimensional representations or their actions on finite objects. How much understanding can one gain about an infinite group in this way? Sometimes little, sometimes a lot. Can one construct algorithms that can recognise when two finitely presented groups have the same finite quotients,or decide whether a group has any finite-dimensional representations? Which groups that arise in nature are completely determined by their finite images? I shall sketch some of the rich history of these problems and explain how input from geometry and low-dimensional topology have transformed the subject in recent years.

Everyone is invited! Join us for coffee and tea in the common room (B448) half an hour before the talk.

Does the N-clock model approximate the XY model?MI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

In this seminar we will investigate the relationship between the N-clock model and the XY model (at zero temperature) through a Gamma-convergence analysis as both the number of particles and N diverge. The N-clock model is a two-dimensional nearest neighbors ferromagnetic spin system, in which the values of the spin field are constrained to lie in a set of N equispaced points of the unit circle. For N large enough, it is usually considered as an approximation of the XY model, for which the spin field is allowed to attain all the values of the unit circle. By suitably rescaling the energy of the N-clock model, we will illustrate how its thermodynamic limit strongly depends on the rate of divergence of N with respect to the number of particles. We shall see that the N-clock model turns out to be a good approximation of the XY model only for N sufficiently large; in other regimes of N, we will show with the aid of cartesian currents that its asymptotic behavior can be described by an energy which may concentrate on geometric objects of different dimensions. The results presented in the talk are based on a work in collaboration with M. Cicalese and M. Ruf.

Balancing power systems with electric vehicles – large-scale stability perspectiveMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

The talk focuses on the provision of power system frequency control with electric vehicles. Frequency control is the primary control function for balancing the electrical system and is normally provided by conventional power plants. Grid operators are confident with the response of conventional units and control loops are designed taking their characteristics in mind. However, conventional units are getting displaced by renewable sources, meaning that there are fewer sources to actually provide balancing services to the grid. Storage system, and specifically electric vehicles, are candidates to fulfill such a role.

A key issue to keep in mind, however, is that electric vehicles are primarily designed for driving, not for providing grid services. This creates uncertainties in grid operators who need to procure such balancing reserve with units that are not designed for that.

The seminar presents both field experiment results and theoretical investigations, taking the power system of the Danish island of Bornholm as test case.

Recent Developments in Quantum Error Correcting CodesMI 00.06.11 HS3 (Boltzmannstr. 3, 85748 Garching)

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 geometryMI 00.06.11 HS3 (Boltzmannstr. 3, 85748 Garching)

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.

Mixing time of the upper triangular matrix walk over Z/mZ. 2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

We study a natural random walk over the upper triangular matrices, with entries in Z/mZ, generated by steps which add or subtract row $i + 1$ to row $i$. We show that the mixing time of the lazy random walk is $O(n^2m^2)$ which is optimal up to constants. This generalizes a result of Peres and Sly and answers a question of Stong and of Arias-Castro, Diaconis and Stanley. This is joint work with Allan Sly.

Existence and regularity for a non-linear Koiter shell interacting with an incompressible fluidMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

We study the unsteady Navier Stokes equations in three dimensions interacting with a non-linear flexible shell of Koiter Type. The latter one constitutes a moving part of the boundary of the physical domain of the fluid. This leads to a coupled system of PDEs. The fluid-structure interaction is captured in form of a weak momentum equation, where the space of testfunctions is part of the concept of solutions. We introduce new methods that allow to prove higher regularity estimates for the shell. Due to the improved regularity estimates it is then possible to extend the known existence theory to weak solutions for non-linear Koiter shell models. This is a work that was achieved in collaboration with B. Muha (Univ. of Zagreb).

Cluster expansion for Poisson-saddlepoint approximation B 252 (Theresienstr. 39, 80333 München)

The intensity of a Gibbs point process is an intractable function in the model parameters. Therefore, approximations such as the Poisson-saddlepoint approximation are needed. With the help of cluster expansion, we show that the Poisson-saddlepoint approximation converges to the true intensity if the dimension of the underlying metric space goes to infinity. Cluster expansion is a method originally established in mathematical physics. If the interactions are weak, the hope is that the Gibbs point process is close to the non-interacting Poisson point process and we may hopefully capture the correction terms by a convergent power series in the activity parameter z. The correction terms of the intensity depend on a sum over all connected graphs. The correction terms of the Poisson-saddlepoint approximation, on the other hand, depend only on the sum over all trees. We show that as the dimension goes to infinity, the contributions of the connected graphs, which are not trees, vanishes.

Particle coherent structures in incompressible fluid flowsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

The motion of small particles in incompressible steady flows can be strongly characterized by the creation of particle attractors. Several dissipative mechanisms may lead to the accumulation of particles, such as inertial forces due to the particle-fluid density mismatch or to finite-size effects in the bulk. Other dissipative effects which may lead to attractors are due to gravity and Basset history forces. The particle accumulation mechanism of interest in this talk relies on the interaction of the particles with undeformable boundaries, very frequent in boundary-driven flows. For dilute suspensions, these accumulation phenomena are successfully modeled as single-particle attractions and are strongly correlated to the topology of the fluid flow without particles. When the accumulations are in form of periodic or quasi-periodic patterns, they are termed finite-size Lagrangian coherent structures, since they are the result of the Lagrangian topology of the fluid flow and of the finite-size effect of the particle near the boundary. Boundary-driven flows will be considered to demonstrate the relative importance of the several dissipation mechanisms which affect the particle motion. A comparison between numerical simulations and experiments will be presented to further strengthen the theoretical understanding of finite-size Lagrangian coherent structures.

Global Mild Solutions for Rough Evolution EquationsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

In this talk we introduce a pathwise approach to investigate mild solutions for parabolic stochastic evolution equations (SEEs) driven by multiplicative rough noise. To this aim we combine techniques from M. Gubinelli and S. Tindel (2010) with arguments employed by M. Garrido, K. Lu and B. Schmalfuß (2015). Regarding the general algebraic framework developed by M. Gubinelli, we derive a Sewing Lemma which ensures the existence of the pathwise integral together with appropriate estimates. Since the ansatz is based on a first order Taylor approximation, these estimates in general are of quadratic type, which means that there is no Gronwall Lemma applicable. Moreover, in the last part of the talk we show how to obtain a local-in-time solution via a fixedpoint argument and finally, explain how to concatenate local-in-time solutions in order to receive a global-in-time solution. This talk is based on a joint work with Alexandra Neamµu (TU Munich).