Weekly Bulletin

Using (almost) toric fibrations and their visible Lagrangians, we can construct many novel and interesting examples of Lagrangian submanifolds of symplectic four-manifolds. Naturally, one can ask whether visible Lagrangians are all the Lagrangians that exist, or, in other words, how faithful the pictures coming from almost toric fibrations are. I will answer this question for Klein bottles in $(S^2 \times S^2,\omega_{\lambda})$, i.e. the product of two spheres where the first factor has area 1 and the other factor has area $\lambda$. In particular, I will first construct a visible Lagrangian Klein bottle when $\lambda < 2$. Then I will show that no Lagrangian Klein bottles exist otherwise. The key input for obstructing the existence of the Klein bottles is Luttinger surgery along with techniques of (compact) pseudoholomorphic curves and Seiberg--Witten theory. This is joint work with J. Evans.

In this talk, I’ll discuss a Liouville theorem for ancient solutions to supercritical Fujita equations, which says if the solution is close to the ODE solution at large scales, then it must be the ODE solution. Then I’ll discuss some application of this Liouville theorem to the analysis of first time singularity in this problem. This is based on a joint work with Juncheng Wei and Ke Wu.

<div data-olk-copy-source="MessageBody"> In this talk, I will start by explaining the basics of Morse theory, a powerful tool of study for smooth manifolds which simply uses a sufficiently generic smooth function on that manifold, which we call a Morse function. I will then explain how you can recover the fundamental group of the manifold from the Morse function, and if time permits, give some other properties of this "Morse fundamental group".</div> <div> </div>

<p><span style="color: #333333;">Cross-sections provide a powerful tool for translating problems about continuous flows into discrete dynamical systems. In this talk, I will introduce the basic theory of cross-sections, illustrating it with examples and applications. I will show how this framework can be used to establish a version of the Lévy–Khintchine theorem for almost every point with respect to a wide class of measures that are singular to Lebesgue measure—in particular, for almost every point in the middle-third Cantor set equipped with its natural self-similar measure. I will then discuss how the notion of cross-section extends to multi-parameter flows, the challenges that arise in this setting, and conclude with an application of this theory to a conjecture of Y. Cheung. <br><br>The talk is based on joint work with Anish Ghosh.</span></p>

The set of finite quotients of a finitely generated group is neatly captured by its profinite completion. A finitely generated (residually finite) group G is called profinitely rigid if whenever another finitely generated (residually finite) group H has isomorphic profinite completion, then H is isomorphic to G. This talk will discuss recent progress on groups that are (are not) profinitely rigid, and in particular describe a construction (Grothendieck Pairs) that produce a finitely presented group that is rigid amongst finitely presented groups, but not amongst finitely generated ones. An emphasis will be placed on the central role that 3-manifold groups play.

The circular β-ensemble (CβE) is a classical model in random matrix theory which generalizes the eigenvalue process of Haar- distributed unitary random matrix. It can be interpreted as a system of two-dimensional point charges at equilibrium on the unit circle. The goal of this talk is to explain how to describe the asymptotics properties of the CβE characteristic polynomial using the theory of orthogonal polynomials on the unit circle (OPUC). I will show that renormalized powers of the characteristic polynomial converge to multiplicative chaos measures. If time permits, I will explain the connection with the eigenvalue counting function, eigenvalue rigidity and previous results on the CβE spectral measure and the Fyodorov- Bouchaud conjecture. This is joint work with Joseph Najnudel (University of Bristol).

<p>I will present a recent result concerning the quasineutral limit for the relativistic Vlasov--Maxwell system, a fundamental model in plasma physics. The quasineutral limit describes the regime in which the Debye length — the characteristic scale of charge separation — becomes negligible compared to the macroscopic scale of the plasma, so that the plasma behaves as if it were electrically neutral. While this singular limit has been extensively studied in the electrostatic case (i.e., the Vlasov--Poisson system), the full electromagnetic setting introduces new difficulties due to the coupling with Maxwell’s equations and the presence of a magnetic field, which give rise to additional oscillatory phenomena. I will explain how, in this context, one can rigorously establish strong convergence to a limiting electron magnetohydrodynamics (e-MHD) system, and clarify the precise sense in which this limit holds. This is based on a joint work with A. Gagnebin, M. Iacobelli, and A. Rege.</p>

<p>V-states are uniformly rotating vortex patch solutions to the 2D Euler equations. Namely, the vorticity is given by the characteristic function of a domain which rotates around the origin with constant angular velocity. Examples of V-states include Kirchhoff ellipses and m-fold symmetric patches which bifurcate from the disk. I will describe how, for any V-state satisfying a certain non-degeneracy condition, there exist smooth rigidly rotating solutions to the 2D Euler equations approximating it arbitrarily well in the natural Hölder spaces. I will then argue that all but countably many Kirchhoff ellipses, as well as all m-fold symmetric V-states near the disk satisfy this non-degeneracy.</p>

The goal of this talk is to present recent results in the field of time-inconsistent stopping and control problems driven by stochastic differential equations. The talk considers the weak equilibrium approach. While solving these problems interesting novel type of strategies appear that are not present in the time consistent version. In particular for time-inconsistent stopping problems, a class of mixed (i.e., randomized) stopping times based on a local time construction of the stopping intensity appears. The counterpart for singular time-inconsistent control problems is given by a control which results in an absolutely continuous rate that creates an inaccessible boundary. Additionally, can be shown that the classical solution approach to these problems used in the time-consistent version (pure stopping times with no mixing and reflection for the SSC) does not always result in an equilibrium making these type of novel strategies necessary. This creates further insight into the strategy space that should be considered in order to prove general existence of equilibria, which is an open problem for most time-inconsistent problems.

Definite integrals where the integrand contains an exponential function are a common sight in mathematics and mathematical physics. In particular cases, the values of these integrals turn out to be expressible as definite integrals with an algebraic integrand, that means, they are classical periods. A very well known example for this is the identity $$\int_0^\infty t^{-1} \sin(t) dt = \pi/2 = \int_{-1}^1 \sqrt{1- t^2}dt.$$ Another example is given by moment integrals of Bessel functions, which turn out to be multiples of pi by an algebraic number in the simplest cases. In more elaborate cases however, the formulas for Bessel moments start to involve special values of L-functions, elliptic integrals and values of hypergeometric functions, all of which are classical periods associated with algebraic varieties defined over number fields. I will explain why it is no accident that Bessel moments, and indeed many other similar definite integrals produce period numbers, and which algebraic varieties they come from.

The goal of this talk is to discuss the geometry and intersection theory of Le Potier’s moduli space of 1-dimensional sheaves on the projective plane. We focus on two aspects: first, the "P=C" conjecture relating two filtrations of highly different nature on cohomology, which can be viewed as a del Pezzo analog of the celebrated P=W conjecture; second, the so-called χ-independence phenomenon, which stems from enumerative geometry and predicts surprising consequences on the cohomology of the moduli space. After surveying known results, I will explain how these two aspects are linked via an “associated graded” χ-independence conjecture (work in progress). This talk is primarily based on joint work with Yakov Kononov, Woonam Lim, and Miguel Moreira.