Operads are a tool that formalise the definitions of algebras of different kinds. They allow to define associative, commutative, dg-, Lie and other algebras uniformly as algebras over certain operads. Some earliest instances of explicitly defined operads occurred in topology, where they have been introduced to describe the algebraic structure of iterated loop spaces elegantly. The relations these adhere to have led to the construction of A∞-algebras, E∞-algebras and the like. We shall see that instances of these naturally arise in topological and algebraic contexts and that they nicely complement the theory of dga-algebras.

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Title & abstracts to be found on the webpage:

https://www.groups.ma.tum.de/probability/veranstaltungen/women-in-probability-2020/

organized by Noam Berger (TUM), Diana Conache (TUM), Nina Gantert (TUM), Silke Rolles (TUM) and Sabine Jansen (LMU).

The Hastings-Levitov process, introduced by Hastings and Levitov in 1998, is a planar aggregation process in which at every time a new particle attaches itself to the existing cluster at a point which is determined by the harmonic measure. This model was studied extensively in recent years. The main advantage of this model is that its direct connection to complex analysis makes it tractable. The main disadvantage is some non-physical behaviour of the particle sizes. In this talk I will present a new half-plane variant of the Hastings-Levitov model, and will demonstrate that our variant, called the Stationary Hastings-Levitov, maintains the tractability of the original model, while avoiding the non-physical behavior of the particle sizes. The talk is based on joint work with Jacob Kagan, Eviatar Procaccia and Amanda Turner.

We study the fundamental group of certain random 2-dimensional cubical complexes which contain the complete 1-skeleton of the d-dimensional cube, and where every 2-dimensional square face is added independently with probability p. These are cubical analogues of Linial–Meshulam random simplicial complexes, and also simultaneously are 2-dimensional versions of bond percolation on the hypercube. Our main result is that if p ≤ 1/2, then with high probability the fundamental group of a random cubical complex is nontrivial, and if p > 1/2 then with high probability it is trivial. As a corollary, we get the same result for homology with any coefficient ring. We also study the structure of the fundamental group below the transition point, especially its free factorization.

TBA

In this talk, we discuss the relative interlevel set homology associated to a continuous function. This invariant is a continuously indexed version of the Mayer–Vietoris pyramid introduced by Carlsson, de Silva, and Morozov. We will discuss how the extended persistence diagram can be obtained from relative interlevel set homology and show that the isomorphism class of relative interlevel set homology is uniquely determined by the extended persistence diagram, due to Cohen-Steiner, Edelsbrunner, and Harer. To this end, we will discuss a decomposition theorem for relative interlevel set homology. The results presented are joint work with Ulrich Bauer and Magnus Botnan and are closely related to two articles by Bendich, Edelsbrunner, Morozov, and Patel as well as Berkouk, Ginot, and Oudot.

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Details im Link: https://www.fm.mathematik.uni-muenchen.de/teaching/teaching_summer_term_2020/seminars/oberseminar_finanz_2019/index.html

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We study how asset prices depend on “liquidity”, that is, the ease with which the assets can be traded. Equilibrium prices and the corresponding trading strategies can be characterised by coupled systems of forward-backward differential equations in this context. We present some first well-posedness results and discuss tractable approximations in the large-liquidity limit. These allow to study how liquidity or liquidity risk are priced over time and across different assets, and also provide testable implications for the impact a transaction tax would have on market volatility.

We consider random walks in balanced i.i.d. non-elliptic random environments (RWBRE). Similar as Brownian motion is related to the heat equation, RWBRE is related to random difference equations. We discuss a parabolic Harnack principle for these equations. The talk is based on joint work with Noam Berger.

I will start with a general motivation for cause-effect estimation and describe common challenges such as identifiability. We will then take a closer look at the instrumental variable setting and how an instrument can help for identification. Most approaches to achieve identifiability require one-size-fits-all assumptions such as an additive error model for the outcome. Instead, I will present a framework for partial identification, which provides lower and upper bounds on the causal treatment effect. Our approach leverages advances in gradient-based optimization for the non-convex objective and works in the most general case, where instrument, treatment and outcome are continuous. Finally, we demonstrate on a set of synthetic and real-world data that our bounds capture the causal effect when additive methods fail, providing a useful range of answers compatible with observation as opposed to relying on unwarranted structural assumptions.

We consider the following martingale dispersion result proved by Peres, Schapira and Sousi: If, up to time n, the jumps of a martingale are bounded from above by log(n)^a (with some positive a < 1) and the conditional variance of each jump is at least 1, then P(M_n = 0) gets arbitrarily small for sufficiently large n. After presenting the main ideas of the proof, we discuss why there is no such dispersion result in the case a=1. Peres, Schapira and Sousi used a more general version of this dispersion result to show that the following random walk W=(X,Y,Z) in Z^3 is transient: When visiting a vertex for the first time, Z changes by +/-1, while on later visits (X,Y) changes by (+/-1,0) or (0,+/-1). We look at this application as well and present the main ideas of the proof that W is transient. Everything is based on the following paper: Yuval Peres, Bruno Schapira, and Perla Sousi. Martingale defocusing and transience of a self-interacting random walk. Ann. Inst. H. Poincaré Probab. Statist., 52(3):1009–1022, 2016.