Schedule for: 24w5502 - Noncommutative Geometry Meets Topological Recursion

A set dinner is served daily between 5:30pm and 7:30pm in the Xianghu Lake National Tourist Resort.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

In the beginning of the 1990s, Dubrovin introduced the notion of Frobenius manifold, now also known as Dubrovin--Frobenius manifold, to give a geometric abstraction of the Witten--Dijkgraaf--Verlinde--Verlinde equations that appeared in the study of 2D topological field theories. It soon became one of the central subjects in the studies of quantum cohomology, singularity theory, integrable systems, and etc. In this mini-course, we give a brief introduction to the theory of Dubrovin--Frobenius manifolds, with a focus on the interplay between these geometric structures and integrable hierarchies of evolutionary partial differential equations (principal hierarchies and Dubrovin--Zhang hierarchies). Applications to the Gaussian Unitary Ensemble (GUE) correlators and Gromov--Witten invariants of the complex projective line will be considered.

In the beginning of the 1990s, Dubrovin introduced the notion of Frobenius manifold, now also known as Dubrovin--Frobenius manifold, to give a geometric abstraction of the Witten--Dijkgraaf--Verlinde--Verlinde equations that appeared in the study of 2D topological field theories. It soon became one of the central subjects in the studies of quantum cohomology, singularity theory, integrable systems, and etc. In this mini-course, we give a brief introduction to the theory of Dubrovin--Frobenius manifolds, with a focus on the interplay between these geometric structures and integrable hierarchies of evolutionary partial differential equations (principal hierarchies and Dubrovin--Zhang hierarchies). Applications to the Gaussian Unitary Ensemble (GUE) correlators and Gromov--Witten invariants of the complex projective line will be considered.

Lunch is served daily between 11:30am and 1:30pm in the Xianghu Lake National Tourist Resort

We study the large-$N$ limit of a family of quartic Dirac ensembles based on $(0, 1)$ fuzzy geometries that are coupled to fermions. These Dirac ensembles are examples of single-matrix, multi-trace matrix ensembles with non-polynomial action. Additionally, they can serve as examples of integer-valued $\beta$-ensembles. Convergence of the spectral density in the large $N$ limit for a large class of such matrix ensembles is proven. The main results we obtain are the addition of a fermionic contribution in the matrix ensemble and the investigation of spectral estimators for such finite dimensional spectral triples.

In this talk, I will start by illustrating matrix models relative to free, infinitesimal free, monotone, and c-free independencies. Notions of independence play a key role in studying joint distributions of non-commutative random variables and hence in studying limiting distributions of the associated random matrix models. I will illustrate in particular recent results relative to the monotone case. Just as in the classical setting, to each notion of independence corresponds a central limit theorem. The second part of the talk will focus on the operator-valued setting, where I show quantitative results for the relative operator-valued central limit theorems. Based on joint works with Arizmendi, Gilliers, Mai, & Tseng.

In this talk, I will explain the AGT correspondence in the context of Argyres-Douglas theories, a class of 4d N=2 supersymmetric theories that defy Lagrangian formulation, making them challenging to study. However, these theories can be captured by the Hitchin system with irregular punctures. By interpreting the states associated with these irregular punctures through representations of W-algebra, we can leverage the AGT correspondence to compute partition functions. This approach offers new insights into the structure and properties of Argyres-Douglas theories.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

In the beginning of the 1990s, Dubrovin introduced the notion of Frobenius manifold, now also known as Dubrovin--Frobenius manifold, to give a geometric abstraction of the Witten--Dijkgraaf--Verlinde--Verlinde equations that appeared in the study of 2D topological field theories. It soon became one of the central subjects in the studies of quantum cohomology, singularity theory, integrable systems, and etc. In this mini-course, we give a brief introduction to the theory of Dubrovin--Frobenius manifolds, with a focus on the interplay between these geometric structures and integrable hierarchies of evolutionary partial differential equations (principal hierarchies and Dubrovin--Zhang hierarchies). Applications to the Gaussian Unitary Ensemble (GUE) correlators and Gromov--Witten invariants of the complex projective line will be considered.

