Let X be a complex algebraic variety containing a point x. One of the central ideas of deformation theory is that the local structure of X near the point x can be encoded by a differential graded Lie algebra. In this talk, Jacob Lurie will explain this idea and discuss some generalizations to more exotic contexts.

# Past Colloquia

One of the big ideas in linear algebra is {\em eigenvalues}. Most matrices become in some basis {\em diagonal} matrices; so a lot of information about the matrix (which is specified by $n^2$ matrix entries) is encoded by by just $n$ eigenvalues. The fact that lots of different matrices can have the same eigenvalues reflects the fact that matrix multiplication is not commutative.

I'll look at how to make these vague statements (``lots of different matrices...") more precise; how to extend them from matrices to abstract symmetry groups; and how to relate abstract symmetry groups to matrices.

Eigenvectors of square matrices are central to linear algebra. Eigenvectors of tensors are a natural generalization. The spectral theory of tensors was pioneered by Lim and Qi around 2005. It has numerous applications, and ties in closely with optimization and dynamical systems. We present an introduction that emphasizes algebraic and geometric aspects

The question of deriving Fluid Mechanics equations from deterministic

systems of interacting particles obeying Newton's laws, in the limit

when the number of particles goes to infinity, is a longstanding open

problem suggested by Hilbert in his 6th problem. In this talk we shall

present a few attempts in this program, by explaining how to derive some

linear models such as the Heat, acoustic and Stokes-Fourier equations.

This corresponds to joint works with Thierry Bodineau and Laure Saint

Raymond.

Most networks and graphs encountered in empirical studies, including internet and web graphs, social networks, and biological and ecological networks, are very sparse. Standard spectral and linear algebra methods can fail badly when applied to such networks and a fundamentally different approach is needed. Message passing methods, such as belief propagation, offer a promising solution for these problems. In this talk I will introduce some simple models of sparse networks and illustrate how message passing can form the basis for a wide range of calculations of their structure. I will also show how message passing can be applied to real-world data to calculate fundamental properties such as percolation thresholds, graph spectra, and community structure, and how the fixed-point structure of the message passing equations has a deep connection with structural phase transitions in networks.

The talk will consider three well-defined problems which can be interpreted as mathematical tests of the physical reality of black holes: Rigidity, stability and formation of black holes.

There has been a great deal of attention paid to "Big Data" over the last few years. However, often as not, the problem with the analysis of data is not as much the size as the complexity of the data. Even very small data sets can exhibit substantial complexity. There is therefore a need for methods for representing complex data sets, beyond the usual linear or even polynomial models. The mathematical notion of shape, encoded in a metric, provides a very useful way to represent complex data sets. On the other hand, Topology is the mathematical sub discipline which concerns itself with studying shape, in all dimensions. In recent years, methods from topology have been adapted to the study of data sets, i.e. finite metric spaces. In this talk, we will discuss what has been

done in this direction and what the future might hold, with numerous examples.

I will review Bott's classical periodicity result about topological K-theory (with period 2 in the case of complex K-theory, and period 8 in the case of real K-theory), and provide an easy sketch of proof, based on the algebraic periodicity of Clifford algebras. I will then introduce the `higher real K-theory' of Hopkins and Miller, also known as TMF. I'll discuss its periodicity (with period 576), and present a conjecture about a corresponding algebraic periodicity of `higher Clifford algebras'. Finally, applications to physics will be discussed.

Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! Various techniques, usually going under the common name of “renormalisation” have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will tip our toes into some of the mathematical aspects of these techniques and we will see how they have recently been used to make precise analytical statements about the solutions of some equations whose meaning was not even clear until recently.