L3

Part of the series "What do historians of mathematics do?"  
In the last year of 14th century, a French mathematician/geometer Jean Mignot, was called from Paris to help with the construction of the Cathedral of Milan. Thus was created one of the most famous stories about how mathematics literally supports great works of art, helping them stand the test of time. This talk will look at some patterns that begin to become apparent in the investigations of the relationship between architecture and mathematics and the creativity that is common to the pursuit of both. I will present the case on how this may matter to someone who is interested in the history of mathematics. To make this more intelligible, I will partly talk also of my personal journey in investigating this relationship and the issues I have researched and written about, and how these in turn changed my view of the nature of mathematics education. 

Part of the series "What do historians of mathematics do?"  

I will address this question by turning to another: "What is algebra?"  In answering this second question, and surveying the way that the answer changes as we move through the centuries, I will highlight some of the problems that face historians of mathematics when it comes to interpreting historical mathematics, and give a flavour of what it means to study the history of mathematics.

Localization and holography are powerful approaches to the computation of supersymmetric observables. The computations may, however, include divergences. Therefore, one needs renormalization schemes preserving supersymmetry. I will consider minimal gauged supergravity in five dimensions to demonstrate that the standard holographic renormalization scheme breaks supersymmetry, and propose a set of non-standard boundary counterterms that restore supersymmetry. I will then show that for a certain class of solutions the improved on-shell action correctly reproduces an intrinsic observable of four-dimensional SCFTs, the supersymmetric Casimir energy.

We study D3 brane theories that are described as deformations of N=2 SCFTs. They arise at the self-intersection of a 7-brane in F-Theory. As we shall explain, the associated string junctions and their monodromies can be studied via torus knots or links. The monodromy reduces (potentially different) flavor algebras of dual deformations of N=2 theories and projects out charged states, leading to N=1 SCFTs. We propose an explanation for these effects in terms of an electron-monopole-dyon condensate.

Regenerative medicine offers great hope in curing many currently untreatable diseases. Tissue engineering and stem cell therapy are the two main components of regenerative medicine. In this talk, I will discuss how engineering can make contributions to this highly interdisciplinary field, including biomaterials as 3D scaffolds, bioreactor design, and stem cell bioprocessing.

Motivated by a problem in quasiconformal mapping, we introduce a new type of problem in complex analysis, with its roots in the mathematical physics of the Bose-Einstein condensates in superconductivity.The problem will be referred to as \emph{geometric zero packing}, and is somewhat analogous to studying Fekete point configurations.The associated quantity is a density, denoted  $\rho_\C$ in the planar case, and $\rho_{\mathbb{H}}$ in the case of the hyperbolic plane.We refer to these densities as \emph{discrepancy densities for planar and hyperbolic zero packing}, respectively, as they measure the impossibility of atomizing the uniform planar and hyperbolic area measures.The universal asymptoticvariance $\Sigma^2$ associated with the boundary behavior of conformal mappings with quasiconformal extensions of small dilatation is related to one of these discrepancy densities: $\Sigma^2= 1-\rho_{\mathbb{H}}$.We obtain the estimates$2.3\times 10^{-8}<\rho_{\mathbb{H}}\le0.12087$, where the upper estimate is derived from the estimate from below on $\Sigma^2$ obtained by Astala, Ivrii, Per\"al\"a,  and Prause, and the estimate from below is much more delicate.In particular, it follows that $\Sigma^2<1$, which in combination with the work of Ivrii shows that the maximal fractal dimension of quasicircles conjectured by Astala cannot be reached.Moreover, along the way, since the universal quasiconformal integral means spectrum has the asymptotics$\mathrm{B}(k,t)\sim\frac14\Sigma^2 k^2|t|^2$ for small $t$ and $k$, the conjectured formula $\mathrm{B}(k,t)=\frac14k^2|t|^2$ is not true.As for the actual numerical values of the discrepancy density $\rho_\C$, we obtain the estimate from above $\rho_\C\le0.061203\ldots$ by using the equilateral triangular planar zero packing, where the assertion that equality should hold can be attributed to Abrikosov. The values of $\rho_{\mathbb{H}}$ is expected to be somewhat close to the value of $\rho_\C$.

If we fix a rectangle in the affine real space and if we choose at random a real polynomial with given degree d, the probability P(d) that a component of its vanishing locus crosses the rectangle in its length is clearly positive. But is P(d) uniformly bounded from below when d increases? I will explain a positive answer to a very close question involving real analytic functions. This is a joint work with Vincent Beffara.

 

The Ising model is one of the most classical statistical mechanics model, which has seen spectacular mathematical and physical developments for almost a century. The description of its scaling limit at the phase transition is at the center of a fascinating (conjectured) connection between statistical mechanics and field theories. I will discuss how recent mathematical progress allows one to make the connection between the two-dimensional Ising model and Conformal Field Theory rigorous. If time allows, I will discuss the insight this gives one into related models and field theories.

Based off joint works with S. Benoist, D. Chelkak, H. Duminil-Copin, R. Gheissari, K. Izyurov, F. Johansson-Viklund, K. Kytölä, S. Park and S. Smirnov

If a dynamical system has a conservation law, i.e. a constant along the trajectory of the motion, the study of its evolution along the trajectories of a perturbed system becomes interesting. Conservation laws can be seen everywhere, especially at the level of probability distributions of a reduced dynamic.  We explain this with a number of models, in which we see a singular perturbation problem and identify a conservation law, the latter is used to seek out the correct scale to work with and to reduce the complexity of the system. The reduced dynamic consists of a family of  ODEs with rapidly oscillating right hands side from which in the limit we obtain a Markov process. For stochastic completely integrable system, the limit describes the evolution of the level sets of the family of Hamiltonian functions over a very large time scale.

The Yang-Mills heat equation is the gradient flow corresponding to the Yang-Mills functional. It was initially introduced by S. K. Donaldson to study the existence of irreducible Yang-Mills connections on the projective plane. In this talk, we will consider this equation over compact three-manifolds with boundary. It is a nonlinear weakly parabolic equation, but we will see how one can prove long-time existence and uniqueness of solutions by gauge symmetry breaking. We will also demonstrate some strong regularization results for the solution and see how they lead to detailed short-time asymptotic estimates, as well as the long-time convergence of the Wilson loop functions.