Colloquia 2018

Colloquia Spring 2018
Daniel Persson (Chalmers/GU)

The dark side of the moon: new connections in geometry, number theory and physics
The word ‘moonshine’ refers to surprising and deep connections between various parts of mathematics and physics. The most famous example is known as monstrous moonshine and was discovered in the late 70’s, but took about 20 years to be fully understood. In a nutshell, the monster describes the intricate symmetries of a gigantic mathematical object, and moonshine shows that it somehow knows about special types of functions in a completely different field, namely number theory. What is even more striking is that the explanation for this connection comes from physics, more precisely string theory. In recent years we have uncovered a host of new examples of such moonshine phenomena that add additional layers to the story, in particular pointing to connections with complex geometry. In this talk I will give a general introduction to moonshine and describe some of these exciting new developments. Is there an underlying principle that unifies all moonshine phenomena?

Ruth Baker (Oxford)

Collective cell invasion: mathematical models and biological insights
Cell invasion is fundamental to embryonic development, tissue regulation and regeneration, and the progression of many diseases, including cancer; yet the mechanisms that drive populations of cells to invade distinct targets are poorly understood. This talk will utilise the embryonic neural crest as an exemplar to study mechanisms of collective cell invasion. I will present results from an interdisciplinary collaboration that demonstrate the roles of population heterogeneity, and cell-cell and cell-microenvironment interactions, in driving invasion. I will also discuss future theoretical challenges in the field, and outline progress my group has made towards tackling them.

Anna-Karin Tornberg (KTH)

Highly accurate integral equation based methods for surfactant laden drops in two and three dimensions
In micro-fluidics, at small scales where inertial effects become negligible, surface to volume ratios are large and the interfacial processes are extremely important for the overall dynamics. Integral equation based methods are attractive for the simulations of e.g. droplet-based microfluidics, with tiny water drops dispersed in oil, stabilized by surfactants.

We have developed highly accurate numerical methods for drops with insoluble surfactants, both in two and three dimensions. In this talk I will discuss some fundamental challenges that we have addressed, that are also highly relevant for other applications.

18-04-23 (N.B. in Pascal)
Julie Rowlett (Chalmers/GU)

The theory of games and microbe ecology
Microbes are critically important to life on Earth. In particular, planktonic microbes collectively generate as much organic matter and oxygen as all terrestrial plants combined. Understanding the abundance, distribution and diversity of microbes in general and plankton in particular is critical to predicting their globally important functions!

Now, this is a math talk, not a biology talk. The motivation for the mathematics, however, comes from biology. The talk is suitable for all mathematicians, from second-year undergraduates to emeritus professors. I will explain a simple game-theoretic model my co-author, Susanne Menden-Deuer, and I have been investigating to understand competition among microbes for limited resources. We've made an interesting discovery: a mathematical theorem. This theorem may provide new explanatory power for some of the rather unique and puzzling ecological features of microbes. I will explain the theorem and its proof, which relies only on elementary mathematics, yet nonetheless appears to offer some new insights.

18-06-04 (N.B. at 1315 in Pascal)
Valentino Tosatti (Northwestern University)

Calabi-Yau manifolds 
Calabi-Yau manifolds are a class of compact complex manifolds that enjoys remarkable geometric properties, which makes them widely-studied objects in several areas of mathematics.  They owe their name to a conjecture posed by E. Calabi in 1954  which was finally solved by S.-T. Yau in 1976. I will give a gentle introduction to these objects and their basic properties, and describe some lines of current research and open problems about Calabi-Yau manifolds.

Colloquia Autumn 2018
Carl Lindberg (Chalmers/GU)

The mathematical challenges of Deep Learning
AI, and in particular its subfield Deep Learning, has experienced an extreme hype over the last decade. We will describe the historical development of the field up to present day, and indicate some very exciting mathematical challenges at its frontier.

18-09-17 (N.B. in Pascal)
Annika Lang (Chalmers/GU)

Simulation of random models: where stochastic analysis meets high performance computing
Many phenomena in natural sciences can be expressed in mathematical formulas via models, e.g. by partial differential equations. To account for uncertainties in nature, noise is added to the models via random variables, random fields, or stochastic processes. Computing statistical properties of such noisy models requires both theoretical mathematical analysis and big computing power. In this presentation, the interaction between basic mathematical concepts, efficient algorithms, and parallel computing is discussed. These relations need to be properly established in order to obtain reliable results in acceptable computing time. The stochastic heat equation serves as example that will guide us through the talk.

18-10-07 (N.B. in Pascal)
Gunnar Carlsson (Stanford)

Topological Methods in Machine Learning and Artificial Intelligence
Since roughly the year 2000, it has been recognized that topology (the mathematical study of shape) provides a collection of powerful techniques for modeling and analysing complex data. The methodology that has been developed has been applied to a wide range of domains, including financial risk modeling, text analysis of various kinds of corpora, model evaluation and improvement, genomics, materials science in industrial processes, cybersecurity, drug discovery, and many others. Additionally, it can be applied to make "black box" AI and ML methodologies more transparent and accessible. I will discuss these methods, with numerous examples.

Tomas Persson (Lund University)

Potentials, energies and Hausdorff dimension
The Hausdorff dimension of a set is a number, not necessarily an integer, which measures the fractal dimension of the set. There is a classical connection between Riesz-potentials, Riesz-energies and Hausdorff dimension. Otto Frostman (Lund) proved in his 1935 thesis that if E is a set and μ is a measure with support in E, then the Hausdorff dimension of E is at least s if the s-dimensional Riesz-energy of μ is finite. I will first define the above mentioned concepts and then give Frostman's result and some of its applications.

Later in the talk, I will mention some new methods where Hausdorff dimension is calculated using potentials and energies with inhomogeneous kernels. Some applications are in stochastic geometry and dynamical systems.

Sandra Di Rocco (KTH)

Algebraic Numerics and Sampling: Algebraic Geometry with a view towards applications
Many problems in science can be described by polynomial equations. The solution set of the corresponding polynomial system is referred to as an algebraic geometrical model for the problem.  When the solution set consists of isolated points the model is easy to describe and even to visualise. For higher dimensional solution spaces, deeper and more sophisticated geometrical and numerical techniques are required. Algebraic Geometry is an area of mathematics with a long standing tradition whose main objects of study are algebraic geometrical models. 

This talk will present two examples where classical algebraic geometry has played and plays a fundamental role: The 6 revolute chain linkage (robot arm) and algebraic sampling (data clouds). Classical and modern theory as well as future challenges will be explained.

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