# College of Arts & SciencesDepartment of Mathematics

## Colloquia 2013

Departmental Colloquia

## SPRING 2013

• Title: Enumerative Geometry: from Classical to Modern.
Speaker: Yu-Jong Tzeng (Harvard Univ.)
Time and Place: Thursday, Jan. 17 at 3:30 pm in LC 412.
Abstract: The subject of enumerative geometry goes back at least to the middle of the 19th century. It studies the number of geometric objects in a given class that satisfy a number of incidence or tangency conditions. For example,
(a) How many lines pass through distinct 2 points?
(b) How many lines pass through 4 general lines in 3-space?
(c) How many rational cubics pass through 9 points on the projective plane?

The field has been very active in the last twenty years due to its interaction with physics, which provides motivation for many theories and conjectural formulas.

One of the most famous problems in enumerative geometry is computing Severi degrees, which are the numbers of degree d plane curves with a given number of nodes that pass through an appropriate number of points in general position. The computation of Severi degrees, as well as counting nodal curves on other surfaces are great examples of classical problems which were open for long time and can finally be solved by modern techniques. Furthermore, a recent conjecture of Gottsche states that there should be a universal formula that computes the number of nodal curves in a sufficiently ample linear system on any surface.

In this talk I will first give an indication of what enumerative geometry is about, then give a survey about counting curves on several surfaces. Finally I will explain why Gottsche's conjecture is true and discuss its generalizations.

• Title: The Hassett-Keel Program in Genus Four.
Speaker: David Jensen (Stony Brook Univ.)
Time and Place: Tuesday, Jan. 22 at 3:30 pm in LC 412.
Abstract: The Hassett-Keel program aims to give modular interpretations of log canonical models for the moduli spaces of curves. The program, while relatively new, has attracted the attention of a number of researchers, and has rapidly become one of the most active areas of research concerning the moduli of curves, with applications to geometry, number theory, and physics. After introducing the basic theory of algebraic curves and moduli spaces, we will turn to the genus four case, in which we construct several of these models as geometric invariant theory quotients of a single, elementary space.

• Title: Modeling the Electrical Activity in Cardiac Tissue
Speaker: Joyce Lin (University of Utah)
Time and Place: Wednesday, January 23 3:30 pm in LC 412
Abstract: Electrical stimulation of cardiac cells causes an action potential wave to propagate through myocardial tissue, resulting in muscular contraction and pumping blood through the body. Approximately two thirds of unexpected, sudden cardiac deaths, presumably due to ventricular arrhythmias, occur without recognition of cardiac disease. While conduction failure has been linked to arrhythmia, the major players in conduction have yet to be well established. Additionally, recent experimental studies have shown that ephaptic coupling, or field effects, occurring in microdomains may be another method of communication between cardiac cells, bringing into question the classic understanding that action potential propagation occurs primarily through gap junctions. In this talk, I will introduce the mechanisms behind cardiac conduction, give an overview of previously studied models, and present and discuss results from a new model for the electrical activity in cardiac cells with simplifications that afford more efficient numerical simulation, yet capture complex cellular geometry and spatial inhomogeneities that are critical to ephaptic coupling.

• Title: Mathematical modeling of the mucus barrier in human lungs
Speaker: Paula Vasquez (University of North Carolina at Chapel Hill)
Time and Place: Thursday, January 24 at 3:30 pm in LC 412
Abstract: The first line of defense of human lungs against inhaled pathogens is mucus. Inhaled viruses, bacteria and particulates land on the mucus layer and diffuse within. These foreign particles are cleared if flow of the mucus layer toward the larynx dominates particle diffusion through the layer. Yet the flow and diffusive transport properties of mucus remain very poorly understood. In this lecture I will survey the mathematical challenges in characterizing mucus transport properties. The talk will focus on progress of our research group understanding remarkable nonlinear phenomena in the oscillatory flow of viscoelastic fluids, and explaining why dynamic and spatial microstructural characterization is essential in the understanding of mucus transport. These research projects involve collaborations at UNC that are either inspired by or directly related to experiments carried out in the Physics Department and Cystic Fibrosis Center.

