Abstract: I will explain the combinatorics behind the cluster algebra structure on braid and Richardson varieties. This generalizes the previously known constructions for positroid varieties and double Bruhat cells. The cluster algebra is described in terms of a 3-dimensional generalization of Postnikov’s plabic graphs, and the underlying quiver is induced by the intersection form of the Goncharov–Kenyon conjugate surface associated to a 3D plabic graph. Joint work with T. Lam, M. Sherman-Bennett, and D. Speyer

The cohomology of moduli spaces of 1-dimensional sheaves on del Pezzo surfaces carries a perverse filtration, from which curve counting invariants of CY 3-folds can be extracted. Conjecturally, the perverse filtration matches a second filtration, defined in terms of tautological classes, called the Chern filtration. I will explain this conjecture, what is currently known, and some consequences. In the second part of the talk I will explain some new ideas to prove properties of the Chern filtration by using parabolic sheaves. In particular, I will discuss the top Chern degree and chi-independence phenomena. The talk is based on joint work in progress with W. Lim and W. Pi.

We study a notion of extended positive representations of surface groups. Examples include maximal representations (Burger—Iozzi—Wienhard), positive representations (Fock—Goncharov), and cusped Hitchin representations (Canary—Zhang—Zimmer). We discuss conditions under which these representations are Anosov or relatively Anosov. We prove that extended positivity is a closed condition, and open in certain subspace of the character variety. Moreover, we describe the boundary of the closure of extended positive representations into a semisimple Lie group G in the real spectral compactification of Hom(\Gamma,G) (introduced by Burger—Iozzi—Parreau—Pozzetti), showing that it consists of extended positive representations into extensions of G over real closed fields. This is joint work with Xenia Flamm, Nicolas Tholozan, and Tengren Zhang.

Abstract: I will explain a proof of the BCFW triangulation conjecture which states that the cells appearing in the Britto–Cachazo–Feng–Witten (BCFW) recursion triangulate the amplituhedron (in full generality at all loop levels). The key ingredient is a relation to origami crease patterns which are planar graphs with faces colored black and white, embedded in the plane so that the sum of black (equivalently, white) angles at each vertex is 180°. Along the way, we prove conjectures of Chelkak–Laslier–Russkikh and Kenyon–Lam–Ramassamy–Russkikh on the existence of such origami embeddings of arbitrary planar graphs, which originated from the works of Kenyon and Smirnov on the conformal invariance of the dimer and Ising models.

Seminar talk is supported in part by the Mrs. Hepsa Ely Silliman Memorial Fund.

Abstract: I will explain the combinatorics behind the cluster algebra structure on braid and Richardson varieties. This generalizes the previously known constructions for positroid varieties and double Bruhat cells. The cluster algebra is described in terms of a 3-dimensional generalization of Postnikov’s plabic graphs, and the underlying quiver is induced by the intersection form of the Goncharov–Kenyon conjugate surface associated to a 3D plabic graph. Joint work with T. Lam, M. Sherman-Bennett, and D. Speyer.

Abstract: We formulate a stable Bernstein problem in two types of positively curved manifolds. We completely solved for one type in all dimensions and partially solved for the other in low dimensions, leading to several interesting corollaries.

In 2009, Kahn and Markovic proved the Surface Subgroup Theorem, and they constructed a ubiquitous collection of \pi_1-injective immersed surfaces in closed hyperbolic 3-manifolds. Hamenstadt later showed that any cocompact lattice of some simple rank 1 Lie group other than SO(2m,1) (m >= 1) has a surface subgroup. Recently, we constructed \pi_1-injective immersed surfaces in closed hyperbolic 2n-manifolds, which addresses the cases missing from Hamenstadt’s work. This is joint work with Jeremy Kahn.

We have impromptu and (sometimes) scheduled talks, on topics in probability, combinatorics, geometry, and dynamics.

Everyone is welcome! 

Abstract:  The uniqueness of tangent flows is central to understanding singularity formation in geometric flows. A foundational result of Colding and Minicozzi establishes this uniqueness at cylindrical singularities under the Type I assumption in the Ricci flow. In this talk, I will present a strong uniqueness result for cylindrical tangent flows at the first singular time. Our proof hinges on a Łojasiewicz inequality for the pointed $\mathcal{W}$-entropy, which is established under the assumption that the local geometry near the base point is close to a standard cylinder or its quotient. This is joint work with Yu Li.

