Events

<p class="x_elementToProof"><span data-ogsc="rgb(0, 0, 0)" data-olk-copy-source="MessageBody">In the 1970s, Bernstein-Gelfand-Gelfand (BGG) constructed a remarkable resolution of highest weight representations of a semisimple Lie algebra in terms of Verma modules. Its geometric incarnation, the dual BGG complex introduced by Faltings in 1983, plays an important role in the study of automorphic forms and the cohomology of Shimura varieties.</span></p> <p class="x_elementToProof"><span data-ogsc="rgb(0, 0, 0)">In this talk, I will try to explain both the representation-theoretic and geometric sides of the BGG story. We will begin with the classical BGG resolution and then see how, via certain bundle functors, it gives rise to a complex of automorphic vector bundles on Shimura varieties whose differentials are natural differential operators on automorphic forms. If time permits, I will also discuss my recent work revisiting the dual BGG complex and constructing these BGG differential operators geometrically, extending Coleman's 1996 construction.</span></p>

<p><span style="caret-color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">For higher rank center free simple Lie groups \(G\) and discrete subgroups \(\Gamma\) of them, we quantify the growth of injectivity radius in \(G/\Gamma\) as a function of the covolume of \(\Gamma\) in \(G\) (which may be infinite). </span><span style="caret-color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">This result is used to prove quantitative versions of three results in the field of discrete subgroups of Lie groups and invariant measures: Stuck-Zimmer theorem, the (quite) recent Fraczyk-Gelander theorem, and the Benjamini-Schramm convergence of symmetric spaces to the universal cover. This also gives a new proof of Stuck Zimmer theorem. I</span><span style="caret-color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">n particular, we prove that whenever \(\Gamma\) is not a lattice, one can find a point (in fact high measure of points) in the ball of radius \(R\) in \(G\) where the injectivity radius of \(G/\Gamma\) is at least \(\log \log \log \log(R)\), thereby quantifying the result of Fraczyk-Gelander which states unboundedness without any rate. </span><span style="caret-color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">This growth of injectivity radius with the volume is proved using a new structure theory for almost stationary measures (itself a new notion), and a corresponding quantitative version of the Nevo-Zimmer structure theorem.</span></p> <p><br style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"><span style="caret-color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">In the talk we will to discuss the main theorem, how to derive its applications, and some of the methods used to prove it. </span><br style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"><br style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"><span style="caret-color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; display: inline !important; float: none;">This talk is based on two joint works with Ilya Gekhtman, Simon Machado and Omri Solan.</span></p>

Moduli spaces often admit several natural compactifications. The comparison of different compactifications is referred to as wall-crossing. Some compactifications are simpler than others, making wall-crossing one of the most powerful established techniques, for example, in the study of moduli spaces of sheaves. I will discuss wall-crossing phenomena for moduli spaces of maps (Gromov–Witten theory) and other related spaces. The first glimpses of wall-crossing in Gromov–Witten theory go back to its beginnings, namely to Givental’s proof of mirror symmetry for the quintic in 1990s. Later, independently of Givental’s work, Manin and Bayer were the first to explore the idea of wall-crossing in the setting of maps, extending Hassett’s ideas on weighted marked curves. Only with the work of Ciocan-Fontanine, Kim, and Maulik from 2010s, which in turn builds on the work of many others, did it become possible to exploit different compactifications of moduli spaces of maps with considerable flexibility for GIT quotients, thereby revealing Givental’s proof as a wall-crossing calculation. Zhou then made the next major step by proving the Ciocan-Fontanine–Kim wall-crossing formulas for GIT quotients in full generality. The goal is to explain how the ideas of Ciocan-Fontanine, Kim, and Zhou can be applied in three different settings: configuration spaces of points, maps to GIT quotients, and Hurwitz theory. Based on these examples, I will try to formulate some general principles underlying such wall-crossing phenomena. The final goal is to combine all three settings to provide a wall-crossing between spaces of maps and Hilbert schemes associated to threefolds of the form Surface × Curve.

When does a sequence of hyperbolic 3-manifolds with volume going to infinity have exponentially growing torsion homology? For arithmetic towers, the work of Bergeron-Sengun-Venkatesh suggests a set of conditions that conjecturally imply exponential growth of torsion homology. Their work relies on Cheeger-Mueller's theorem, linking torsion homology and analytic torsion. For nice sequences of hyperbolic 3-manifolds we use a different approach to find a condition implying exponential torsion homology growth: we give a condition on the spectrum of the Laplacian. I will give several motivations for this condition and show how to construct concrete examples of sequences satisfying it. This is based on joint work with Anshul Adve, Vikram Giri, Ben Lowe and Jonathan Zung.

