Veranstaltungen

I will define a certain type of properadic algebra called a Y-infinity algebra, which is an A-infinity algebra together with higher structure maps encoding a certain type of Poincaré duality structure. In particular, the algebra of chains on the based loop space of any oriented manifold is canonically endowed with this type of structure. By using the formalism of properadic Kaledin classes, we can study these algebraic structures and detect their formality, or lack thereof. I will also explain the relation between these algebraic structures and string topology operations, and how Y-infinity formality plays a role in understanding these operations. This talk is about joint works with M. Rivera, Z. Wang and C. Emprin.

The goal of this series of talks is to present a proof, at a higher level of detail than usual, of the following theorem. Let H be any smooth function on R^4 and let Y be any compact and regular level set. Then Y admits an infinite family of proper compact subsets that are invariant under the Hamiltonian flow, which in addition have dense union in Y. This improves on a 2018 result by Fish-Hofer, which showed the existence of one proper invariant subset. Related results were proved jointly with Dan Cristofaro-Gardiner for area-preserving surface diffeomorphisms and 3D Reeb flows. I will mention them if there is time.

The goal of this series of talks is to present a proof, at a higher level of detail than usual, of the following theorem. Let H be any smooth function on R^4 and let Y be any compact and regular level set. Then Y admits an infinite family of proper compact subsets that are invariant under the Hamiltonian flow, which in addition have dense union in Y. This improves on a 2018 result by Fish-Hofer, which showed the existence of one proper invariant subset. Related results were proved jointly with Dan Cristofaro-Gardiner for area-preserving surface diffeomorphisms and 3D Reeb flows. I will mention them if there is time.

The goal of these two talks is to discuss results on equivariant Gromov-Witten invariants of Hilbert schemes of points on C^2. In the first talk, we present a summary of results in genus 0 due to Okounkov-Pandharipande. We also discuss some aspects of determining higher genus theory using Givental-Teleman reconstruction. In the second talk, we discuss recent progress on a detailed study of genus 1 theory.

The cop numbers are two new integer-valued invariants of finitely generated groups defined in terms of a two player pursuit game played on Cayley graphs. We use the cop numbers to give a new characterization of hyperbolic groups and virtually free groups. We discuss potential applications and recent progress on remaining open questions. This is joint work with Kevin Klinge.

I will discuss a microlocal analysis approach to spectral theory on asymptotically Minkowski spaces both for scalar wave operators and also for Dirac type operators. This in turn gives rise to complex powers of the operators, allowing for the analysis of a spectral zeta function, relating its residues to geometric information. This is joint work with Nguyen Viet Dang and Michal Wrochna. Special time: Wednesday, 16:15 Special room: ETH LFW B3

Turbulent mixing induced by hydrodynamic instabilities occurs when two fluids of different densities, velocities, and viscosities interact. Theoretical, experimental, and numerical efforts to understand and predict the dynamics of hydrodynamic instabilities are very important for science and engineering applications. Statistical convergence and turbulence quantification are crucial for achieving reliable and accurate modeling and simulations. In this talk, we present an increasingly accurate and robust front-tracking/ghost-fluid method with higher-order weighted essentially non-oscillatory schemes used for the numerical simulations of Rayleigh-Taylor and Richtmyer-Meshkov Instabilities. We investigate the time evolution of velocity fields and fluctuations for different configurations to explore the scaling law of the energy spectrum.

A Theorem of Vitale, Molchanov and Wespi, states that the expected absolute determinant of a random matrix with independent columns is equal to the mixed volume of some convex bodies called zonoids. In the case where the columns of the matrix are centered Gaussian, the corresponding zonoids are ellipsoids and this was studied by Kabluchko and Zaporozhets. The case of non centered Gaussian vectors, however, remains relatively unstudied. The convex bodies we obtain in this case, which I call Gaussian zonoids, are not ellipsoids. In this talk, I will show you what a Gaussian zonoid looks like and how one can approximate it with an ellipsoid. At the level of random determinants, this allows to approximate expectation of non centered Gaussian determinants with centered ones. If time allows, I will show how this applies to Gaussian random fields. Namely, I will show how one can give a quantitative estimate of the concentration of a small Gaussian perturbation of a hypersurface.

