We will prove Theorem 3.8 of Preiss's paper: roughly
speaking, given a uniformly distributed measure and a point in its
support, it is possible to find a nearby point for which an
appropriate scaling of the measure looks like Lebesgue measure on a
hyperplane. The proof will involve a Taylor expansion of the Gaussian
moments studied a few weeks ago.

Special Lagrangian cones play a central role in the understanding of the SYZ conjecture, an important conjecture in mathematics based upon mirror symmetry and certain string theory models in physics. According to string theory, our universe is a product of the standard Minkowski space with a Calabi-Yau 3-fold. Strominger, Yau, and Zaslov conjectured that Calabi-Yau 3-folds can be fibered by special Lagrangian 3-tori with singular fibers. To make this idea rigorous one needs control over the singularities, which can be modeled by special Lagrangian cones. In this talk, we discuss special Lagrangian cones, the difficulties involved in defining and computing invariants of them, and the hope that these invariants may offer in understanding the SYZ conjecture.

Title: Anomalous blow-up with vanishing energy in 1-D Perona-Malik diffusion

Date: 11/14/2016

Time: 4:02 PM - 4:52 PM

Place: C517 Wells Hall

Speaker: Seonghak Kim, Renmin University of China

We consider the initial-Neumann boundary value problem of non-parabolic equations in one space dimension. In particular, we mainly focus on the problem with a diffusion flux function of the Perona-Malik type in image processing, which is a well-known example of forward-backward parabolic problems. For this problem, we will discuss the existence of weak solutions that converge uniformly to the initial mean value as time approaches a certain final value while the spatial derivative blows up and the associated energy vanishes in some sense. The method is a combination of a classical parabolic theory, the convex integration method in Baire's category setup and the almost transition gauge invariance. This is a joint work with Baisheng Yan(MSU).

Title: Math Curriculum Designed for Instructors and Students

Date: 11/15/2016

Time: 10:00 AM - 11:00 AM

Place: C304 Wells Hall

Speaker: Scott Baldridge, Louisiana State University

In designing new courses and curricula, it is essential to begin with a
good sense of four features: the purpose, the content, the audience, and the
narrative. In the usual approach, students are assumed to be the audience.
However, research on pedagogical content knowledge suggests that instructors
should equally be considered an audience of the content. In fact, instructors
are often the more important audience due to the critical role they play in engaging
students with the mathematics. In this talk, I will present two projects
that focused on instructor’s mathematical knowledge for teaching from the
onset. I will end by discussing a new paper on instructor-level knowledge of
quantities and rates, and its implications for the creation of instructor-student
curricula for beginning-level university math courses.

Our idea of constructing a cofinite graph starts by defining a uniform topological graph Gamma. This idea is motivated by cofinite groups structure, due to B. Hartley.
We define and establish a theory of cofinite connectedness of a cofinite graph. We found that if G is a cofinite group and Gamma = Gamma(G;X) is the Cayley graph with respect to a generating set X of G, then Gamma can be given a suitable cofinite uniform topological structure so that X generates G topologically if and only if Gamma is cofinitely connected.
Next we develop group actions on cofinite graphs. Defining the action of an abstract group over a cofinite graph in the most natural way we characterize a unique way of uniformizing an abstract group with a cofinite structure, obtained from the cofinite structure of the graph in the underlying action, so that the aforesaid action becomes uniformly continuous. We then apply the general theory to additional structure such as groupoids,
thus leading to the notions of cofinite groupoids.
This is a joint work with Dr. J.M.Corson (University of Alabama, Department of Mathematics) and Dr. B. Das (University of North Georgia, Department of Mathematics).

Speaker: Ricardo Nemirovsky, Manchester Metropolitan University, UK

In this talk I will distinguish ‘emergent learning’ from ‘teleological learning,’ which is learning for the sake of passing predefined tests and goals. While teleological learning may succeed or fail, emergent learning is always going on in ways that are under-determined and largely unpredictable. Emergent learning involves getting engaged in events and places and participating in the circulation of affects imbuing them. I will describe ongoing work towards a new theory of emergent learning as self-sustained development in the thick of circulating affects. My talk will be divided in two parts: 1) Articulation of a theoretical/philosophical framework about the nature of affects; and, 2) Exploration of the socio-historical and material dynamics of affects and emergent learning in the course of a videotaped 10-minute episode during a field trip by 5th graders to a science museum.

