Conformal Field Theories (CFT) are an important concept in physics. In two dimensions, there exists a mathematical rigorous description for them in terms of vertex operator algebras (VOA). In the first part of my talk, I will discuss some work on such VOAs, with a particular focus on VOAs with an infinite number of modules, which are not very well understood. I will also briefly discuss applications to Calabi-Yau geometry and to quantum gravity. In the second part of my talk, I will discuss ideas of how to generalize VOAs and other concepts from two to higher dimensions. I will namely discuss so-called Cauchy fields and the conformal bootstrap program.

# Vertex Operator Algebras and Conformal Field Theories

# System of points in saturated dg-categories

In this work we consider the question of realizing triangulated dg-categories by derived categories of algebraic varieties. For this, we introduce the notion of "system of points" in saturated dg-categories. We show that given such a system on a dg-category T, we can construct an algebraic space M, of finite type, smooth and separated, together with a dg-functor from T to a certain twisted dg-category of sheaves on M. We prove that this functor is furthermore an equivalence if and only if M is proper. All along this work we study t-strutcures on algebraic families of objects in T, which might be of independant interest.

Ref: http://arxiv.org/abs/1504.07748

(Joint work with Bertrand Toen)

# The dual view of Markov Processes I

This is the first of two talks on a dualized view of Markov processes. The first part will describe how to view a probabilistic transition system as a transformer of functions rather than as a transformer of probability distributions. A Markov process is normally viewed as a Markov kernel i.e. a map from S x Σ → [0,1] where S is a state space and Σ is a σ-algebra on S. These Markov kernels are morphisms in the Kleisli category of the Giry monad. In recent work by Chaput, Danos, Panangaden and Plotkin, Markov processes were reinterpreted as linear maps on the space of positive L1 functions on S. This is analogous to taking the predicate transformer view of Markov processes. A number of dualities and isomorphisms emerge in this picture. Most interestingly conditional expectation can be understood functorially.

# Mechanizing Meta-Theory in Beluga

Mechanizing formal systems, given via axioms and inference rules, together with proofs about them plays an important role in establishing trust in formal developments. In this talk, I will survey the proof environment Beluga. To specify formal systems and represent derivations within them, Beluga provides a sophisticated infrastructure based on the logical framework LF; in particular, its infrastructure not only supports modelling binders via binders in LF, but extends and generalizes LF with first-class contexts to abstract over a set of assumptions, contextual objects to model derivations that depend on assumptions, and first-class simultaneous substitutions to relate contexts. These extensions allow us to directly support key and common concepts that frequently arise when describing formal systems and derivations within them.

# Y -Meshes and generalized pentagram maps

This will be a review of the recent paper with this title by Max Glick and Pavlo Pylavskyy (arXiv:1503.02057 ) To quote from their abstract:

"We introduce a rich family of generalizations of the pentagram map sharing the property that each generates an infinite configuration of points and lines with four points on each line. These systems all have a description as Y - mutations in a cluster algebra and hence establish new connections between cluster theory and projective geometry. Our framework incorporates many preexisting generalized pentagram maps due to M. Gekhtman, M. Shapiro, S. Tabachnikov, and A. Vainshtein and also B. Khesin and F. Soloviev. In several of these cases a reduction to cluster dynamics was not previously known."

# K3 surfaces, moonshine, and string theory

Moonshine is a mysterious relationship in mathematics between finite groups and modular forms, which appears to have deep connections to physics and string theory. The most famous example is monstrous moonshine, which relates the coefficients of the modular J-function to dimensions of representations of the largest sporadic simple group, the monster group.

I will discuss "umbral moonshine," a new moonshine phenomenon which relates mock modular forms to automorphism groups of Niemeier lattices, the first case of which was first discovered as a connection between the elliptic genus of a K3 surface and the Mathieu group M24. I will explain how moonshine has connections to symmetries of sigma models arising from string compactification on K3 surfaces, and thus many interesting applications in mathematics and physics, including number theory, enumerative geometry, supersymmetric black holes, string dualities, and AdS/CFT, among other things.

# Introduction to *-Autonomous categories, continued

Next time, I will finish proving things about sigma and tau; in particular that tau is left adjoint to sigma. They will be used to show that the inclusions of the categories of weak, resp. strong, objects has a left, resp. right, adjoint and then use generalities on adjoints to show that they are equivalent categories. In future lecture(s), at least one and at most two, I will show that the chu category is equivalent to the category of weak objects, and therefore to the category of strong ones. Since the chu category is *-autonomous, so are they. Then I will discuss examples. One is groups. The others are actually all examples of one rather general situation. Let K be a spherically complete field (this was new to me, see: https://en.wikipedia.org/wiki/Spherically_complete_field), which includes all locally compact fields, then there are *-autonomous categories starting with the normed K-spaces. This includes the case that K is discrete.

# The fundamental theorem of algebra, complex analysis and ... astrophysics

The fundamental theorem of algebra, complex analysis and ... astrophysicsThe Fundamental Theorem of Algebra first rigorously proved by Gauss states that each complex polynomial of degree $n$ has precisely $n$ complex roots. In recent years various extensions of this celebrated result have been considered. We shall discuss the extension of the FTA to harmonic polynomials of degree $n$. In particular, the theorem of D. Khavinson and G. Swiatek that shows that the harmonic polynomial $ar{z}-p(z), deg , p=n>1$ has at most $3n-2$ zeros as was conjectured in the early 90's by T. Sheil-Small and A. Wilmshurst. L. Geyer was able to show that the result is sharp for all $n$. G. Neumann and D. Khavinson proved that the maximal number of zeros of rational harmonic functions $ar{z}-r(z), deg ,r =n>1$ is $5n-5$. It turned out that this result confirmed several consecutive conjectures made by astrophysicists S. Mao, A. Petters, H. Witt and, in its final form, the conjecture of S. H. Rhie that were dealing with the estimate of the maximal number of images of a star if the light from it is deflected by $n$ co-planar masses. The first non-trivial case of one mass was already investigated by A. Einstein around 1912. We shall also discuss the problem of gravitational lensing of a point source of light, e.g., a star, by an elliptic galaxy, more precisely the problem of the maximal number of images that one can observe. Under some more or less ``natural'' assumptions on the mass distribution within the galaxy one can prove (A.Eremenko and W. Bergweiler - 2010, also, K - E. Lundberg - 2010) that the number of visible images can never be more than four in some cases and six in the other. Interestingly, the former situation can actually occur and has been observed by astronomers. Still there are much more open questions than there are answers.

# Beyond delta hedging

Industry best practices for the hedging of derivatives consist in neutralizing the Greek letters of a derivatives portfolio. Delta hedging which is extensively used in practice is a specific case of this approach. However, many alternative methodologies to hedge derivatives were developed in the literature. The aim of the presentation is to give an overview of those alternative methods which can be used to increase the performance of hedges in terms of risk reduction.

# Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing nonlinearity

We consider a positive solution to a nonlinear elliptic equation on a punctured ball. The linear part is the classical Laplacian. When the nonlinear part is positive and critical, this is similar to the classical problem studied by Caffarelli-Gidas-Spruck. When the nonlinear part is negative and a pure power, the problem is associated to a natural convex functional and the singularities are completely understood. In the present work, we mix the two nonlinearities. We show the existence of several potential behaviors. Two of them are natural extensions of the case of constant-sign nonlinearity. Two other behaviors are arising from the interaction of the two nonlinearity. In this talk, I will describe all the possible behaviors and I will show how the methods of apriori analysis in nonlinear elliptic problems are helping understanding this problem.

This is joint work with Florica Cirstea (Sydney)