Title:   A new approach to small black holes in string theory    

Abstract:   In this talk, I will introduce a new approach for studying $d+1$ dimensional Euclidean Schwarzschild black holes with Hawking temperature near the Hagedorn temperature, as well as the Horowitz-Polchinski (HP) solutions. The worldsheet theory that describes some of these backgrounds is strongly coupled. We use its underlying affine $SU(2)_L\times SU(2)_R$ symmetry to continue to weak coupling, by varying the level of the current algebra from the small value relevant for black holes and HP solutions to a large value. In this limit, the dynamics can be captured by a solvable effective field theory, closely related to previous work on the non-abelian Thirring model. I will also discuss several interesting properties of the resulting solutions that have implications for the black hole/fundamental string transition.

Stefano Filipazzi, Duke University
Baily--Borel compactifications of period images and the b-semiampleness conjecture

A fundamental tool to study algebraic varieties is given by morphisms to projective space. A line bundle is called semiample if some positive tensor multiple provides a morphism to projective space. Unlike the case of ample line bundles, there are no general numerical criteria to determine whether a line bundle is semiample. In particular, proving the semiampleness of line bundles can be a challenging task.
In this talk, we will report on recent developments regarding the semiampleness of line bundles coming from Hodge theory. As consequence, we will obtain functorial compactifications of general period images, generalizing classic work of Baily--Borel and thus confirming a conjecture of Griffiths. This talk is based on joint work with Bakker, Mauri, and Tsimerman.

Ben Bakker, UIC

Abstract:

The fundamental group π1(X) is an important invariant of a complex algebraic variety X. Despite its topological nature, it is closely connected to the geometry of many algebraic structures on X. In this talk I want to discuss two elementary questions involving the fundamental group: (1) How many simply-connected algebraic subvarieties does X contain? (2) How many global holomorphic functions does the universal cover X̃  have? The second question is related to a famous conjecture of Shafarevich, which asserts that in the case X is a smooth projective variety, X̃  has enough global holomorphic functions to separate points up to compact ambiguity. I will describe joint work with Brunebarbe and Tsimerman which gives a complete answer to questions (1) and (2) provided π1(X) admits a faithful linear representation, using techniques coming from non-abelian Hodge theory.

Jeffery Diller, University of Notre Dame
Equilibrium measure and entropy for unstabilizable rational surface maps

Let f be a rational self-map on a complex projective surface and λ1,λ2 be its so-called first and second dynamical degrees. The topological entropy of f is known to be bounded above by the larger of logλ1 and logλ2. It is conjectured that the bound is actually an equality. However, when logλ1 predominates, approaches to proving equality have strongly relied on the additional hypothesis that f is algebraically stable. In this talk, I will discuss recent work with Roland Roeder in which we prove the conjecture for many unstabilizable rational maps. These include some for which λ1 is a transcendental number.

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Title:   The initial boundary value problem for the Einstein vacuum equations with umbilic boundary
Speaker:   Grigorios Fournodavlos
Abstract:   In this talk, we will discuss the initial boundary value problem in general relativity from a mathematical point of view. We will first make the analogy to the classical Cauchy problem for the Einstein equations, first resolved by Yvonne Choquet-Bruhat in 1952, and highlight some desired properties that a potential treatment of timelike boundaries should have. Then, we will go over the various technical difficulties that arise in such an endeavor. The main result that we will present in the end concerns the local well-posedness of the problem for homogeneous Neumann conditions, in geometric terms, when the boundary is umbilic.

Title:   Thermodynamics of gravity at finite cutoff
Speaker:   Batoul Banihashemi
Abstract:   Defining thermodynamic ensembles for gravitational systems is tied to specifying boundary conditions. In this talk I'll first briefly review how introducing a Dirichlet timelike boundary clarifies thermodynamics of de Sitter space. Then I'll discuss geometries subjected to conformal boundary conditions, where the conformal class of the boundary metric and the trace of the extrinsic curvature K are held fixed. In the high temperature limit the series of subextensive terms in the free energy are compared to predictions from thermal effective field theory, and we observe agreement in all considered cases. Ensembles with negative K include solutions with cosmic-type horizons, where the system boundary is smaller than the horizon. In some regions of parameter space, these solutions are the dominant phase and are necessary for consistency with thermal effective field theory.

Seraphina Lee, Harvard University
TBA

Title:   Timelike Dirichlet Boundaries and the End of Time
Speaker:   Don Marolf 
Abstract:   TBA

Title:   Quantum aspects of time-like boundaries
Speaker:   Gonzalo Torroba (Discussion)
Abstract:   Timelike boundaries in gravitational systems provide a powerful framework for understanding quasilocal holography, cosmological horizons, and the microscopic structure of spacetime in the absence of asymptotic regions. This discussion session will focus on quantum aspects of timelike boundaries, including possible UV completions, lessons from holography, and effects from quantum fields. Participants are invited to share perspectives, open problems, and recent progress in this area.