This mini-course introduces basic ideas and various recent mathematical developments about quantization that arises from topologically/holomorphically twisted quantum field theory. The focus is on homological method of BRST-BV and its applications in geometry and topology. We illustrate some applications in topological/chiral algebraic index, topological B-model and mirror symmetry.

The generalised Brézin-Gross-Witten (BGW) tau function is a KdV tau function depending on one parameter which deforms the BGW tau function. It is known that the BGW tau function is related to intersection numbers over the moduli space of curves, and that it is also related to the moduli space of super curves. It is natural to ask whether the same is true of the generalised BGW tau function. In this talk I will discuss joint work with Alexander Alexandrov where we show that indeed the generalised BGW tau function is related to intersection numbers over the moduli space of curves, and I will describe a conjectural relationship with the moduli space of super curves, proven in some cases.

I will discuss multi-orientable tensor models with a mixed U(N) and O(D) symmetry. A particular U(N)^2 \otimes O(D) order-3 tensor model can be viewed as a complex multi-matrix model with D copies of complex matrices. This model admits an expansion in two parameters owing to the presence of N and D, and yields a more refined classification of Feynman graphs generated by the model. Such a refined expansion has piqued our curiosity in the possibility of finding a new universality class for this tensor model. In the double-scaling limit, we achieved the complete recursive characterisation of the Feynman graphs of arbitrary genus ($\ell=0$), and further extended the analysis further and classified higher grade $\ell$ [arXiv:2310.13789]. In the triple-scaling limit, we found two universality classes, namely branched polymers for 2PR and Liouville quantum gravity (Brownian sphere) for 2PI graphs [arXiv:2003.02100]. Furthermore, we examined multi-orientable tensor models in a more general setting for arbitrary order of tensors and counted U(N)^{\otimes p} \otimes O(N)^{\otimes q} invariants [arXiv:2404.16404]. By providing a systematic way to count tensor invariants, we newly discovered integer sequences added to OEIS and unveiled a new correspondence between the number of multiorientable-model observables and topological quantum field theory (TQFT) on particular cellular complexes.

In linear algebra, it is well-known that the rank of matrices is subadditive, which means that for any matrices $A$ and $B$, we have $rank(A+B)\leq rank(A)+rank(B)$. This property leads to the question: for any polynomial $p$ over non-commuting indeterminates $x$ and $y$, is there an upper bound for $rank(p(A,B))$ when considering matrices $A$ and $B$? Furthermore, can this upper bound be achieved in a dimensionless optimal sense? We address these questions with the aid of free probability theory. Our approach is based on a recent joint-work with Octavio Arizmendi, Guillaume Cébron and Roland Speicher. Our findings are grounded in the universality property of freely independent random variables with respect to the von Neumann rank. This universality reveals that the free independent copy of matrices can provide an upper bound on the rank. Interestingly, this upper bound can be saturated by matrices, owing to the relationship between free random variables and random matrices. Additionally, we will present precise formulas for the commutator and anticommutator as illustrative examples.

This mini-course introduces basic ideas and various recent mathematical developments about quantization that arises from topologically/holomorphically twisted quantum field theory. The focus is on homological method of BRST-BV and its applications in geometry and topology. We illustrate some applications in topological/chiral algebraic index, topological B-model and mirror symmetry.

The lectures will introduce to the theory of Topological Recursion, with a special account of recent discoveries in this theory including: — irregular and degenerate topological recursion; — xy swap duality; — KP integrability. The lectures are based on a sequence of recent joint papers with A.Alexandrov, B.Bychkov, P.Dunin-Barkowsky, S.Shadrin

The lectures will introduce to the theory of Topological Recursion, with a special account of recent discoveries in this theory including: — irregular and degenerate topological recursion; — xy swap duality; — KP integrability. The lectures are based on a sequence of recent joint papers with A.Alexandrov, B.Bychkov, P.Dunin-Barkowsky, S.Shadrin

The lectures will introduce to the theory of Topological Recursion, with a special account of recent discoveries in this theory including: — irregular and degenerate topological recursion; — xy swap duality; — KP integrability. The lectures are based on a sequence of recent joint papers with A.Alexandrov, B.Bychkov, P.Dunin-Barkowsky, S.Shadrin

This mini-course introduces basic ideas and various recent mathematical developments about quantization that arises from topologically/holomorphically twisted quantum field theory. The focus is on homological method of BRST-BV and its applications in geometry and topology. We illustrate some applications in topological/chiral algebraic index, topological B-model and mirror symmetry.