Boyer, Gordon, and Hu proposed a relative version of the L-space conjecture using slope detections and conjectured that this should be equivalent to the L-space conjecture for toroidal manifolds. We establish the equivalence for the foliation-detection part.

Abstract: The famous Aharonov–Bohm effect demonstrates the non-trivial quantum mechanical effect of (singular) magnetic potentials. Although the single pole case has been studied widely using rotational and scaling invariance, the technique fails to work for the Aharonov–Bohm Hamiltonian with multiple poles. In this talk, we will discuss a new framework to study the resolvent, resonances, and wave asymptotics of the Aharonov–Bohm Hamiltonian with multiple poles. This results in an interesting behavior where scattering and wave asymptotics depend on the magnetic fluxes, interpolating the odd and even dimensional Euclidean scattering. This talk includes joint works with Tanya Christiansen and Kiril Datchev. 

Representation theoretic multiplicities are at the heart of many open
problems in algebraic combinatorics. At the same time these
quantities appear in Geometric Complexity Theory in the search for
multiplicity obstructions for separating computational complexity
classes like VP vs VNP. Most recently they have also been considered
in quantum computing.
In this talk we will introduce the objects and problems, explain how
formalization through computational complexity theory could answer
some of the open problems in the negative. We will also explain their
role in GCT and quantum computing with a mixture of positive and
negative answers.

Abstract: The Robinson–Schensted (RS) correspondence admits diagrammatic interpretations via Fomin’s growth diagrams and Viennot’s shadow line construction. Works of Spaltenstein, Springer, Steinberg, and van Leeuwen connected this combinatorial construction to the relative position map of Springer fibers. Motivated by Kazhdan–Lusztig cell theory, various generalizations of the Robinson–Schensted correspondence into the affine type A setting have been studied. Prominent examples include Shi’s insertion algorithm and the affine matrix ball construction by Chmutov–Pylyavskyy–Yudovina. In particular, using the affine matrix ball construction, Boixeda–Ying–Yue showed that the two-sided cells and S-cells agree when the nilpotent is of rectangular type.

In this talk, we introduce a new combinatorial construction of the affine RS correspondence via growth diagrams and shadow lines that is in a sense dual to Shi’s insertion and the affine matrix ball construction, and geometrically natural in terms of relative positions of affine flags.

Ongoing joint work with Sylvester Zhang.

The goal of this work is to explore and develop the machinery of branched bending in finite-volume hyperbolic n-manifolds as a means of explaining the flexibility of these manifolds. Here, we define branched bending deformations as deformations supported on a piecewise totally geodesic complex of (n-1)-dimensional faces meeting along (n-2)-dimensional branching loci. We establish a lower bound on the dimension of the deformation space of such manifolds containing a branched complex, and establish some facts about the Borromean rings as a special example (recovering a result of Menasco and Reid).

Joint work with Amir Mohammadi, Zhiren Wang, and Lei Yang

Let Q be an indefinite ternary quadratic form. In the 1980s Margulis proved the longstanding Oppenheim Conjecture, stating that unless Q is proportional to an integral form, the set of values Q attains at the integer points is dense in R. We give quantitative results to that effect.
In particular, if the coefficients of Q are algebraic, but Q is not proportional to an integral form, and if $(\alpha,\beta)$ a fixed interval, the number of integer points v in a ball of radius R for which $\alpha<Q(v)<\beta$ is asymptotic to $c(Q,\alpha,\beta)R$ as $R\to\infty$ with an effective power saving error term (but $c(Q,\alpha,\beta)$ might not be what you expect!).

Our work is based on a quantitative equidistribution result for unipotent flows, as well as upper bound estimates by Eskin-Margulis-Mozes and Wooyeon Kim. 

Abstract: Pour a drop of milk into a cup of coffee. We know from experience that after a few moments of stirring, the mixture becomes homogeneous. In this talk, we study the advection of a passive scalar (the concentration of milk) under a random fluid (coffee) velocity on the torus. We prove, when the velocity solves the 2d stochastic Navier–Stokes equations on the torus, that the scalar converges exponentially to its mean. Our result applies even when only finitely many (as few as 4) Fourier modes are randomly forced. Joint with Keefer Rowan (EPFL). 

We have impromptu and (sometimes) scheduled talks, on topics in probability, combinatorics, geometry, and dynamics.

Everyone is welcome!