Department Seminars & Colloquia
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In this talk we review some of the recent progresses on the mathematical theory of the surface superconducting states of type 2 superconductors and of the surface smectic states of liquid crystals, including the recent work with S. Fournais and A. Kachmar on the conjecture of surface smectic states. We discuss the effects of domain geometry and physical parameters to the characterization of the phase transitions with emphasis on the analogies between the mathematical descriptions of superconductors and liquid crystals. Some observations and questions on the related topics will also be presented.
In this talk, we would like to discuss about an algorithm to numerically solve a normal
ow equation in level set method on a polyhedron mesh in 3D. The equation has been
extensively used in image processing and surface evolution. Unlike to commonly used a
structured mesh in level set method, it is very challenging to obtain a high order scheme in
a polyhedron mesh. We propose a cell-centered gradient dened by
ux signs to design a
robust scheme considered as an extension of well-known Rouy-Tourin scheme into 3D with
the second order upwind dierence. A high order of convergence, performance in parallel
computation, and a recovery of signed distance function from a sparse data are illustrated
in numerical examples.
I have pioneered a couple of mathematical and computational approaches for free
boundary problems and optimization; 1) The immersed boundary (IB) method for
advection-electrodiffusion, 2) an IB method for non-Newtonian two-phase viscoelastic
fluids and gels, 3) an extended finite element method for phonon Boltzmann transport
and shape/topology optimization by adjoint method, 4) minimum attention in motor
control by one shot method.
Let us galvanize a couple of applications in mechanobiology; 1) Cardiac differentiation
and dendritic spine motility by the IB methods, 2) collective cell migration of wound
healing and cancer metastasis, 3) the mechanics of pulmonary arterial hypertension from
micro CT image-segmented vascular network, ventricular-vascular interaction in
coronary impedance matching, and renal peristaltic concentration.
Siegel pioneered the generalization of the theory of elliptic modular functions to the modular functions in several variables, which are called Siegel modular functions. Siegel modular functions are of fundamental importance in number theory and algebraic geometry. However, we know relatively little about Siegel modular functions until now because it is difficult to find attractive examples that can be handled. In this talk, we construct explicit generators of Siegel modular function field of higher genus and level in terms of multi-variable theta constants.
This is a joint work with Dale Rolfsen (University of British Columbia).
Let G be a group with a strict total ordering < of the elements of G.
If < is invariant under the left-multiplication,
(i.e, g < h implies fg < fh for all elements f, g and h in G)
then we call (G, <) a left-ordered group.
If the ordering < is also invariant under the right-multiplication, we call
(G,<) a bi-ordered group.
In this talk, we consider which groups of links in the 3-sphere
(that is the fundamental groups of the complements of links) are bi-orderable.
We focus on the links obtained from braids together with the braid axis.
We prove that groups of some of interesting links are bi-orderable.
Our examples include the minimally twisted 4- and 5-chain links, the Whitehead link.
We also prove that the group of the (-2,3,8)-pretzel link can not be bi-orderable.
Cohomology jump loci are topological invariants generalizing the usual singular cohomology groups. I will give a survey on the theory of cohomology jump locus, especially the structure theorems developed by Simpson, Schnell, Budur and myself. As an application, I will give an example of non-Kahler Calabi-Yau symplectic-complex manifold. This example is joint work with Lizhen Qin.
Graphical models capture the conditional independence structure among random variables via the existence of edges among vertices. One way of inferring a graph is to use a partial correlation coefficient using the fact that a zero partial correlation coefficient is equivalent to conditional independence under the Gaussian assumption. In order to relax the distributional assumption, we propose kernel partial correlation which is a new conditional independence measure. It is a direct nonparametric extension of the partial correlation coefficient and estimated using a combination of two statistical methods. First, a support vector regression is employed to separate non-random components of conditional distributions, and then the dependence between remaining random components is assessed through a kernel-based association measure. The proposed method is not only a flexible conditional independence measure but also can be estimated robustly under high levels of noise owing to the robustness of the employed nonparametric approaches. Upon comparisons to existing approaches, our method outperforms others when it is applied to simulated data as well as real data from single-cell RNA-sequencing experiments.
A Gaussian graphical model (GGM) is one of the most widely-used tools for statistical network analysis, partly because it turns estimation of a graph into sparse estimation of a high-dimensional precision matrix. To relax the multivariate Gaussian assumption, there have been many developments in non-Gaussian graphical models, through the use of either nonparametric copula transformation or generalized additive models. However, each of these methods has limitations, where the copula transformation cannot capture general non-linear associations and the generalized additive model is not able to detect dependences through conditional variances. In this talk, I will discuss two recent graphical model approaches that extend the GGM approach. The first one is an additive semigraphoid model approach that infers a graph based on additive conditional independence (ACI) relation. The ACI is different from conditional independence, but it satisfies the axioms for a semigraphoid that capture the essence of a graph. The second one is a sparse quantile-based graphical model (SQGM) that infers a graph based on the components of quantile regression. As an inverse mapping function of a conditional probability function, conditional quantile is able to characterize a complete picture of conditional dependence. Utilizing this feature, SQGM is able to broaden the scope of graphical models. Finally, the applications of these approaches to real biological datasets from genomics will be presented.
We consider a class of stochastic processes, which can be regarded as
the perturbation of deterministic dynamics in a potential field. These
processes exhibit a phenomenon known as metastable behavior if the
potential field has several local minima. Metastable behavior is the
phenomenon in which a process starting from one of local minima
arrives at the neighborhood of the global minimum after a sufficiently
long time scale. The precise asymptotic analysis of this transition
time has been known only for the reversible dynamics, based on the
potential theory of reversible Markov processes. In this presentation,
we review this metastability theory for reversible dynamics, and
introduce our recent generalization to the non-reversible dymamics.
(joint work with C. Landim)
This talk is a part of lecture series introducing vertex algebras. Last time, I explained motivations and definitions of vertex algebras. In this lecture, I will explain more about the locality axiom using formal distributions and the decomposition theorem. Also, I will introduce normally ordered products of a vertex algebra.
BDDC(Balancing Domain Decomposition by Constraints) methods based on an adaptive selection of primal constraints for problems posed in H(div) are introduced. Our methods are fully algebraic and deal with highly oscillating coefficients. Bounds on the condition number of the preconditioned linear system are also provided which are independent of the values and jumps. This is joint work with Olof Widlund(Courant Institute), Stefano Zampini(KAUST), and Clark Dohrmann(Sandia National Lab).