Math Department Events Listing

I will talk about some questions that arise in attempting to apply conformal mapping techniques to free boundary problems in 2D electrostatics and fluid flows. This is ongoing work in collaboration with Stuart Kent.

p-adic heights are analogues of the classical heights and can be used to formulate an analogue of the Birch and Swinnerton-Dyer conjecture. Among the different definitions of p-adic heights there is the Coleman-Gross definition for Jacobians of curves and the Mazur-Tate height for elliptic curves (and more general Abelian varieties). They are known to be equal under some restrictions using global methods, even though both decompose as the sum of local terms. We give a proof of their equality using p-adic Arakelov geometry. As a corollary, we obtain a very simple proof of a recent theorem of Minhyong Kim, characterizing integral points on some elliptic curves.

Somitogenesis, the formation of the body's primary segmental structure common to all vertebrate development, requires coordination between biological mechanisms at several scales. Explaining how these mechanisms interact across scales and how events are coordinated in space and time is necessary for a complete understanding of somitogenesis and its evolutionary flexibility. So far, mechanisms of somitogenesis have been studied independently. To test the consistency, integrability and combined explanatory power of current prevailing hypotheses, we built an integrated clock-and-wavefront model including submodels of the intracellular segmentation clock, intercellular segmentation-clock coupling via Delta/Notch signaling, an FGF8 determination front, delayed differentiation, clock-wavefront readout, and differential-cell-cell-adhesion-driven cell sorting. We identified inconsistencies between existing submodels and gaps in the current understanding of somitogenesis mechanisms, and proposed novel submodels and extensions of existing submodels where necessary. For reasonable initial conditions, 2D simulations of our model robustly generate spatially and temporally regular somites, realistic dynamic morphologies and spontaneous emergence of anterior-traveling stripes of Lfng. In this talk, I will introduce the current prevailing models of somitogenesis as well as our own novel or extended submodels, discuss how we implemented previous and novel submodels in our integrated computational model, and highlight our key results. This is joint work with Julio M. Belmonte, J. Scott Gens, Sherry G. Clendenon, and James A. Glazier.

We have all heard rumors of online classes allowing departments to reduce costs and campuses to reduce congestion, but what if it could go further than that? What if there was a way to teach online that increased student engagement and participation, all while allowing the instructor to tailor the material to each individual student's particular needs? We will discuss a new model that is being attempted and the positive results that are coming out.

Nakajima defined an affine-Kac-Moody algebra action on the cohomology of instanton moduli spaces on orbifolds. Such instantons were constructed by Kronheimer and Nakajima in terms of quivers. I begin by presenting a generalization of their instanton construction. My generalization is naturally formulated in terms of bows. Next I present an M-theoretic argument relating the instanton moduli spaces to affine-Kac-Moody algebras. In the simpler setup such argument leads to the geometric Langlands correspondence for curves. Thus in this new setup is natural to view this as a geometric langlands correspondence in one dimension higher.

Systems of interacting particles are difficult to analyze and understand which makes it a field rich in research questions. To try and grasp at understanding, various assumptions and simplifications have been attempted to make these systems easier to manage. In the case of interacting Boson systems, this simplification takes us from large linear systems of symmetric tensors to small nonlinear systems of C^2 functions. As a result, it is necessary to try and solve the nonlinear differential equations analytically. It turns out the internet sources like Wolfram Alpha are a viable source of solutions if you are willing to mathematically verify it. In this talk, I will present how to estimate a solution, then solve the differential equation using using Jacobi elliptic functions. If there is time, I will also present results that follow from this solution.

The arrangement of seeds, florets, and petals on a sunflower has been a matter of scientific inquiry for centuries. As such, there are many different theories as to its origin. I will explore a few of these theories and do my best to show that the story is not yet complete.

In the process, some obvious (and not so obvious) questions may arise: 1) Are the seeds really arranged in spirals? 2) If so, how might we find the center of the pattern? 3) Or the order in which the seeds developed? 4) Where do Fibonacci and the golden ratio come in? 5) Finally, how might we rectify this with the fact that nature has no protractor?

The systematic study of integral geometry was initiated in the 1930s by W. Blaschke, who recognized that the famous formulas of Buffon, Steiner and Crofton may all be viewed as "kinematic formulas": expressions for the averages of certain geometric measurements, applied to the intersection of two subspaces X, Y of a homogeneous space M, over all possible relative positions of X and Y. The measurements involved are "valuations", which are essentially finitely additive measures. A kinematic formula typically gives this average in terms of valuations applied to X and Y separately.

S. Alesker's work on valuation theory, in particular his construction of a multiplication on the vector space of valuations, has thoroughly revolutionized this subject in recent years. This has led to the calculation of kinematic formulas in a number of previously baffling cases--- most recently, in joint work with A. Bernig and G. Solanes, the complex space forms. Despite these successes, however, several basic and mind-boggling questions persist.

Diffusive Shock Acceleration is believed to be the mechanism responsible for the acceleration of charged particles at very high energies, which is a phenomenon observed in a wide variety of astrophysical environments. Standard DSA theory is based on solutions of the Parker transport equation (a type of Fokker-Planck, or Kolmogorov forward equation) where the advection speeds and/or diffusion coefficients are discontinuous functions of position.

The presence of these discontinuities creates problems with Monte-Carlo simulations of shock acceleration by significantly slowing down run times. In this talk, I discuss how one can efficiently model shock acceleration by simulating an equivalent stochastic process, called Skew Brownian Motion, and present a numerical algorithm that allows one to correctly treat particle behavior at a shock in a physically relevant way.

Random search strategies occur throughout nature as a means of efficiently searching large areas for one or more targets of unknown location, which can only be detected when the searcher is within a certain range. Examples include animals foraging for food or shelter, the motor-driven transport and delivery of macromolecules to particular compartments within cells, and a promoter protein searching for a specific target site on DNA. One particular class of model, which can be applied both to foraging animals and active transport in cells, treats a random searcher as a particle that switches between a slow motion (diffusive) or stationary phase in which target detection can occur and a fast motion "ballistic" phase; transitions between bulk movement states and searching states are governed by a Markov process.

In this talk we review recent work on the analysis of random intermittent search models of motor-driven transport in the dendrites of neurons. The stochastic search process is modeled in terms of a system of Chapman-Kolmogorov equations, which are reduced to a scalar Fokker-Planck equation using perturbation methods. The reduced FP equation is used to compute various quantities that characterise the efficiency of the search process, including the mean first passage time (MFPT) to target detection. We then consider a number of applications. First, we analyze the effects of dendritic branching on the efficiency of motor-driven transport and show that bidirectional rather than unidirectional transport is more effective. Second, we analyze a biophysical model of bidirectional transport, in which opposing motors compete in a "tug-of-war," and use this to explore how local signaling mechanisms could regulate the delivery of molecular cargo to subcellular targets such as synapses. We end by discussing extensions to higher dimensional searches and models of multiple searchers.