We discuss optimal insurance contract design where an individual's preference is of
the rank-dependent expected utility (RDU) type. Although this problem has been studied
in the literature, their contracts suffer from a problem of moral hazard for paying
more compensation for a smaller loss. Our project addresses this setback by exogenously
imposing the constraint that both the indemnity function and the insured's retention
function be increasing with respect to the loss. We characterize the optimal
solutions via calculus of variations, and then apply the result to obtain explicitly expressed
contracts for problems with Yaari's dual criterion and general RDU.

The theory of complex projective plane curves has a long history. However, curves of higher genus are rarely studied. It turns out that Heegaard-Floer theory can be effectively used to obtain constraints on possible cusp types of such curves. In fact, restricting ourselves to the case of curves with one cusp having a torus knot link, one can obtain an almost complete classification of possible torus knot types for infinitely many curve genera. The proof is a nice interplay of the theory of numerical semigroups, generalized Pell equations and birational transformations.
These results were obtained in a joint work with Daniele Celoria and Marco Golla. Independently, similar work was done by Maciej Borodzik, Matthew Hedden and Charles Livingston.

Currently there is a new generation of large astronomical telescope under construction, e.g. the European Extremely Large Telescope (E-ELT) of the European Southern Observatory (ESO) with a mirror diameter of 39 meters or the Thirty Meter Telescope (TMT), build by a consortium headed by Caltech. The operation of those huge telescopes require new mathematical methods in particular for the Adaptive Optics systems of the telescopes.
The image quality of ground based astronomical telescopes suffers from turbulences in the atmosphere. Adaptive Optics (AO) systems use wavefront sensor measurements of incoming light from guide stars to determine an optimal shape of deformable mirrors (DM) such that the image of the scientific object is corrected after reflection on the DM. The solution of this task involves several inverse problems: First, the incoming wavefronts have to be reconstructed from wavefront sensor measurements. The next step involves the solution of the Atmospheric Tomography problem, i.e., the reconstruction of the turbulence profile in the atmosphere. Finally, the optimal shape of the mirrors has to be determined. As the atmosphere changes frequently, these computations have to be done in real time. In the talk we introduce mathematical models for the elements of different Adaptive Optics system such as Single Conjugate Adaptive Optics (SCAO) or Multi Conjugate Adaptive Optics (MCAO) and present fast reconstruction algorithms as well as related numerical results for each of the sub-tasks that achieve the accuracy and speed required for the operation of ELTs.

Title: Computational Modeling in Studying Blood Clot Formation

Date: 11/18/2016

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

Speaker: Zhiliang Xu, University of Notre Dame

Blood clotting is a multiscale process involving blood cells, fibrinogen polymerization, coagulation reactions, ligand-receptor interactions and blood plasma flow. Detailed multiscale models of blood clotting to cover all aspects of clotting are, if not possible, extremely difficult to develop. Models focusing on specific events across one or two spatial-temporal scales seem to be plausible. In this talk, ligand-receptor binding kinetics model, computational model of fluid-structure interaction (FSI) for simulating flow-elastic membrane with mass and a continuum model for studying the structural stability of clots will be presented. The binding kinetics model revealed that platelet αIIbβ3 integrin and fibrin interacts through a two-step mechanism. The new FSI model is derived by using the energy law and distributed-Lagrange-multiplier/fictitious-domain (DLM/FD) formulation. The continuum model for studying the structural stability of clots utilized the phase field and energetic variational approaches. Simulation results show that rheological response of the blood clot to the flow is determined by mechanical and structural properties of its components. Two main mechanisms are shown to significantly affect volume of the already formed clot: dynamic balance between platelet adhesion and platelet removal by the flow on the blood clot surface and removal of parts of the clot through rupture.