Title:   Euclidean well-posedness and Lorentzian ill-posedness of generalized conformal boundary conditions
Speaker:   Xiaoyi Liu
Abstract:   We study four-dimensional Einstein gravity with a finite boundary. We introduce a one-parameter family generalization of Anderson’s conformal boundary condition and show that these boundary conditions lead to a well-posed elliptic system in Euclidean signature. We further discuss the associated thermodynamic properties. In Lorentzian signature, we analyze the corresponding initial-boundary value problem (IBVP) and demonstrate that these boundary conditions render the system ill-posed at the linearized level.

Title:   A theory of two-sided de Sitter space with matter and the cosmological future wedge
Speaker:   Gauri Batra
Abstract:   We show how to capture the states in the spectrum of two-sided de Sitter space with timelike boundaries, taking into account the causal connection between the two boundaries in states with matter. To do so, we review the construction of the theory describing one static patch with a timelike boundary via a deformation of AdS/CFT. We outline how this leads to a microstate count of the de Sitter horizon. We then study the two-sided theory by constructing the thermofield double for this system. We show how to consistently combine the one-sided theories to produce a theory of the two-sided system, where the two boundaries are in causal contact. This theory reproduces some expected properties of the two-sided system. We end with some exploration of bulk reconstruction in this framework, enabling us to study the cosmological future wedge

Title:   TBA
Speaker:   Kuroush Allameh
Abstract:   TBA

Title:    Discussion
Speaker:   Chawakorn Maneerat/Andrew Svesko (Discussion)
Abstract:   TBA

Title:   Geometric boundary conditions for the IBVP of vacuum Einstein equations
Speaker:   Zhongshan An
Abstract:   In general relativity, it is of great interest to construct spacetimes satisfying the vacuum Einstein equations. While the Cauchy problem of vacuum Einstein equations has been well studied since the work of Choquet-Bruhat, the initial boundary value problem (IBVP) remains much less understood. To establish a well-posed IBVP, one needs to impose appropriate boundary conditions on the time-like boundary. However, due to the complexity of energy estimate and gauge issue, so far there has not been a canonical choice of boundary conditions. In this talk, I will discuss the local-in-time well-posedness of various choices of geometric boundary conditions for the IBVP based on a series of works joint with Michael Anderson

Title:   TBA
Speaker:   Pau Figueras
Abstract:   TBA

Title:   Student presentations
Speaker:   Samanta Saha (Case Western U.),  Antonia Seifert (Perimeter Institute),Xinkang Wang (Chicago U.), Themistoklis Zikopoulos (Perimeter Institute)
Abstract:   See Webpage under Student Presentations for titles and abstracts

Title:   TBA
Speaker:   Simone Speziale
Abstract:   TBA

Title:   Weak turbulence on Schwarzschild-AdS spacetime
Speaker:   Christopher Kehle
Abstract:   TBA

Title:   TBA
Speaker:   Amr Ahmadain 
Abstract:   TBA

Title:   TBA
Speaker:   Silvia Georgescu
Abstract:   TBA

Title:   Gravity on a finite region with fixed-area boundary conditions
Speaker:   Tom Hartman
Abstract:   Gravity can be formulated on a finite region of Euclidean spacetime, with boundary conditions that fix the areas or bending angles at extremal surfaces on the boundary. The path integral with these boundary conditions computes the statistics of black hole operators. I will discuss this approach in general dimensions, and then focus on the semiclassical geometries in 3D gravity with negative cosmological constant, which are built from generalized hyperbolic tetrahedra. These are the geometries underlying recent topological calculations in Virasoro TQFT and Conformal Turaev-Viro theory.

Title:   Towards Hamiltonian String Field Theory, and Dirichlet Walls
Speaker:   Aron Wall
Abstract:   I will present preliminary work towards formulating string field theory in a Hamiltonian (equal time slice) formalism. This problem is closely related (via a Wick rotation) to the problem of finding consistent End-Of-The-World boundary conditions for string scattering, in which the metric satisfies Dirichlet boundary conditions (rather than Neumann). In defining the transition amplitudes, two fundamental problems arise: (i) string worldsheets have tadpoles which transgress across the boundary infinitely often, and (ii) when a string does cross the boundary, it can cross multiple times. However, I believe that both of these problems can be overcome (at least at tree level) by clever resummation tricks.