The time at which renewable (e.g., solar or wind) energy resources produce electricity cannot generally be controlled. In many settings, consumers have some flexibility in their energy consumption needs, and there is growing interest in demand-response programs that leverage this flexibility to shift energy consumption to better match renewable production -- thus enabling more efficient utilization of these resources. We study optimal demand response in a model where consumers operate home energy management systems (HEMS) that can compute the "indifference set" of energy-consumption profiles that meet pre-specified consumer objectives, receive demand-response signals from the grid, and control consumer devices within the indifference set. For example, if a consumer asks for the indoor temperature to remain between certain upper and lower bounds, a HEMS could time use of air conditioning or heating to align with high renewable production when possible. Here, we show that while price-based mechanisms do not in general achieve optimal demand response, i.e., dynamic pricing cannot induce HEMS to choose optimal demand consumption profiles within the available indifference sets, pricing is asymptotically optimal in a mean-field limit with a growing number of consumers. Furthermore, we show that large-sample optimal dynamic prices can be efficiently derived via an algorithm that only requires querying HEMS about their planned consumption schedules given different prices. We demonstrate our approach in a grid simulation powered by OpenDSS, and show that it achieves meaningful demand response without creating grid instability. Mohammad Mehrabi, Omer Karaduman, Stefan Wager https://arxiv.org/abs/2409.07655

The time at which renewable (e.g., solar or wind) energy resources produce electricity cannot generally be controlled. In many settings, consumers have some flexibility in their energy consumption needs, and there is growing interest in demand-response programs that leverage this flexibility to shift energy consumption to better match renewable production -- thus enabling more efficient utilization of these resources. We study optimal demand response in a model where consumers operate home energy management systems (HEMS) that can compute the "indifference set" of energy-consumption profiles that meet pre-specified consumer objectives, receive demand-response signals from the grid, and control consumer devices within the indifference set. For example, if a consumer asks for the indoor temperature to remain between certain upper and lower bounds, a HEMS could time use of air conditioning or heating to align with high renewable production when possible. Here, we show that while price-based mechanisms do not in general achieve optimal demand response, i.e., dynamic pricing cannot induce HEMS to choose optimal demand consumption profiles within the available indifference sets, pricing is asymptotically optimal in a mean-field limit with a growing number of consumers. Furthermore, we show that large-sample optimal dynamic prices can be efficiently derived via an algorithm that only requires querying HEMS about their planned consumption schedules given different prices. We demonstrate our approach in a grid simulation powered by OpenDSS, and show that it achieves meaningful demand response without creating grid instability. Mohammad Mehrabi, Omer Karaduman, Stefan Wager https://arxiv.org/abs/2409.07655

Many pension plans and private retirement products contain annuity factors, converting the funds at some future time into lifelong income. In general model settings like for example the Li-Lee mortality model, analytical values for the annuity factors are not available and one has to rely on numerical techniques. Their computation typically requires nested simulations as they depend on the interest rate level and the mortality tables at the time of retirement. We exploit the flexibility and efficiency of feed-forward neural networks to value the annuity factors at the time of retirement. In a numerical study, we compare our deep learning approach to (least-squares) Monte-Carlo (LSMC) which can be represented as a special case of the neural network (NN).

The goal of these two talks is to discuss results on equivariant Gromov-Witten invariants of Hilbert schemes of points on C^2. In the first talk, we present a summary of results in genus 0 due to Okounkov-Pandharipande. We also discuss some aspects of determining higher genus theory using Givental-Teleman reconstruction. In the second talk, we discuss recent progress on a detailed study of genus 1 theory.