We study space-time Galerkin--Petrov formulations for parabolic evolution problems and their relation to classical implicit time-stepping schemes. Although such schemes are stable in the usual time-stepping sense, their interpretation as space-time operator equations may lead to conditional stability, with constants depending on the relation between temporal and spatial mesh sizes. We revisit this phenomenon for the continuous Galerkin method of Aziz and Monk, which yields the Crank--Nicolson scheme in the lowest-order case, and provide a detailed space-time error analysis for solutions of both high and low regularity. In particular, the space-time framework allows us to analyze the deteriorated behaviour of classical time-stepping methods for nonsmooth initial data. By applying integration by parts in time, we derive an adjoint space-time formulation that incorporates the initial condition in a natural variational way. In the lowest-order case, this formulation leads to a Rannacher-type smoothing of the initial data. The theoretical results are complemented by numerical experiments. Joint work with R. Löscher and M. Reichelt.

We discuss a class of graph embeddings into Minkowski space R^{2,2} = C^{1,1}, called t-surfaces, which arise in the study of the planar dimer model. A t-surface consists of a (perfect) t-embedding together with its associated origami map. Perfect t-embeddings were recently introduced as a key tool for proving that the gradient of the dimer height function converges to that of the Gaussian Free Field in a canonically associated metric, under suitable technical assumptions. After introducing the notion of a t-embedding, we describe a construction of perfect t-embeddings for regular hexagons of the hexagonal lattice. For which the corresponding t-surfaces converge to space-like maximal surfaces in Minkowski space R^{2,1}. As a consequence, this construction yields a new proof of the convergence of fluctuations of the dimer height function to the Gaussian Free Field in the conformal structure induced by the limiting maximal surface.

Short after the definition of the Khovanov homology at the very end of the 20th century by Mikhail Khovanov, Jacob Rasmussen defined a knot invariant using a deformation of Khovanov homology, namely Lee-Khovanov homology. The Milnor conjecture concerning the slice genus of torus knots, namely g₄(T_(p,q)) = ½(p-1)(q-1), which was originally proved using gauge theory by Kronheimer and Mrowka was reproved using the Rasmussen invariant. This first purely combinatorial proof is considered to be the first application of Khovanov homology.

Stably exotic fillings of a 3-manifold Y are 4-manifolds bounding Y that are homeomorphic but not stably diffeomorphic rel. Y. We show that every closed, orientable 3-manifold admits such fillings, as do certain families of nonorientable 3-manifolds. In contrast we show that for a 3-manifold containing a 2-sided RP², any two smooth, homeomorphic fillings are stably diffeomorphic. This is joint work with Patrick Orson, Mark Powell, and Arunima Ray.

<p>In this talk, we follow in the footsteps of Leonardo da Vinci. He observed that air bubbles can spiral as they rise through water and described this motion in his notes and sketches. We revisit this phenomenon in an idealised fluid model: the three-dimensional, two-phase incompressible Euler equations with surface tension.</p> <p>We construct bubbles and drops whose centroids trace helical paths, with irrotational flow in each fluid phase. These solutions are steady in a frame translating along and rotating about a fixed axis. The proof is a symmetry-breaking bifurcation argument around the stationary spherical solution. I will explain the main ideas behind the construction and discuss a complementary rigidity result highlighting the role of surface tension.</p> <p>This is joint work with Björn Gebhard and Christian Seis.</p>

In this talk, we consider a time-homogeneous d-dimensional diffusion with no drift coefficient. We give a necessary and sufficient condition for the convexity in the starting position of the distribution of its solution on the path-space when the codomain is endowed with the convex order. To do so, we introduce and check the seemingly stronger notion of coupling-convexity in the starting position.

The universal moduli space of slope-semistable (with respect to the canonical polarization) vector bundles on stable curves is not compact. There are two natural ways to compactify it. The first is to add slope-semistable torsion-free sheaves, while the second is to consider vector bundles on semistable curves. In this talk, we briefly review the compactification by torsion-free sheaves and then focus on the second approach, discussing its main properties and its relationship with the former.