• Title: Mirror symmetry, Landau-Ginzburg models, and algebraic geometry.
Speaker: Matthew Ballard (Univ. of Wisconsin)
Time and Place: Friday, Jan. 25 at 3:30 pm in LC 412.
Abstract: The phenomenon of mirror symmetry, first noticed in theoretical physics a couple decades ago, provides a powerful and tantalizing framework to relate two very different branches of mathematics: symplectic topology and algebraic geometry. Many years after its introduction, Kontsevich proposed an overarching general framework using categories. This relationship is now known as Homological Mirror Symmetry (HMS). Instead of just studying algebraic varieties, we can also add a regular function, to form a Landau-Ginzburg model, and extend the reach of HMS.

In this talk, we will review the basics of HMS and how it inspires one to study derived categories of coherent sheaves on varieties. We will explain how these Landau-Ginzburg models yield categories whose objects are natural generalizations of Eisenbud's matrix factorizations. Then, we will discuss how physical intuition leads to predictions of equivalences of these categories and present rigorous mathematical results concerning the equivalences.

• Title:Title: The Cancer identity and roles of macrophages
Speaker: Duan Chen (The Ohio State University)
Time and Place: Tuesday, January 29 3:30 pm in LC 412
Abstract: Hypoxia, acidosis, and strong reducing capacity are distinguish properties of solid tumors from healthy tissues. These parameters change along cancer development and are among the most critical parameters for optimization of anti-cancer therapies and screening of anti-cancer drugs. Multi-scale models are established to explore explanations, correlations and impacts of the three properties in tumor growth with corresponding chemotherapies, from the aspects of tumor-immune system interactions at the tissue level, chemical interactions inside individual cancer cells and proton transport through membrane proteins. High-performance numerical algorithms are developed to handle computational challenges in solving the PDE systems and the models are validated by the comparisons between numerical simulations and experimental data. Additionally, the models can be summarized as different types of free boundary problems and the well-posedness of their solutions are analyzed.

• Title: Mathematical modeling of renal hemodynamics: Feedback dynamics and coupled oscillators.
Speaker: Anita Layton (Duke University).
Time and place: Tuesday, April 9 3:30 pm in LC 412.
Abstract: We have formulated a mathematical model for the rat afferent arteriole (AA), glomerulus, and short loop of Henle, and used that model to study the interactions between the tubuloglomerular feedback (TGF) and myogenic mechanism, the two key mechanisms that mediate renal autoregulation. Blood flow is described by Poiseuille flow. The AA model consists of a series of arteriolar smooth muscle cells, each of which represents ion transport, cell membrane potential, cellular contraction, gap junction coupling, and wall mechanics. The myogenic response representation is based on the hypothesis that the voltage dependence of calcium channel openings responds to transmural pressure so that the vessel constricts when pressure increases. The glomerular filtration model is based on the model by Deen et al. (AJP 1972). The TGF model represents the pars recta, descending limb, and thick ascending limb, and predicts tubular fluid flow rate and [Cl-] along the loop. The model can be used as a fundamental component in a multi-scale renal microvasculature model for investigations of pathogenesis of hypertensive renal diseases. This research was supported in part by NIH grant DK-89066 and NSF grant DMS-0715021.

## FALL 2013

• Title: Tuesday, November 5 at 4:30 in LC 412.
Speaker: Ameera Chowdhury, UCLA.
Time and Place: Tuesday, November 5 at 4:30 in LC 412..
Abstract: Let V be an n-dimensional vector space over a finite field. Assign a real-valued weight to each 1-dimensional subspace in V so that the sum of all weights is zero. Define the weight of a subspace S of V to be the sum of the weights of all the 1-dimensional subspaces it contains. We prove that if n is greater than or equal to 3k, then the number of k-dimensional subspaces in V with non-negative weight is at least the number of k-dimensional subspaces in V that contain a fixed 1-dimensional subspace. This result verifies a conjecture of Manickam and Singhi from 1988.

• Title: Set-theoretic Complete Intersections and Divisor Class Groups.
Speaker: Robin Hartshorne (University of California at Berkeley)
Time and place: Friday, November 8, 2013 at 3:30 pm in LC 412.
Abstract: I will begin by reviewing some of the history of the problem of set-theoretic complete intersections in projective space. Then I will report on some recent work, joint with Claudia Polini, inspired by this problem, which provides a small contribution to the problem itself.
NOTE:This colloquium will be the first talk in the conference, Commutative Algebra -- Algebraic Geometry in the Southeast", organized by Jesse Kass, Andy Kustin, and Adela Vraciu for Nov. 8 - 10, 2013. The conference website is http://www.math.sc.edu/~kustin/CA-AGMeeting.html.

