Karoly Böröczky (Alfréd Rényi Institute of Mathematics)
The Isoperimetric inequality, the Brunn-Minkowski theory, and the Lp Minkowski problem
10:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Christoforos Neofytidis (Ohio State University)
Numerical invariants monotone under the Gromov order abstract
Abstract:
We suggest a Kodaira dimension for geometric manifolds (in the sense of Thurston and more generally) and study its monotonicity with respect to the existence of maps of non-zero degree, as well as its relationship with other monotone invariants, such as the simplicial volume. Part of this talk is based on joint work with Weiyi Zhang.
10:30 • Université de Genève, Conseil Général 7-9, Room 1-05
Pascal Fong (Université Paris-Saclay)
Title T.B.A.
10:30 • Universität Basel, Spiegelgasse 5, SR 05.002
Hamza Hanbali (Monash University, Melbourne, Australia)
Mean-variance longevity risk-sharing for annuity contracts abstract
Abstract:
This paper investigates longevity risk-sharing as a solution to the sustainability and affordability problems in the annuity market, and in particular how much longevity risk could be transferred back to policyholders under a mean-variance utility framework. First, it provides dynamic risk-sharing rules for annuities. Second, it studies the contract properties from the perspectives of both the provider and individual policyholders. Third, in policyholders’ decision, it highlights and accounts for two levels of uncertainty and two levels of correlation induced by systematic longevity risk. Fourth, it provides necessary and sufficient conditions on the premium loading and the share of transferred risk, such that both parties prefer risk-sharing. The analytical and numerical results of the paper offer a deeper understanding of the effects of systematic and diversifiable risks on those preferences, and show that the products presented in this paper are suitable retirement solutions.
11:00 • EPF Lausanne, UniL campus, Extranef - 109
Prof. Dr. Daniel Smania (Instituto de Ciências Matemáticas e de Computação USP)
Deformations of one-dimensional dynamical systems abstract
Abstract:
Perhaps one of the main features of one-dimensional dynamics (either real or complex) is that the theory of deformations is rich. By this we mean that the topological classes of such maps often are infinite dimensional manifolds, but with finite codimension. They are kind of "almost" structurally stable! Moreover for smooth families of maps inside a given topological class the associated family of conjugacies also moves in a smooth way. There are various applications in the study of renormalisation theory and linear response theory. There is a nice theory in complex dynamics but for real maps with finite smoothness on the interval our current understanding is far behind the complex setting. We will discuss recent developments obtained in joint work with Clodoaldo Ragazzo but also some results with Viviane Baladi and Amanda de Lima. Ergodic theory will be a crucial tool.
13:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Goncalo Dos Reis (University of Edinburgh)
High order splitting methods for stochastic differential equations abstract
Abstract:
In this talk, we will discuss how ideas from rough path theory can be leveraged to develop high order numerical methods for SDEs. To motivate our approach, we consider what happens when the Brownian motion driving an SDE is replaced by a piecewise linear path. We show that this procedure transforms the SDE into a sequence of ODEs – which can then be discretized using an appropriate ODE solver. Moreover, to achieve a high accuracy, we construct these piecewise linear paths to match certain “iterated” integrals of the Brownian motion. At the same time, the ODE sequences obtained from this path-based approach can be interpreted as a splitting method, which neatly connects our work to the existing literature. For example, we show that the well-known Strang splitting falls under this framework and can be modified to give an improved convergence rate. We will conclude the talk with a couple of examples, demonstrating the flexibility and convergence properties of our methodology.
