## Juli – August 2018

This seminar is cancelled.

This seminar is cancelled.

Synaptic time-dependent plasticity (STDP) is a biological mechanism which changes the strength of the connections between two neurons depending on the timing of the spikes in the pre- and postsynaptic neurons. Many studies relate STDP to the development of input selectivity and temporal coding, but time and energy efficiency is usually not studied. This work focuses on the property of STDP to reduce latencies, which has only been briefly addressed (Song, Miller & Abbot, Nat. Neuroscience 2001). As a trivial example, suppose three presynaptic neurons consistently trigger a postsynaptic spike; by STDP, their strengths increase to the point where only two synapses are necessary. Since the first two synapses always arrive before the third, the postsynaptic spike is triggered earlier. We extend this notion to populations of neurons and to account for inhibitory plasticity. Our work relates the system-level goal of speeding computation to a mechanistic, neuron level rule.

An elliptic ruled surface is a 4-manifold satisfying the condition that it has a holomorphic fibration over an elliptic curve with fibers that are projective lines. Every elliptic ruled surface is algebraic, and, in particular, a Kaehler surface. In this talk I would like to discuss the symplectomorphism group of elliptic ruled surfaces. More precisely, we will show that every symplectomorphism that is smoothly isotopic to the identity is isotopic to the identity within the symplectomorphism group.

We give examples of smooth plane quartics over Q with complex multiplication over bar{Q} by a maximal order with primitive CM type. We describe the required algorithms as we go: these involve the reduction of period matrices, the fast computation of Dixmier-Ohno invariants, and reconstruction from these invariants. Finally, we discuss some of the reduction properties of the curves that we obtain.

Persistent homology is a tool used to analyse topological features of data. In this talk, I will describe a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations — barcode to path, path to tensor series — results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves high performance on several classification benchmarks.

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY

The uniform probability measure on a convex polytope induces piecewise polynomial densities on the projections of that polytope. For a fixed combinatorial type of simplicial polytopes, the moments of these measures are rational functions in the vertex coordinates. We study projective varieties that are parametrized by finite collections of such rational functions. Our focus lies on determining the prime ideals of these moment varieties. Special cases include Hankel determinantal ideals for polytopal splines on line segments, and the relations among multisymmetric functions given by the cumulants of a simplex. In general, our moment varieties are more complicated, and they offer nice challenges for both numerical and symbolic computing in algebraic geometry. This is joint work with Kathlen Kohn and Boris Shapiro.

We introduce a novel representation of structured polynomial ideals, which we refer to as chordal networks. The sparsity structure of a polynomial system is often described by a graph that captures the interactions among the variables. Chordal networks provide a computationally convenient decomposition into simpler (triangular) polynomial sets, while preserving the underlying graphical structure. We show that many interesting families of polynomial ideals admit compact chordal network representations (of size linear in the number of variables), even though the number of components is exponentially large. Chordal networks can be effectively used to obtain several properties of the variety, such as its dimension, cardinality, components, and radical ideal membership. We apply our methods to examples from algebraic statistics and vector addition systems; for these instances, algorithms based on chordal networks outperform existing techniques by orders of magnitude.