I will discuss KP integrability of topological recursion which is very natural in the context of the x-y swap relation. It can be described by certain integral transforms, leading to the Kontsevich-like matrix models. This allows us to establish general KP integrability properties of the TR differentials for genus zero spectral curves. This talk is based on a joint work with Boris Bychkov, Petr Dunin-Barkowski, Maxim Kazarian, and Sergey Shadrin also discussed in the mini-course of Maxim Kazarian.

I will discuss an extension of the Remodeling Conjecture of Bouchard-Klemm-Mariño-Pasquetti to all-genus descendant mirror symmetry of toric Calabi-Yau 3-folds, that is, the all-genus descendant Gromov-Witten invariants can be produced from the Laplace transform of the Eynard-Orantin topological recursion on the mirror curve. The result is based on a correspondence between equivariant line bundles supported on toric subvarieties and relative homology cycles on the mirror curve. The Laplace transform along the cycles gives genus-zero descendant Gromov-Witten invariants with certain Gamma class insertions. This talk is based on joint work in progress with Bohan Fang, Chiu-Chu Melissa Liu, and Zhengyu Zong.

We discuss families of geometric structures that progressively generalise Frobenius manifolds, under the assumption of regularity and in relation to integrable hierarchies.

I'll review the refined topological recursion recently formulated for (smooth) degree two genus zero curves. I'll discuss the computation of the corresponding free energy for some simple cases, and some subtleties which arise in comparison to the unrefined case. Time permitting, I'll mention the relation to refined BPS structures. Based on joint work with K. Osuga.

Weighted Hurwitz numbers were introduced by Harnad and Guay-Paquet as objects covering a wide class of Hurwitz numbers of various types. A particularly strong property of Hurwitz numbers is that they are governed by the celebrated topological recursion of Chekhov--Eynard--Orantin. The program of understanding how TR can be used to compute different types of Hurwitz numbers was carried out over the last two decades by considering each case separately, and finally, the general case of rationally-weighted (or even more than that) Hurwitz numbers was recently proved by Bychkov--Dunin-Barkowski--Kazarian--Shadrin. We will discuss a more general case of weighted $b$-Hurwitz numbers. We show that their generating function can be associated with the Whittaker vector for a $\mathcal{W}$-algebra of type A. In particular it satisfies explicit $W$-constraints. Our result gives a new explanation of remarkable enumerative properties of Hurwitz numbers and extends it to the $b$-deformed case. This is a joint work with Nitin Chidambaram and Kento Osuga.

We introduce a family of polynomials, which arise in three distinct ways: in the large $N$ expansion of a matrix integral, as a weighted enumeration of factorisations of permutations, and via the topological recursion. More explicitly, we interpret the complex Grassmannian $Gr(M,N)$ as the space of $N\times N$ idempotent Hermitian matrices of rank $M$ and develop a Weingarten calculus to integrate products of matrix elements over it. In the regime of large $N$ and fixed ratio $M / N$, such integrals have expansions whose coefficients count factorisations of permutations into monotone sequences of transpositions, with each sequence weighted by a monomial in $t = 1 - N / M$. This gives rise to the desired polynomials, which specialise to the monotone Hurwitz numbers when $t = 1$. Furthermore, we conjecture on the basis of overwhelming empirical evidence that the deformed monotone Hurwitz numbers are real-rooted polynomials whose roots satisfy remarkable interlacing phenomena. The whole story has an analogue involving integration on real Grassmannians. This is joint work with Brice Arrigo, Xavier Coulter and Ellena Moskovsky.