2022-2023

Following hal.archives-ouvertes.fr/hal-03502404v3 with L. Han, Y. Li, S. Sun, I will show how partial theta series, i.e. functions of the form Θ(𝜏):=∑𝑓(𝑛)𝑒^{𝑖𝜋𝜏𝑛^2/𝑀} with 𝑓:ℤ→ℂ an 𝑀-periodic function (or the product of a power of 𝑛 by such function), give rise to divergent asymptotic series at every rational point of the boundary of their domain of definition {ℑ𝜏>0}. I will discuss the summability and resurgence properties of these series by means of explicit formulas for their formal Borel transforms, and the consequences for the modularity properties of Θ, or its “quantum modularity” properties in the sense of Zagier’s recent theory. Écalle’s “Alien calculus” allows one to encode this phenomenon in a kind of “Bridge equation”.
Interesting examples stem from the study of Gukov-Pei-Putrov-Vafa invariants and Witten-Reshetikhin-Turaev invariants for the Poincaré homology sphere (cf. [Gukov-Putrov-Marino, arXiv:1605.07615]) or more generally Seifert homology 3-spheres ([Andersen-Mistegård, J. Lond. Math. Soc. 2022]).

Following hal.archives-ouvertes.fr/hal-03502404v3 with L. Han, Y. Li, S. Sun, I will show how partial theta series, i.e. functions of the form Θ(𝜏):=∑𝑓(𝑛)𝑒^{𝑖𝜋𝜏𝑛^2/𝑀} with 𝑓:ℤ→ℂ an 𝑀-periodic function (or the product of a power of 𝑛 by such function), give rise to divergent asymptotic series at every rational point of the boundary of their domain of definition {ℑ𝜏>0}. I will discuss the summability and resurgence properties of these series by means of explicit formulas for their formal Borel transforms, and the consequences for the modularity properties of Θ, or its “quantum modularity” properties in the sense of Zagier’s recent theory. Écalle’s “Alien calculus” allows one to encode this phenomenon in a kind of “Bridge equation”.
Interesting examples stem from the study of Gukov-Pei-Putrov-Vafa invariants and Witten-Reshetikhin-Turaev invariants for the Poincaré homology sphere (cf. [Gukov-Putrov-Marino, arXiv:1605.07615]) or more generally Seifert homology 3-spheres ([Andersen-Mistegård, J. Lond. Math. Soc. 2022]).

In this talk, we will give a brief account of the relationship between radial positive-definite functions on free groups and the moments of probability measures on the interval [−1,1]. The case for the commutative setting is proven by Bochner. Meanwhile, Haagerup and Knudby proved the case for l1 radial positive definite function. We explore the case for l2 radial positive definite functions (completely positive Fourier multipliers) on free groups.

It is known in the Euclidean distance geometry tracing
back to Schoenberg (1935-37) that a graph admits a quadratic
embedding in a Euclidean space if and only if the distance matrix
$D=[d(x,y)]$ is conditionally negative definite. This condition,
being equivalent to that the q-matrix $Q=[q^{d(x,y)}]$ is positive
definite for all $0\le q\le 1$, appears also in quantum probability.
A new numerical invariant of a graph called the quadratic embedding
constant (QEC) was introduced in 2017 for a quantitative approach.
In this talk we will review the basic ideas and recent results and
propose some questions.
References: arXiv:2207.13278, arXiv:2206.05848, arXiv:1904.08059

In the talk, we recall a characterisation of well-posedness for abstract divergence form problems that provides a more detailed picture than the classical Lax—Milgram theorem. With this we analyse well-posedness of 1-dimensional divergence form problems with bounded, measurable, possibly sign-changing coefficients and $H^{-1}$-right-hand sides. Lifting this result to an operator equation in $L_2$, we provide a solution theory for multi-d divergence form problems with real coefficients piecewise constant on slabs. As it turns out, the condition for continuous invertibility of the corresponding operator equation in $L_2$ can be conveniently characterised looking at the zeros of a certain explicit polynomial expression. This leads to a genericity result for solving these kind of equations.

Painlevé equations are six nonlinear second order ODEs whose solutions are thought to be the special functions of the 21st century, and have already appeared in countless works in integrable systems, combinatorics, and random matrix theory among many others. In this talk, I will focus my attention on one equation, Painlevé III (PIII). While its generic solutions are transcendental, it is known to possess families of special–function solutions: solutions written in terms of elementary and/or classical special functions. In the first half of the talk, I will focus on the rational solutions of PIII, highlighting the analysis of these solutions near the origin, a singular point of the differential equation. These results rely on a characterization of solutions of PIII in terms of a 2×2 Riemann-Hilbert problem. The leading term of the asymptotics turns out to solve the “degenerate” version of PIII, often known as PIII(D8). In the second half of the talk, I will describe how this procedure can be carried out more generally to realize a confluence from solutions of the generic PIII(D6) to solutions of PIII(D8). This is joint work with Oleg Lisovyy, Peter Miller, and Andrei Prokhorov.