• Title: Discontinuous Galerkin Methods: Algorithm Design and Applications.
Speaker: Xinghui Zhang, Michigan State.
Time and Place: Monday, December 2 at 4:30 in LC 210A.
Abstract: In this talk, we discuss discontinuous Galerkin (DG) methods with emphasis on their algorithm design targeted towards applications for shock calculation and plasma physics. DG method is a class of finite element methods that has gained popularity in recent years due to its flexibility for arbitrarily unstructured meshes, with a compact stencil, and with the ability to easily accommodate arbitrary h-p adaptivity. However, some challenges still remain in specific application problems. In the first part of my talk, we design a new limiter using weighted essentially non-oscillatory (WENO) methodology for DG methods solving conservation laws, with the goal of obtaining a robust and high order limiting procedure to simultaneously achieve uniform high order accuracy and sharp, non-oscillatory shock transitions. The main advantage of this limiter is its simplicity in implementation, especially on multi-dimensional unstructured meshes. In the second part, we propose energy-conserving numerical schemes for the Vlasov-AmpÃ¨re (VA) and Vlasov-Maxwell systems. Those equations are fundamental models in the simulation of plasma physics. The total energy is an important physical quantity that is conserved by those models. Our methods are the first Eulerian solver

• Title: Monte Carlo Simulations at the Interface Between Statistical Physics and Biochemistry.
Speaker:David Landau, University of Georgia.
Time and Place: Tuesday, December 3 at 4:30 in LC 412.
Abstract: Systems with complex free energy landscapes are important in nature and present particular problems for both theory and experiment. Mathematical physicists have made heroic efforts to find exact solutions of diverse models but with limited success. Because of the huge number of possible states of the system, stochastic sampling seems advantageous; but because of the exponential variation of the weights of states with different energies, importance sampling Monte Carlo algorithms must be used. While many models have been examined, traditional Monte Carlo algorithms fail for complex systems because of the long time scales that result at low temperatures where â€œinteresting behaviorâ€ occurs. We shall describe a non- traditional approach, now termed â€œWang-Landau samplingâ€, that is highly successful. Then, results for prototypical models at the interface between statistical physics and biochemistry, obtained using Wang-Landau sampling, will be shown to demonstrate advances in our understanding of the behavior of diverse systems which possess rough energy landscapes.

• Title: Proper Orthogonal Decomposition Reduced-order Models of Complex Systems.
Speaker:Zhu Wang, University of Minnesota.
Time and Place: Wednesday, December 4 at 4:30 in LC 210A.
Abstract: Many grand challenge problems at the frontier of computational science, such as energy efficient building design and control, require repeated numerical simulations of large-scale dynamical systems. To alleviate the tremendous computational cost needed by direct numerical simulations, model reduction techniques have been widely used to achieve computationally efficient reduced-order models. Among them, the proper orthogonal decomposition method has been commonly applied to generate reduced-order models for turbulent flows dominated by coherent structures. However, to achieve a balance between the low computational cost required by a reduced-order model and the complexity of the targeted turbulent flows, appropriate closure modeling strategies are needed. In this talk, we present several new nonlinear closure methods for proper orthogonal decomposition reduced-order models, which synthesize ideas originating from large eddy simulation. We develop rigorous error estimates, design efficient algorithms, and perform the numerical validation and verification of the new models in challenging computational settings. The applications of these models in realistic problems will also be discussed.

• Title: Sessile Drop Oscillations: Contact Line Dynamics and Symmetry Breaking.
Speaker:Joshua Bostwick, Northwestern University.
Time and Place: Thursday, December 5 at 4:30 in LC 412.
Abstract: Oscillations of a sessile drop are of interest in a number of industrial applications, such as ink-jet printing and drop atomization. We generalize Rayleighâ€™s stability analysis for the free drop, focusing on the wetting properties of the solid substrate and mobility of the three-phase contact-line. We report oscillation frequencies and modal structures which display spectral splitting similar to the Bohr model of the atom in quantum mechanics and compare with relevant experiments. An analogy is made between the spectrum of these 'broken' states and the filling order of the periodic table by energy level. In addition to the oscillatory spectrum, we report a new hydrodynamic instability that has fundamental implications for fluid transport.