Math Colloquium

### Speaker: Marston Conder, University of Auckland

### Title: Experimental Algebra and Combinatorics

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

Time: Thursday 13 July at 11:00

### Abstract:

Some 40 years after the computer-based proof of the 4-Colour Theorem by Appel and Haken, there is still a degree of healthy skepticism about the use of computers to prove nice theorems in mathematics. But there is a distinction between proofs that are highly dependent on computation (verifiable or otherwise), and the use of computer-based experimentation to analyse and construct examples, to produce data that might exhibit patterns from which conjectures can be drawn and tested, or to investigate a range of possible scenarios — subsequently leading to theorems that can be proved by hand.

In this talk I will describe a range of instances of experimental computations involving finite and infinite groups that have led to unexpected but theoretically provable discoveries about discrete objects possessing a high degree of symmetry. These include discoveries about the genus spectra of particular classes of regular maps on surfaces, the smallest regular and chiral polytopes, and various kinds of edge-transitive graphs. Such examples highlight the value of experimental computation, and the surprising outcomes it can often produce.

Math Colloquium

### Speaker: Daniel Fernandez Moreno, University of Iceland

### Title: Entanglement entropy at non-equilibrium in holography

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

Time: Friday 7 July at 13:20

### Abstract:

In recent years, holographic models have proved to be successful at studying far-from-equilibrium physics. This provides a new approach to studying quantum quenches in strongly coupled systems. In this talk, based on ArXiv:1705.04696, I will focus on the local quench-like time evolution obtained by joining two 1+1 dimensional heat baths at different temperatures. I will present results for the entanglement entropy of different entangling regions obtained by adapting the time-dependent Hubeny-Rangamani-Takayanagi prescription. The interest of this study relies on the presence of emergent collective behavior, which can provide insight into the interplay between quantum effects and out of equilibrium physics.

Math Colloquium

### Speaker: Christopher Kellett, University of Newcastle (Australia)

### Title: Back and Forth in Lyapunov’s Second Method: Non-Uniform Subtleties

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

Time: Thursday 6 July at 13:20

### Abstract:

Lyapunov’s second or direct method provides an easy-to-check sufficient condition for stability properties of equilibria. The converse question – given a stability property, does there exist an appropriate Lyapunov function? – has been fundamental in differentiating and classifying different stability properties, particularly with regards to “uniform” stability.

In this talk, I will review the usual textbook definitions for Lyapunov functions for time-varying systems and describe where they are deficient. Some interesting new sufficient (and probably necessary) conditions pop up along the way.

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.