# Seminars & Colloquia

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The branched virtual fibering theorem by Sakuma states that every closed orientable 3-manifold with a Heegaard surface of genus g has a branched double cover which is a genus g surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minimal entropy of pseudo-Anosov monodromies of all hyperbolic, genus g surface bundles as branched double covers of the 3-sphere behaves like 1/g. We also give an alternative construction of surface bundles over the circle in Sakuma's theorem when closed 3-manifolds are branched double covers of the 3-sphere branched over links. A feature of surface bundles coming from our construction is that their monodromies can be read off the braids obtained from the links as the branched set.

The branched virtual fibering theorem by Sakuma states that every closed orientable 3-manifold with a Heegaard surface of genus g has a branched double cover which is a genus g surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minimal entropy of pseudo-Anosov monodromies of all hyperbolic, genus g surface bundles as branched double covers of the 3-sphere behaves like 1/g. We also give an alternative construction of surface bundles over the circle in Sakuma's theorem when closed 3-manifolds are branched double covers of the 3-sphere branched over links. A feature of surface bundles coming from our construction is that their monodromies can be read off the braids obtained from the links as the branched set.

Thanks to Gromov's link condition, it is easy to construct many CAT(0) cube complexes. On the contrary, constructing hyperbolic cube complexes is often a delicate matter. In this talk I will briefly explain the standard technique that is used to show that some Right Angled Coxeter Groups are hyperbolic and I will then introduce a new technique which applies to a larger class of cube complexes (cube complexes with coupled links).

In many applications, the dynamics of gas and plasma can be accurately modeled using kinetic Boltzmann equations. These equations are integro-differential systems posed in a high-dimensional phase space, which is typically comprised of the spatial coordinates and the velocity coordinates. If the system is sufficiently collisional the kinetic equations may be replaced by a fluid approximation that is posed in physical space (i.e., a lower dimensional space than the full phase space). The precise form of the fluid approximation depends on the choice of the moment-closure. In general, finding a suitable robust moment-closure is still an open scientific problem.

In this work we consider two specific closure methods: (1) a regularized quadrature-based closure (QMOM) and (2) a nonextensible entropy-based closure (QEXP).

In QMOM, the distribution function is approximated by Dirac deltas with variable weights and abscissas. The resulting fluid approximations have differing properties depending on the detailed construction of the Dirac deltas. We develop a high-order discontinuous Galerkin scheme to numerically solve resulting fluid equations. We also develop limiters that guarantee that the inversion problem between moments of the distribution function and the weights and abscissas of the Dirac deltas is well-posed.

In QEXP, the true distribution is replaced by a Maxwellian distribution multiplied by a quasi-exponential function. We develop a high-order discontinuous Galerkin scheme to numerically solve resulting fluid equations. We break the numerical update into two parts: (1) an update for the background Maxwellian distribution, and (2) an update for the non-Maxwellian corrections. We again develop limiters to keep the moment-inversion problem well-posed.

Imaginary geometry [Miller-Sheffield '16] provides a coupling between Schramm-Loewner evolutions (SLE) and Gaussian free field (GFF), and even can be combined with Liouville quantum gravity (LQG) through mating of trees [Duplantier-Miller-Sheffield '18]. This talk gives a brief survey on this program. Then we suggest a construction of north-going flow line in imaginary geometry from alternating west and east-going flow lines, using an excursion theory for planar Brownian motions. This leads a convergence of multiple trees in peanosphere which has been employed in specific settings [Gwynne-Holden-Sun '16, Li-Sun-Watson '17]. Joint work with E. Gwynne, N. Holden, X. Sun, and S. Watson.

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E6-1, ROOM 3434
Discrete Math
Tillmann Miltzow (Université Libre de Bruxelles, Brussels)
The Art Gallery Problem is ∃R-complete

We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classical problem in computational geometry. Given a simple polygon P and an integer k, the goal is to decide if there exists a set G of k guards within P such that every point p∈P is seen by at least one guard g∈G. Each guard corresponds to a point in the polygon P, and we say that a guard g sees a point p if the line segment pg is contained in P. The art gallery problem has stimulated extensive research in geometry and in algorithms. However, the complexity status of the art gallery problem has not been resolved. It has long been known that the problem is NP-hard, but no one has been able to show that it lies in NP. Recently, the computational geometry community became more aware of the complexity class ∃R. The class ∃R consists of problems that can be reduced in polynomial time to the problem of deciding whether a system of polynomial equations with integer coefficients and any number of real variables has a solution. It can be easily seen that NP⊆∃R. We prove that the art gallery problem is ∃R-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the art gallery problem, and (2) the art gallery problem is not in the complexity class NP unless NP=∃R. As a corollary of our construction, we prove that for any real algebraic number α there is an instance of the art gallery problem where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many geometric approaches to the problem. This is joint work with Mikkel Abrahamsen and Anna Adamaszek.