Quantum Markov semigroups (QMS) are noncommutative generalizations of
classical Markov semigroup, where the underlying probability spaces are replaced by operator
algebras. Quantum Markov semigroups model open quantum systems weakly interacting
with an environment. As such, they have received considerable attention from the community
of quantum information science. One of the fundamental functional inequalities for
QMS is the noncommutative analog of the (modified-)logarithmic Sobolev inequality ((M-
)LSI). It is well-known that classical LSI (resp. MLSI) satisfies the so-called tensorization
property. This property is crucial in showing the equivalence between LSI and Talagrand’s
concentration inequality. It turns out tensorization is not satisfied in general by the quantum
analog of LSI and MLSI. One partial solution is to introduce the concept of a complete
modified logarithmic Sobolev inequality (CLSI). This definition naturally satisfies tensorization
property. Two central problems in the study of CLSI are: (1) obtain tight estimates of
CLSI constant, and (2) calculate CLSI constant in realistic physics models. In this talk, we
will focus on solving one instance of the second problem: the so-called Dicke’s superradiance
model in quantum optics. Mathematically, this model is intimately related to the tensor
representation of SU(2). We will provide a framework to calculate the CLSI constant for
this model. Using the same framework, we will also obtain CLSI constant for some other
interesting models including models involving QMS on type III von Neumann algebras. The
last example opens the door to study second quantized open quantum systems. This talk is
based on the manuscript: https://arxiv.org/abs/2209.11099.

Given a compact set $K$ in $\mathbb{R}^n$ and a continuous, real valued function $f$ on $K$, when is there a $C^{m,\omega}$ function $F$ defined on $\mathbb{R}^n$ such that $F|_K = f$? ($C^{m,\omega}$ is the space of $C^m$ functions whose $m$th order derivatives are uniformly continuous with modulus of continuity $\omega$.) Whitney famously answered this question in 1934 when extra data is provided on the derivatives of the extension on $K$, and less famously answered the original question for subsets of $\mathbb{R}$. The question was answered in full by Charles Fefferman in 2009.

The Heisenberg group $H$ is $\mathbb{R}^3$ with a sub-Riemannian and metric structure generated by a class of admissible curves. We will consider Whitney’s original question for curves in $H$: given a compact set $K$ in $\mathbb{R}$ and a continuous, map $f:K \to H$, when is there a $C^{m,\omega}$ admissible curve $F$ such that $F|_K = f$? I will present a project with Andrea Pinamonti and Gareth Speight in which we address this question.

In this talk, we consider the transport properties of the class of limit-periodic continuum Schrödinger operators whose potentials are approximated exponentially quickly by a sequence of periodic functions. For such an operator H, and X_H(t) the Heisenberg evolution of the position operator, we show the limit of (1/t)X_H(t)ψ as t→∞ exists and is nonzero for ψ≠0 belonging to a dense subspace of initial states which are sufficiently regular and of suitably rapid decay. This is viewed as a particularly strong form of ballistic transport, and this is the first time it has been proven in a continuum almost periodic non-periodic setting. In particular, this statement implies that for the initial states considered, the second moment grows quadratically in time.

If a random variable, X, has finite moments of all orders, then its moments can be recovered from its number operator N. In this talk we study the random variables X, for which there exist two polynomials, P and Q, of degree at most 2, such that:

[[P(N),X],X] = Q(X),

where [A, B] denotes the commutator of the operators A and B. These random variables include the Gaussian, Gamma, and beta distributions, which correspond to the Hermite, Laguerre, and Jacobi orthogonal polynomials.

I will show a new, elementary and geometric proof of the classical theorem of Alexandrov about the second order differentiability of convex functions. The same method leads to new proofs of recent results about Lusin approximation of convex functions and convex bodies by $C^{1,1}$ convex functions and convex bodies. Finally, I will discuss recent results about Lusin approximation of strongly convex functions by $C^2$ strongly convex functions. The talk is based on my work with Azagra, Cappello and Drake.

Recurrence coefficients coming from the three-term recurrence relation satisfied by orthogonal polynomials appear in various applications. After mentioning one particular application to Lanczos’ tridiagonalization algorithm I will proceed to introduce the Riemann-Hilbert method of Fokas-Its-Kitaev for studying orthogonal polynomials and their recurrence coefficients. In the second part I will talk about some more recent developments on orthogonal polynomials with weights functions having logarithmic singularities and present some result on their recurrence coefficients. This part is a collaboration with Percy Deift.

The Kawahara and the modified Kawahara equations are two nonlinear dispersive equations known to model shallow water waves. In this talk, we focus on their well-posedness theory, in particular the topic of unconditional uniqueness. We present arguments in this direction based on the method of normal form reductions. One of these results is based on joint work with Bai Lin.

