Effective webpages are an increasingly important medium for disseminating information related to classes and research. For TAs, they provide a critical way to share material with your class whereas for those entering the job market, they are often checked by potential employers interested in obtaining more information about candidates. Hence it is important to construct…
The Triangle Lectures in Combinatorics is a series of combinatorial workshops held each semester on a Saturday in the Research Triangle region of North Carolina, funded by the National Science Foundation. The workshop this spring, the 15th installment of the Triangle Lectures in Combinatorics (TLC), will be hosted by North Carolina State University in Raleigh,…
The flow polytope associated to an acyclic graph is the set of all nonnegative flows on the edges of the graph with a fixed netflow at each vertex. We will examine flow polytopes arising from permutation matrices, alternating sign matrices and Tesler matrices. Our inspiration is the Chan-Robins-Yuen polytope (a face of the polytope of…
In general, solutions to the Laplacian equation enjoy relatively high smoothness. However, they can exhibit singular behaviors at domain corners or points where boundary conditions change type. In this talk, I will focus on the mixed Dirichlet-Neumann boundary conditions for Laplacian equation, and discuss how singularities in this case adversely affect the accuracy and convergence…
In joint work with Rafal Komendarczyk and Ismar Volic, we study the space of braids, that is, the loopspace of the configuration space of points in a Euclidean space. We relate two different integration-based approaches to its cohomology, both encoded by complexes of graphs. On the one hand, we can restrict configuration space integrals for…
One of the primary goals of text mining is to group together, organize and summarize textual information. In social network analysis, we typically tie individuals together through social connections — can we connect documents and their contents in a similar fashion? What can social network analysis tell us about the content of a document or…
The evolution, through spatially periodic linear dispersion, of rough initial data leads to surprising quantized structures at rational times, and fractal, non-differentiable profiles at irrational times. Such phenomena, also known as the Talbot effect, have been observed in dispersive waves, optics, and quantum mechanics, and lead to intriguing connections with exponential sums arising in number…
In this talk we discuss an algorithm to overcome the curse of dimensionality, in possibly non-convex/time/state-dependent Hamilton-Jacobi partial differential equations. They may arise from optimal control and differential game problems, and are generally difficult to solve numerically in high dimensions. A major contribution of our works is to consider an optimization problem over a single…
Poroelastic models have been widely used in Biomechanics. For example, modeling brain edema and cancellous bones. We aim at solving the Biot model under the MAC Finite Difference discretization and the stabilized finite element discretizations. To solve the resulting saddle point linear systems, some iterative methods are proposed and compared. In these methods, the outer iteration solver can be the GMRES…
The configuration space of some $n$ points in a space $X$ is a well-studied object in topology, geometry, and combinatorics. We present a generalization, the simplicial configuration space $M(S,X)$, which takes as its data a simplicial complex $S$ on $n$ points. In this talk, we will describe how $M(S,X)$ gives rise to polynomial and homological invariants of…
Through the actuation of elastic filaments, microorganisms can move through a viscous fluid or generate a flow within a complex environment. In this talk, we will focus on simple models of two such systems. First, as a step towards understanding the chemotactic behavior of bacteria within micropores, we consider a single elastic helical flagellum confined…
Any 7th or 8th grade girl interested in math Saturday, April 7, 2018 | 10:00am – 3:00pm SAS Hall at NCSU RSVP: Space is limited! Please reserve your spot by April 2nd by emailing graduate student Katherine Harris at keharri4@ncsu.edu. Please include any dietary allergies/restrictions in your email. PDF
We derive quantitative estimates proving the propagation of chaos for large stochastic systems of interacting particles. We obtain explicit bounds on the relative entropy between the joint law of the particles and the tensorized law at the limit. We have to develop for this new laws of large numbers at the exponential scale. But our…
Using the knot filtration on the Heegaard Floer chain complex, Ozsváth and Szabó defined an invariant of knots in the three sphere called τ(K) and showed that it is a lower bound for the 4-ball genus. Generalizing their construction, I will show that for a (not necessarily null-homologous) knot, K, in a rational homology sphere, Y, we obtain…
The behavior of thin layers of viscous fluids coating solid surfaces is of importance in many industrial and natural applications. Interfacial instabilities and pattern formation can occur in films on water-repellent (also called hydrophobic or partially wetting) surfaces. Lubrication models asymptotically reduce the governing equations for the free-surface flow to a fourth-order nonlinear parabolic partial differential equation for the evolution…
We will talk about bijections between various families of combinatorial objects counted by OEIS A071356. These include certain underdiagonal lattice paths, pattern avoiding permutations, pattern avoiding inversion sequences, and posets. Connections will be made to combinatorial Hopf algebras using some lattice theory. OEIS A071356 is closely related to the Catalan numbers.