14:00 • Université de Genève, Conseil Général 7-9, Room 1-05
Finite sample rates for logistic regression with small noise or few samples abstract
Abstract:
The logistic regression estimator is known to inflate the magnitude of its coefficients if the sample size n is small, the dimension p is (moderately) large or the signal-to-noise ratio 1/\\sigma is large (probabilities of observing a label are close to 0 or 1). With this in mind, we study the logistic regression estimator with p << n/\\log n, assuming Gaussian covariates and labels generated by the Gaussian link function, with a mild optimization constraint on the estimator\'s length to ensure existence. We provide finite sample guarantees for its direction, which serves as a classifier, and its Euclidean norm, which is an estimator for the signal-to-noise ratio. We distinguish between two regimes. In the low-noise/small-sample regime (n\\sigma <= p\\log n), we show that the estimator\'s direction (and consequentially the classification error) achieve the rate (p\\log n)/n - as if the problem was noiseless. In this case, the norm of the estimator is at least of order n/(p\\log n). If instead n\\sigma >= p\\log n, the estimator\'s direction achieves the rate \\sqrt{\\sigma p\\log n/n}, whereas its norm converges to the true norm at the rate \\sqrt{p\\log n/(n\\sigma^3)}. As a corollary, the data are not linearly separable with high probability in this regime. The logistic regression estimator allows to conclude which regime occurs with high probability. Therefore, inference for logistic regression is possible in the regime n\\sigma >= p\\log n. In either case, logistic regression provides a competitive classifier.This is joint work with Sara van de Geer.
14:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.2
Marc Abboud (Université de Rennes 1)
Dynamical degrees of endomorphisms of affine surfaces abstract
Abstract:
Let $f: \\mathbf C^2 \\rightarrow \\mathbf C^2$ be a polynomial transformation. The dynamical degree of $f$ is defined as $\\lim_n (\\text{deg} f^n)^{1/n}$, where $\\text{deg} f^n$ is the degree of the $n$-th iterate of $f$. In 2007, Favre and Jonsson showed that the dynamical degree of any polynomial endomorphism of $\\mathbf C^2$ is an algebraic integer of degree $\\leq 2$. For any affine surface, there is a definition of the dynamical degree that generalizes the one on the affine plane. We show that the result still holds in this case: the dynamical degree of an endomorphism of any complex affine surface is an algebraic integer of degree $\\leq 2$. In this talk, I will give an overview of the recent results obtained on dynamical degrees on algebraic varieties and explain the key tools of the proof.
14:15 • EPF Lausanne, Salle MA A3 30
Peter Friz (TU Berlin)
Rough analysis of local stochastic volatility models abstract
Abstract:
Local stochastic volatility models are a class of (in general non-Markovian) diffusion models that have become standard in the financial industry, largely due to an efficient McKean-Vlasov particle calibration algorithm, mathematically not yet fully understood. In this talk we address the pricing problem, that is, the computation of certain expectations. We proceed by a partial conditioning that exhibits a rough semimartingale & SDE structure, topic of recent investigations with P. Zorin-Kranich, K. Lê , A. Hocquet and the speaker. In particular, we exploit partial (rough) Markovianity to compute certain conditional expectations via rough partial differential equations (RPDEs), later randomized with martingale rough paths. As far as the speaker knows, the resulting (formal) SPDE is beyond existing SPDE theory. Joint work with P. Bank (TU Berlin), C. Bayer and L. Pelizzari (both WIAS Berlin).
15:00 • EPF Lausanne, Bernoulli Center
Dr. Lennart Döppenschmitt (Universität Zürich)
What is... a brane? abstract
Abstract:
I will attempt to give an answer in three parts. First I will argue that branes appear naturally as boundary conditions in quantum field theories, in particular, we will observe how Lagrangian branes are suitable boundary conditions in the topological A-model. We will proceed with a swift introductory "101" on generalized complex geometry to properly fortify our understanding of branes in the mathematical sense. Well-equipped with the proper language, we will go through three and a half examples among which is a new(ish) perspective on Kähler metrics as branes (yes, metrics can be branes). If time permits we will finish with a quick look at the role that branes play in the story of mirror symmetry à la Strominger-Yau-Zaslow.
16:15 • UZH Zentrum, Building KO2, Room F 150