Math Colloquium

### Speaker: Peter J. Olver, University of Minnesota

### Title: Equivalence, Invariants, Puzzles, and Cancer

Location: Tg-227 (Tæknigarður, 2. hæð)

Time: Friday 9 June at 16:00

### Abstract:

A fundamental issue in computer vision is recognizing when two objects in an image are the “same”. The underlying mathematical apparatus for studying such equivalence problems is transformation group (or, more generally, groupoid) theory. Cartan’s solution to the equivalence and symmetry problem for submanifolds relies on the associated geometric invariants, through what is now known as the differential invariant signature. Furthermore, the new equivariant approach to the method of moving frames provides a systematic and algorithmic approach that can be applied to very general Lie group and even Lie pseudo-group actions. The talk will conclude with recent applications to automated assembly of broken objects, such as jigsaw puzzles, and cancer detection.

PhD thesis defense rehearsal

### Speaker: Örn Arnaldsson, University of Minnesota

### Title: Involutive moving frames

Location: Tg-227 (Tæknigarður, 2. hæð)

Time: Tuesday 6 June at 10:50

### Abstract:

Cartan’s equivalence method and the method of the equivariant moving frames are the two best known methods for solving equivalence problems in differential geometry, differential equations, calculus of variations and control theory. My thesis demonstrates how the two methods are really two sides of the same coin and combines them in a powerful hybrid method that increases the computational efficiency of both of its progenitors. This novel viewpoint provides effortless proofs to some previously hard-to-prove results, such as the Lie-Tresse theorem. Furthermore, we obtain proof of a long standing conjecture of Cartan on the termination of his equivalence method.

Math Colloquium

### Speaker: Valentina Giangreco, University of Iceland

### Title: Non-analyticity of holographic Rényi entropy in Lovelock gravity

Location: Tg-227 (Tæknigarður, 2. hæð)

Time: Friday 26 May at 13:20

### Abstract:

In the first part of my talk I will introduce the definition of Rényi entropy, and some basic concepts of the so-called holographic principle (AdS/CFT). The second part of the talk is devoted to the analysis of certain black hole instabilities in gravity theories with higher derivative corrections (Lovelock gravity theories) and their implications for the holographic Rényi entropy.

Math Colloquium

### Speaker: Ayan Mukhopadhyay, Vienna University of Technology

### Title: The mogul pistes of non-equilibrium causal correlations in strongly interacting holographic systems: universal features and how they reveal the microscopic theory.

Location: V-158 (VR-II)

Time: Thursday 11 May at 13:30

### Abstract:

I will report exact calculations of holographic retarded (causal) correlation function away from equilibrium in states driven from one thermal equilibrium to another by a homogeneous energy injection. Our calculations reveal universal features of thermalization of the retarded propagator. Furthermore, I will discuss how the measurement of the non-equilibrium retarded propagator (possible by techniques such as pump-probe spectroscopy) enables us to learn a lot about the dual gravity description (i.e. the microscopic theory) by establishing novel connections between quantum information content of non-equilibrium density matrices and the classical dynamics of the apparent and event horizons in the dual geometries.

Math Colloquium

### Speaker: Jason Smith, University of Strathclyde

### Title: Poset Fibrations and Their Applications to Pattern Posets

Location: Tg-227 (Tæknigarður, 2. hæð)

Time: Friday 21 April at 13:30

### Abstract:

A poset fibration is a rank and order preserving surjective map between posets. It was shown by Quillen that many properties of posets can be maintained across a fibration, we introduce some of these results. Pattern occurrence has been studied on a wide range of combinatorial objects, and using the notion of pattern containment we can define a poset on these objects. Many such pattern posets have been studied in the literature and the results on these posets often have a similar theme. Using poset fibrations we introduce some results that show why such similarities arise between different pattern posets.

Math Colloquium

### Speaker: Friðrik Freyr Gautason, K.U. Leuven

### Title: Large field inflation in string theory

Location: Tg-227 (Tæknigarður, 2. hæð)

Time: Friday 7 April at 13:20

### Abstract:

I start by motivating that certain questions in cosmology, in particular dark energy and large field inflation, should be addressed in a quantum model that includes gravity such as string theory. I will give an overview how these questions are translated to dynamics in the extra dimensions of string theory and what challenges one encounters. I then present a novel model for inflation in string theory and discuss some of the stringent consistency constraints the parameters of the model must satisfy and how these constraints affect cosmological observables.

