Veranstaltungen

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.

Toric contact manifolds provide an interesting class of contact manifolds. In this mini-​course we will introduce them, show how the ones with zero first Chern class can be determined by certain integral convex polytopes, called toric diagrams, and how to directly read relevant contact invariants from these toric diagrams. Plenty of hands-​on examples and some applications will be provided.

More information: https://math.ethz.ch/fim/activities/minicourses.htmlcall_made

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.

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.

Given a space parametrizing branched covers of smooth curves (with some given genus, degree, and ramification profile data), there are two well-known compactifications - the admissible covers of Harris and Mumford and the stable maps of (relative) Gromov-Witten theory. Both of these compactifications give rise to cycles on moduli spaces of stable curves. I will describe work in progress giving a way to translate between the cycles produced by the two compactifications. Part of this talk presents joint work with Q. Zhao.

We will briefly review the representations of groups in infinite-dimensional hyperbolic spaces. In particular, the family of SL2(R) representations will be studied, an operation on them that resembles a convex combination will be introduced and a partial classification will be presented.

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Farewell Lecture

One of the main motivations behind derived differential geometry is to deal with singularities arising from zero loci or intersections of submanifolds. Both cases can be considered as fiber products of manifolds which may not be smooth in classical differential geometry. Thus we need to extend the category of differentiable manifolds to a larger category in which one can talk about "homotopy fiber products". In this talk, we will discuss a solution to this problem in terms of dg manifolds. The talk is mainly based on a joint work with Kai Behrend and Hsuan-Yi Liao.

Toric contact manifolds provide an interesting class of contact manifolds. In this mini-​course we will introduce them, show how the ones with zero first Chern class can be determined by certain integral convex polytopes, called toric diagrams, and how to directly read relevant contact invariants from these toric diagrams. Plenty of hands-​on examples and some applications will be provided.

More information: https://math.ethz.ch/fim/activities/minicourses.htmlcall_made

In the first part of this talk, we will recall the building blocks of deep learning, framing the learning problem as an optimization problem solved in practice by gradient descent. This first part will be very accessible and self-contained. Then we will attempt to convey how surprising it is that deep learning works so well given the extreme complexity of its solution space, pointing toward the existence of an implicit regularization mechanism self-selecting the simpler solutions that generalize best ahead of the more complex ones that do not perform well. At last, we will outline a recent approach attempting to uncover such an implicit regularization mechanism based on the backward error analysis of the gradient descent scheme.

In the recent years, the most important developments in Optimization were related to clarification of abilities of the higher-order methods. These schemes have potentially much higher rate of convergence as compared to the lower-order methods. However, the possibility of their implementation in the form of practically efficient algorithms was questionable during decades. In this talk, we discuss different possibilities for advancing in this direction, which avoid all standard fears on tensor methods (memory requirements, complexity of computing the tensor components, etc.). Moreover, in this way we get the new second-order methods with memory, which converge provably faster than the conventional upper limits provided by the Complexity Theory.

In this talk, I will describe a variant of the Hewitt and Savage theorem for laws of exchangeable finite families of random variables as well as their marginals. This representation reveals the role of some universal polynomials of measures which contain correlated correction terms and capture the geometry of extreme points of symmetric laws and their marginals. I will also describe possible applications to some multi-marginal optimal problems with symmetries. This is based on a joint work with Gero Friesecke and Daniela Vögler (TU Munich).

We will start by defining a notion of geometric complexity for neural networks based on intuitive notions of volume and energy. This will be motivated by the visualization of training sequences in the case of simple 1d neural regressions. Then we will explain why for neural networks the optimization process creates a pressure to keep the network geometric complexity low. Additionally, we will see that many other common heuristics in the training of neural networks (from initialization schemes to explicit regularization strategies) have as a side effect to also keep the geometric complexity of the learned solutions low. We will conclude by explaining how this points toward a preference toward a form of harmonic maps built in the commonly used training and tuning heuristics in deep learning.

I'll discuss efforts to understand the solubility of random Diophantine equations, focusing on recent joint work with Sofos and Teräväinen about the integral Hasse principle for random norm form equations defined over the integers.