I will start with a motivation of what algebraic and model-theoretic properties an algebraically closed field of characteristic 1 is expected to have. Then I will explain how these properties forces one to follow the route of Hrushovski's construction/Schanuel-type conjecture analysis. Then I am able to formulate very precise axioms that such a field must satisfy.  The main theorem then states that under the axioms the structure has the desired algebraic properties.
The axioms have a form of statements about existence of solutions to systems of equations in terms of a 'multi-dimansional' valuation theory and the validity of these statements is an open problem to be discussed. 

 

In this talk, I’ll share the progress that we have made in the field of e-voting, including the proposal of a new paradigm of e-voting system called self-enforcing e-voting (SEEV). A SEEV system is End-to-End (E2E) verifiable, but it differs from all previous E2E systems in that it does not require tallying authorities. The removal of tallying authorities significantly simplifies the election management and makes the system much more practical than before. A prototype of a SEEV system based on the DRE-i protocol (Hao et al. USENIX JETS 2014) has been built and used regularly in Newcastle University for classroom voting and student prize competitions with very positive student feedback. Lessons from our experience of designing, analysing and deploying an e-voting system for real-world applications are also presented.

Every finite regular CW complex is, ipso facto, a cohomologically stratified space when filtered by skeleta. We outline a method to recover the canonical (i.e., coarsest possible) stratification of such a complex that is compatible with its underlying cell structure. Our construction proceeds by first localizing and then resolving a complex of cosheaves which capture local cohomology at every cell. The result is a sequence of categories whose limit recovers the desired strata via its (isomorphism classes of) objects. As a bonus, we observe that the entire process is algorithmic and amenable to efficient computations!

TBA

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. 

One of the challenges of 21st-century science is to model the evolution of complex systems.  One example of practical importance is urban structure, for which the dynamics may be described by a series of non-linear first-order ordinary differential equations.  Whilst this approach provides a reasonable model of urban retail structure, it is somewhat restrictive owing to uncertainties arising in the modelling process.

We address these shortcomings by developing a statistical model of urban retail structure, based on a system of stochastic differential equations.   Our model is ergodic and the invariant distribution encodes our prior knowledge of spatio-temporal interactions.  We proceed by performing inference and prediction in a Bayesian setting, and explore the resulting probability distributions with a position-specific metrolpolis-adjusted Langevin algorithm.