Math Colloquium

### Speaker: Tom Steentjes, Eindhoven University of Technology

### Title: Feedback stabilization of nonlinear systems: “universal” constructions towards real-life applications

Location: Tg-227 (Tæknigarður, 2. hæð)

Time: Friday 17 March at 13:20

### Abstract:

Various feedback stabilizers based on Sontag’s “universal” formula for stabilizing control laws are presented, incorporating restrictions inspired by real-life applications. The first main contribution is an extension of Sontag’s “universal” formula for positive nonlinear control systems. More specifically, an auxiliary function is introduced in the feedback interconnection, such that invariance of the positive orthant is retained for the system in closed loop with the “universal” stabilizer. We further state a “universal” event-based stabilizer for bounded controls and develop an extension of the controller for positive systems. In a motivating case study from systems biology, the methodology is shown to provide clinically realistic control inputs, which can be used for treatment in real life. The second main contribution is the construction of continuous and piecewise affine (CPA) feedback stabilizers for nonlinear control systems affine in the input, motivated by the ease of implementation of the resulting control law. A verification procedure for “universal” CPA stabilizers is provided, together with an alternative computational method for CPA stabilizers via linear programming. Two numerical examples are presented for illustration of the CPA method.

Short bio:

Tom Steentjes was born in Tilburg, the Netherlands, in 1993. He received his BSc degree in Electrical Engineering (Automotive) in 2014, from the Department of Electrical Engineering at Eindhoven University of Technology (TU/e). In 2016, he completed the Systems and Control master’s program at the TU/e. The MSc thesis, entitled “Feedback stabilization of nonlinear systems: ‘universal’ constructions towards real-life applications”, was supervised by dr. Alina Doban and dr. Mircea Lazar. The MSc degree was granted with distinction Cum Laude.

Math Colloquium

### Speaker: Paolo Zanardi, University of Southern California

### Title: Quantum algorithms for topological and geometric analysis of Big Data

Location: V-138 (VR-II)

Time: Monday 20 February at 10:00

### Abstract:

Extracting useful information from large data sets can be a daunting task. Topological methods for analysing data sets provide a powerful technique for extracting such information. Persistent homology is a sophisticated tool for identifying topological features and for determining how such features persist as the data is viewed at different scales. I will discuss quantum machine learning algorithms for calculating Betti numbers – the numbers of connected components, holes and voids – in persistent homology, and for finding eigenvectors and eigenvalues of the combinatorial Laplacian. The algorithms provide an exponential speed-up over the best currently known classical algorithms for topological data analysis.

Reference: Seth Lloyd, Silvano Garnerone e Paolo Zanardi, Quantum algorithms for topological and geometric analysis of data, Nature Communications 7, 10138 (2016). See also: http://news.mit.edu/2016/quantum-approach-big-data-0125

Math Colloquium

### Speaker: Thomas Vallier, University of Iceland

### Title: Discussion on Bootstrap percolation on a random graph coupled with a lattice

Location: Tg-227 (Tæknigarður, 2. hæð)

Time: Friday 3 February at 13:20

### Abstract:

will give an informal seminar on the paper by Janson, Kozma, Ruszinko and Sokolov. In the paper “Bootstrap percolation on a random graph coupled with a lattice”, the authors consider a set of vertices \(N\)x\(N\) on a lattice. On top of that, every vertex shares a link with any other vertex with a probability inversely proportional to there block distance and independently of any other link. That means for two vertices \(u\) and \(v\) at distance \(d P(u,v) = c/(Nd)\) where \(c\) is a constant.

The authors derive the diameter of the of the graph in a very elegant way which we will unfortunately not focus on. Instead, we will discuss the other part which deals with the following cellular automaton with threshold \(k\):

1. Start with a set of active (or infected depending on the terminology you want to use) vertices at time \(0\), \(A(0)\).

2. Any vertex which has at least k active vertices in its closed neighbourhood (including itself) at time \(t\) is set as active at time \(t+1\). Notice that two vertices are neighbours if they share an edge in common either from the lattice or from the random connections.

Otherwise, if it has less than \(k\) active vertices in the closed neighbourhood then the vertex is set as inactive at time \(t+1\).

3. We repeat the process over time.

The authors study the “mean field approximation” which simplifies the problem by averaging the links over space. Under this assumption, the problem simplifies into a 1-dimensional dynamical system. The authors derive the fixed points of the dynamical system for

\(k \leq 3\).

There are good indications that the process becomes more interesting (that means more complicated) for larger \(k\). I would like to discuss that subject with you all if you’re curious about it. Anyone who’s interested in talking and sharing ideas is very welcome.

Math Colloquium

### Speaker: Benedikt Stufler, École normale supérieure de Lyon

### Title: The asymptotic geometric shape of random combinatorial trees

Location: TG-227 (Tæknigarður, 2. hæð)

Time: Friday 27 January at 13:20

### Abstract:

In his pioneering papers in the early 90s, Aldous established the continuum random tree (CRT) as the scaling limit of random labelled trees. He conjectured that the CRT also arises as scaling limit of trees considered up to symmetry. The convergence of random Pólya trees, that is, unlabelled rooted trees, was confirmed around 20 years later by Haas and Miermont, and we discuss an alternative proof by Panagiotou and the speaker. The second part of the talk treats random unlabelled unrooted trees and discusses a very general result that allows for a transfer of asymptotic properties of rooted trees to unrooted trees, in particular the convergence toward the CRT.