Isomonodromic approach to the Kontsevich-Witten and Kontsevich-Penner tau functions
I will explain the Witten-Kontsevich Theorem and the implications/generalizations of this connection between Algebraic Geometry and Integrable Systems. I will explain what is meant by isomonodromic approach and what are the applications. This talk is based on recent works by Prof. Bertola, Prof. Cafasso, Prof. Dubrovin, and Prof. D. Yang. The discussion will be as elementary as possible and no particular prerequisite is needed.
Critical points of polynomial functions
From linear algebra we know that a generic real symmetric matrix A=(aij)i,j=1n has n distinct real eigenvalues. A geometric reason for this is that the set of symmetric matrices with repeated eigenvalues has codimension two in the space of all symmetric matrices. Let's denote by fA(x)=xtAx=Σi,j=1n aijxixj the quadratic form associated to the symmetric matrix A =(aij). Then critical points and critical values of fA : Sn-1 → K(R) (the restriction of fA to the unit sphere) are exactly unit eigenvectors and eigenvalues of A. In particular, the number of critical points of fA | Sn-1 equals 2n for any generic A. If, now, f=Σi1,...,id=1n ai1... id xi1... id is a homogeneous form of degree at least d ≤ 3 the number C(f) of critical point of f|Sn-1 is not generically constant anymore. However, there is an upper bound on this number:C(f) ≤ 2 (d-1)n-1 (d-2)-1 where f is a generic degree d ≥ 1 form in n ≥ 2 variables. I will explain how to construct for any d and n generic forms attaining this bound. Moreover, I will show that the bound is attained by harmonic forms, known as spherical harmonics. The presentation will be as elementary as possible.
Blowup Equations for Refined Topological Strings
One main goal to study Calabi-Yau manifolds is to determine their enumerative invariants, Gromov-Witten, Gopakumar-Vafa, Donaldson-Thomas and so on. In physics, these invariants are encoded in the partition function of the topological string theory on such manifolds. For the so called local Calabi-Yau, the enumerative invariants can be further generalized to refined BPS invariants. The traditional technique to compute such invariants includes the refined topological vertex in A model and refined holomorphic anomaly equations in B model. Recently a new method called blowup equations was proposed, which was generalized from the Gottsche-Nakajima-Yoshioka K-theoretic blowup equations for supersymmetric gauge theories. These blowup equations also play an important role on the quantization of mirror curve of local Calabi-Yau and result in the equivalence between Nekrasov-Shatashivili quantization and the Grassi-Hatsuda-Marino conjecture. As I will explain from the very beginning, no prerequisites on Calabi-Yau are needed.
Moduli of semistable sheaves as quiver moduli
I will discuss how moduli spaces of Gieseker-semistable sheaves on certain projective varieties can be realized as quiver moduli spaces by using t-structures and exceptional collections in the derived category. This construction can be used to prove easily some of the geometric properties of the moduli space of sheaves, and to do some explicit computations. I will first give an introduction to all these notions to make the talk as self-contained as possible.
Gross-Pitaevskii: an effective theory in Quantum Mechanics
During the talk we will briefly look at what the main definitions are in (the mathematics of) Quantum Mechanics and then we will dig deeper in the Gross-Pitaevskii effective theory for Bose-Einstein Condensates. In particular we will also focus on the GP variational problem and how this relates to Superfluidity in BEC.
Categorical Webs and S-duality
We review the categorical approach to the BPS sector of a 4d N=2 QFT, clarifying many tricky issues and presenting a few novel results. To a given N=2 QFT one associates several triangulated categories: they describe various kinds of BPS objects from different physical viewpoints (e.g. IR versus UV). These diverse categories are related by a web of exact functors expressing physical relations between the various objects/pictures. The aim of this talk is to give an introduction to the above topics, focussing more on examples, ideas and applications rather than technicalities: no particular prerequisites are needed.