The standard action of a torus $T := (\mathbb{C}^{\ast})^n$ on the complex vector space $\mathbb{C}^n$ induces an action of $T$ on the flag variety $\mathcal{F}\ell(\mathbb{C}^n)$. The Schubert variety $X_w$ associated to a permutation $w \in \mathfrak{S}_n$ admits the action of $T$. In this talk, we study the closure $Y_w$ of a generic $T$-orbit in the Schubert variety $X_w$. We associate a graph $\Gamma_w(u)$ to each $u \leq w$, and show that $Y_w$ is smooth at the fixed point $uB$ if and only if the graph $\Gamma_w(u)$ is acyclic. This is joint work with Mikiya Masuda.

There have been many exciting developments on the arithmetic of Shimura varieties in the recent decade, which have contributed to better understandings of the (still largely conjectural) Langlands programme. In this talk, I’ll try to give an overview of recent developments in mod p reduction of Shimura varieties (in both good reduction and bad reduction cases), and conclude the talk by introducing my joint work with Paul Hamacher on Mantovan’s formula (a cohomological formula conjecturally encoding the local-global compatibility of Langlands correspondences).

We introduce a new method to derive a lower bound of the mean curvature for the solutions of inverse mean curvature flow. This estimate for IMCF, an expanding curvature flow, corresponds to interior curvature estimates of Ecker-Huisken and Caffarelli-Nirenberg-Spruck for shrinking curvature flow and elliptic problem. It yields the smoothness of solutions and the existence of the flow for complete non-compact convex initial hypersurfaces.

In recent years, community detection has been an active research area in various fields including machine learning and statistics. While a plethora of works has been published over the past few years, most of the existing methods depend on a predetermined number of communities. Given the situation, determining the proper number of communities is directly related to the performance of these methods. Currently, there does not exist a golden rule for choosing the ideal number, and people usually rely on their background knowledge of the domain to make their choices. To address this issue, we propose a community detection method that also adaptively finds the number of the underlying communities. Central to our method is fused l1 penalty applied on an induced graph from the given data.

The classical Grunwald–Wang theorem is an example of a local–global principle stating that except in some special cases which are precisely determined, an element m in a number field K is an a-th power in K if and only if it is an a-th power in the completion K℘ of K for all but finitely many primes ℘ of K. In this talk, based on the analogue between the power map and the Carlitz module, I introduce an analogue of the Grundwald-Wang theorem in the Cartlitz module setting.

The original version of Waring's problem asks whether, for every positive integer n, there exists M(n) such that every non-negative integer is of the form a_1^n + ......+ a_{M(n)}^n, where the $a_i$ are non-negative integer. Since 1909, Hilbert proved that such a bound exists. In this talk, I introduce an analogue of Waring's problem for an algebraic group G, which is a generalization of the original Waring's Problem in the algebraic-group setting. (Joint work with Michael Larsen)

Groups of piecewise projective homeomorphisms provide elegant examples of groups that are non amenable, yet do not contain non abelian free subgroups. In this talk I will present a survey of these groups and discuss their striking properties. I will discuss properties such as (non)amenability, finiteness properties, normal subgroup structure, actions by various degrees of regularity and Tarski numbers.

We show that for any connected sum of lens spaces L there exists a connected sum of lens spaces X such that X is rational homology cobordant to L and if Y is rational homology cobordant to X, then there is an injection from H_1(L; Z) to H_1(Y; Z). Moreover, as a connected sum of lens spaces, X is uniquely determined up to orientation preserving diffeomorphism. As an application, we show that the natural map from the Z/pZ homology cobordism group to the rational homology cobordism group has large cokernel, for each prime p. This is joint work with Paolo Aceto and Daniele Celoria.

The Swendsen-Wang dynamics is an MCMC sampler of the Ising/Potts model, which recolors many vertices at once, as opposed to the classical single-site Glauber dynamics. Although widely used in practice due to efficiency, the mixing time of the Swendsen-Wang dynamics is far from being well-understood, mainly because of its non-local behavior. In this talk, we prove cutoff phenomenon for the Swendsen-Wang dynamics on the lattice at high enough temperatures, meaning that the Markov chain exhibits a sharp transition from “unmixed” to “well-mixed.” The proof combines two earlier methods of proving cutoff, the update support [Lubetzky-Sly ’13] and information percolation [Lubetzky-Sly ’16], to establish cutoff in a non-local dynamics. Joint work with Allan Sly.