Using the variational characterization of the smallest eigenvalue below the essential spectrum of a lower semibounded self-adjoint operator, we prove strict domain monotonicity (with respect to changing the finite interval length) of the principal eigenvalue of the Friedrichs extension $T_F$ of the minimal operator for regular four-coefficient Sturm–Liouville differential expressions. As a consequence of the strict domain monotonicity of the principal eigenvalue of the Friedrichs extension in the regular case, and on the basis of oscillation theory in the singular context, in our main result we characterize all lower bounds of $T_F$ as those $\lambda\in \mathbb{R}$ for which the differential equation $\tau u = \lambda u$ has a strictly positive solution $u > 0$ on $(a,b)$.

We consider the periodic Cauchy problem for the Gurevich-Zybin system on the n dimensional torus. The GZ system models dark matter as a collision-free gas in an expanding universe. We establish local in-time well-posedness for classical solutions in Sobolev spaces. Moreover, the local in-time result is extended to all time under additional assumptions. In particular, we consider three cosmological eras determined by the universal expansion parameter. In each era, global in time existence is established if the initial density contrast is small and the rate of change of the initial velocity is sufficiently large (joint work with J. Holmes and R. Thompson).

Important properties of physical systems in the ambit of quantum field theory are often encoded in the spectrum of Laplace-type operators. To analyze such properties the spectrum is organized in the form of so-called spectral functions. The most prominent spectral function employed in mathematical physics is the spectral zeta function. This talk will serve as a gentle introduction to this important function and will illustrate, through simple examples, the analytic techniques used to study it. This talk is designed to provide the audience with a basic understanding of the spectral zeta function and its potential applications and is especially suitable for graduate students and non-experts.

The main result of the talk: Given any $\epsilon>0$, every locally finite subset of $\ell_2$ admits a $(1+\epsilon)$-bilipschitz embedding into an arbitrary infinite-dimensional Banach space.

The result is based on two results which are of independent interest:

(1) A direct sum of two finite-dimensional Euclidean spaces contains a sub-sum of a controlled dimension which is $\epsilon$-close to a direct sum with respect to a $1$-unconditional basis in a two-dimensional space.

(2) For any finite-dimensional Banach space $Y$ and its direct sum $X$ with itself with respect to a $1$-unconditional basis in a two-dimensional space, there exists a $(1+\epsilon)$-bilipschitz embedding of $Y$ into $X$ which on a small ball coincides with the identity map onto the first summand and on a complement of a large ball coincides with the identity map onto the second summand.

The Besicovitch projection theorem asserts that if a subset E of the plane has finite length in the sense of Hausdorff and is purely unrectifiable (so its intersection with any Lipschitz graph has zero length), then almost every linear projection of E to a line will have zero measure. As a consequence, the probability that a randomly dropped line intersects such a set E is equal to zero. This shows us that the Besicovitch projection theorem is connected to the classical Buffon needle problem. Motivated by the so-called Buffon circle problem, we explore what happens when lines are replaced by more general curves. This leads us to discuss generalized Besicovitch theorems and the ways in which we can quantify such results by building upon the work of Tao, Volberg, and others. This talk covers joint work with Laura Cladek and Krystal Taylor.

The Schur product of two n by n matrices is a pointwise product. More precisely, the Schur product of $m= (m_{ij})$ and $A=(a_{ij})$ is the matrix $(m_{ij}a_{ij})$. Let us fix the matrix m and view the Schur product of matrices with m as a map on the matrix algebra, which we call a Schur multiplier. The boundedness of Schur multipliers (with more general indices) naturally connects to the approximation property of group von Neumann algebras as shown in the work of Haagerup, Lafforgue/de la Salle, Parcet/de la Salle/Ricard, and Parcet etc.

In this talk, I plan to introduce an analogue of Marcinkiewicz-multiplier theory for Schur multipliers on Schatten-p classes. The talk will be based on a recent work with ChianYeong Chuah and Zhenchuan Liu.

We analyze non-perturbatively the one-dimensional Schrodinger equation describing the emission of electrons from a model metal surface by a classical oscillating electric field. The amplitude of the external electric field and the frequency are arbitrary.

We prove existence and uniqueness of classical solutions of the Schrodinger equation for general initial conditions. When the initial condition is in L2 the evolution is unitary and the wave function goes to zero for any fixed x as t becomes large. To show this we prove a RAGE type theorem and show that the discrete spectrum of the quasienergy operator is empty.

Due to a number of pathological features of the Hamiltonian (among which unboundedness in the physical as well as the spatial Fourier domain) the existing methods to prove such results do not apply, and we introduce new, more general ones. The actual solution exhibits a very complex behavior, as seen both analytically and numerically: it shows a steep increase in the current as the frequency passes a threshold value (which depends on the strength of the electric field. For small electric field, this represents the threshold in the classical photoelectric effect, as described by Einstein’s theory.

Work in collaboration with Ovidiu Costin, Ian Jauslin and Joel L Lebowitz.

The article is devoted to the existence of solutions of a system of integro-differential equations in the case of the normal diffusion and the influx/efflux terms proportional to the Dirac delta function. The proof of the existence of solutions relies on a fixed point technique. We use the solvability conditions for the non-Fredholm elliptic operators in unbounded domains.