Geometry and arithmetic of integrable hierarchies of KdV type. I. Integrality

For each of the simple Lie algebras $mathfrak{g}=A_l$, $D_l$ or $E_6$, we show that the all-genera one-point FJRW invariants of $mathfrak{g}$-type, after multiplication by suitable products of Pochhammer symbols, are the coefficients of an algebraic generating function and hence are integral. Moreover, we find that the all-genera invariants themselves coincide with the coefficients of the unique calibration of the Frobenius manifold of $mathfrak{g}$-type evaluated at a special point. For the $A_4$ (5-spin) case we also find two other normalizations of the sequence that are again integral and of at most exponential growth, and hence conjecturally are the Taylor coefficients of some period functions.

A curious identity that implies Fabers conjecture

We prove that a curious generating series identity implies Fabers intersection number conjecture (by showing that it implies a combinatorial identity already given in arXiv:1902.02742) and give a new proof of Fabers conjecture by directly proving this identity.

Arithmetic properties of the Herglotz function

In this paper we study two functions $F(x)$ and $J(x)$, originally found by Herglotz in 1923 and later rediscovered and used by one of the authors in connection with the Kronecker limit formula for real quadratic fields. We discuss many interesting properties of these functions, including special values at rational or quadratic irrational arguments as rational linear combinations of dilogarithms and products of logarithms, functional equations coming from Hecke operators, and connections with Starks conjecture. We also discuss connections with 1-cocycles for the modular group $mathrm{PSL}(2,mathbb{Z})$.

Strichartz Estimates with Broken Symmetries

In this note we study the eigenvalue problem for a quadratic form associated with Strichartz estimates for the Schr{o}dinger equation, proving in particular a sharp Strichartz inequality for the case of odd initial data. We also describe an alternative method that is applicable to a wider class of matrix problems.

Nu-invariants of extra-twisted connected sums

We analyse the possible ways of gluing twisted products of circles with asymptotically cylindrical Calabi-Yau manifolds to produce manifolds with holonomy G_2, thus generalising the twisted connected sum construction of Kovalev and Corti, Haskins, Nordstrom, Pacini. We then express the extended nu-invariant of Crowley, Goette, and Nordstrom in terms of fixpoint and gluing contributions, which include different types of (generalised) Dedekind sums. Surprisingly, the calculations involve some non-trivial number-theoretical arguments connected with special values of the Dedekind eta-function and the theory of complex multiplication. One consequence of our computations is that there exist compact G_2-manifolds that are not G_2-nullbordant.

Information thermodynamics of financial markets: the Glosten-Milgrom model

The Glosten-Milgrom model describes a single asset market, where informed traders interact with a market maker, in the presence of noise traders. We derive an analogy between this financial model and a Szilard information engine by {em i)} showing that the optimal work extraction protocol in the latter coincides with the pricing strategy of the market maker in the former and {em ii)} defining a market analogue of the physical temperature from the analysis of the distribution of market orders. Then we show that the expected gain of informed traders is bounded above by the product of this market temperature with the amount of information that informed traders have, in exact analogy with the corresponding formula for the maximal expected amount of work that can be extracted from a cycle of the information engine. This suggests that recent ideas from information thermodynamics may shed light on financial markets, and lead to generalised inequalities, in the spirit of the extended second law of thermodynamics.

Masur--Veech volumes of quadratic differentials and their asymptotics

Based on the Chen--Moller--Sauvaget formula, we apply the theory of integrable systems to derive three equations for the generating series of the Masur--Veech volumes ${rm Vol} , mathcal{Q}_{g,n}$ associated with the principal strata of the moduli spaces of quadratic differentials, and propose refinements of the conjectural formulas given in [12,4] for the large genus asymptotics of ${rm Vol} , mathcal{Q}_{g,n}$ and of the associated area Siegel--Veech constants.

Theta Blocks

We define theta blocks as products of Jacobi theta functions divided by powers of the Dedekind eta-function and show that they give a powerful new method to construct Jacobi forms and Siegel modular forms, with applications also in lattice theory and algebraic geometry. One of the central questions is when a theta block defines a Jacobi form. It turns out that this seemingly simple question is connected to various deep problems in different fields ranging from Fourier analysis over infinite-dimensional Lie algebras to the theory of moduli spaces in algebraic geometry. We give several answers to this question.

Genus-zero and genus-one string amplitudes and special multiple zeta values

In this paper we show that in perturbative string theory the genus-one contribution to formal 2-point amplitudes can be related to the genus-zero contribution to 4-point amplitudes. This is achieved by studying special linear combinations of multiple zeta values that appear as coefficients of the amplitudes. We also exploit our results to relate closed strings to open strings at genus one using Browns single-valued projection, proving a conjecture of Broedel, Schlotterer and the second author.

Cotangent sums, quantum modular forms, and the generalized Riemann hypothesis

We show that an asymptotic property of the determinants of certain matrices whose entries are finite sums of cotangents with rational arguments is equivalent to the GRH for odd Dirichlet characters. This is then connected to the existence of certain quantum modular forms related to Maass Eisenstein series.