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Abstract: Understanding the functioning of ecosystems requires the understanding of the interactions between consumer species and their resources. How do these interactions affect the variations of population abundances? How do population abundances determine the impact of predators on their prey? The authors defend the view that the "null model" that most ecologists tend to use (derived from the Lotka-Volterra equations) is inappropriate because it assumes that the amount of prey consumed by each predator is insensitive to the number of conspecifics. The authors argue that the amount of prey available per predator (rather than the absolute abundance of prey) is the basic determinant of the dynamics of predation. This so-called ratio dependence is shown to be a much more reasonable "null model". Lessons can be drawn from a similar debate that took place in microbiology in the 1950's. Currently, populations of bacteria are known to follow the analogue of ratio dependence when growing in real-life conditions. Three kinds of arguments are developed. First, it is shown that available direct measurements of prey consumption are "in the middle" but most are close to ratio dependence and all are clearly away from the usual Lotka-Volterra relationship; an example is the system of wolves and moose on Isle Royale. Second, indirect evidence is based on the responses of food chains to nutrient enrichment: all empirical observations at the community level agree very well with the ratio-dependent view. Third, mechanistic approaches explain how ratio dependence emerges at the global scale, even when assuming Lotka-Volterra interactions at the local scale; this is illustrated by microcosm experiments, by individual-based models and by mathematical models. Changing the fundamental paradigm of the predator-prey interaction has far-reaching consequences, ranging from the logical consistency of theoretical ecology to practical questions of eco-manipulation, biological control, conservation ecology.
Dr. Lev Ginzburg
Department of Ecology and Evolution
Stony Brook University
will present
Life is 4D: Allometric Slopes Are Understandable when Viewed in 4D
Monday, February 6, 2012, 3:30pm
Ungar Room 402
Refreshments at 3:00pm in Ungar Room 521
All interested persons are welcome to attend.
Professor Roger Arditi
Department of Ecology and Evolution
Université Pierre et Marie Curie, Paris
will present
How Species Interact
Friday, February 3, 2012, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
This work is in collaboration with Lev Ginzburg.Abstract: Part I. I argue that rigorous mathematics gives insight into the current question of whether taxation helps or hinders employment.
Professor Larry Shepp
Patrick T. Harker Professor
Wharton School, University of Pennsylvania
Board of Governor's Professor
Rutgers University
will present
Is Mathematical Modeling Able to Give Insight into Current Questions in Finance, Economics, and Politics?
Thursday, February 2, 2012, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Part II. I compute (within one simple model) how much money future knowledge, obtained either via insider information or via high-frequency trading, of future stock prices brings to a possessor of such knowledge.Abstract: I will talk about my recent work with Rodnianski and Szeftel concerning a solution of the conjecture. I will also compare the result with the other known breakdown criteria in GR.
Professor Sergiu Klainerman
Higgins Professor of Mathematics
Princeton University
will present
On the Bounded L2 Curvature Conjecture as a Breakdown Criteria for the Einstein Vacuum Equations
Wednesday, January 11, 2012, 4:00pm
Ungar Room 402
Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: I will present joint work with Jan Metzger. A basic question in mathematical relativity is how geometric properties of an asymptotically flat manifold (or initial data set) encode information about the physical properties of the space time that it is embedded in. For example, the square root of the area of the outermost minimal surface of an initial data with non-negative scalar curvature provides a lower bound for the "mass" of its associated space time, as was conjectured by Penrose and proven by Bray and Huisken-Ilmanen. Other special surfaces that have been studied in this context include stable constant mean curvature surfaces and isoperimetric surfaces. I will explain why positive mass works to the effect that large stable constant mean curvature surfaces are always isoperimetric. This answers an old conjecture of Bray's and complements the results by Huisken-Yau and Qing-Tian on the "global uniqueness problem for stable CMC surfaces" in initial data sets with positive scalar curvature. Time permitting, I will sketch applications related to G. Huisken's isoperimetric mass and very recent related results with S. Brendle on further isoperimetric features of the exact spatial Schwarzschild metric.
Dr. Michael Eichmair
Massachusetts Institute of Technology
will present
Isoperimetric Structure of Initial Data Sets
Friday, December 16, 2011, 4:00pm
Ungar Room 402
Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: In my talk I will discuss an example of cross-fertilization between medicine and mathematics. Data on the size of outbreaks with methicillin-resistant Staphylococcus aureus (MRSA) in Dutch hospitals were collected to estimate the transmissibility of the two most relevant MRSA strains in the Netherlands. Analysis of these data led to a relation between the epidemiological model for the spread of MRSA in hospitals and queuing theory, and a new estimation method for transmissibility of pathogens in outbreak settings with contact screening. In very recent theoretical work, Amaury Lambert and Pieter Trapman derive an improved estimator if easily collectable data is available. These data are now collected in a currently performed study in the Netherlands.
Dr. Martin Bootsma
Department of Mathematics
Utrecht University, The Netherlands
will present
The Spreading Capacity of Methicillin-resistant Staphylococcus aureus (MRSA)
Tuesday, December 6, 2011, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: Multi-drug resistance for cancer cells as been a serious issue since several decades. In the past, many models have been proposed to describe this problem. These models use a discrete structured for the cancer cell population, and they may include some class of resistant, non resistant, and acquired resistant cells. Recently, this problem has received a more detailed biological description, and it turns out that the resistance to treatments is due in 40% of cancers to a protein called P-glycoprotein (P-gp). Moreover some new biological experiments show that transfers can occur by the mean of Tunneling nanoTubes built in between cells (direct transfers). Transfers can also occur through microparticles (containing P-pg) released by over expressing cells into the liquid surrounding these cells. These microparticles can then diffuse and can be recaptured by the cells (indirect transfers). This transfers turn to be responsible for the acquired resistance of sensitive cells. The goal of this talk is to introduce this problem, and to present a cell population dynamic model with continuous P-gp structure.
Dr. Pierre Magal
University of Bordeaux Segalen, France
will present
P-gp Transfer and Acquired Multi-drug Resistance in Tumors Cells
Monday, November 28, 2011, 4:30pm
Ungar Room 402
Refreshments at 4:00pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: Let (M, g) be a compact, simply connected n-dimensional Riemannian manifold with sectional curvature K. (M, g) is said to be pointwise ε-pinched for some constant 1 ≥ ε > 0 if there is a positive function K0 on M such that K0 ≥ K > εK0.
Professor DaGang Yang
Tulane University
will present
Einstein 4-Manifolds with Pinched Sectional Curvature
Thursday, November 17, 2011, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Question: For what values of ε, can one expect the Ricci flow initiated from g to converge to a metric of constant sectional curvature, and therefore diffeomorphic to the standard sphere Sn?
The 1/4-pinched differentiable sphere theorem, by H.W. Chen for n = 4, and by S. Brendle and R. M. Schoen for n ≥ 5, says that ε = 1/4 is the smallest possible value. For ε < 1/4, P. Petersen and T. Tao have shown that there is a constant εn, 1/4 > εn > 0, for each dimension n such that, if 1 ≥ K > εn, then the Ricci flow initiated from g will converge to a metric which is either of constant sectional curvature or is a compact rank one symmetric space.
It is therefore natural to propose the following question: For each n ≥ 4, what is the smallest pinching constant εn > 0 such that, if K0 ≥ K > εnK0, then the Ricci flow initiated from g can still be expected to converge to a metric of constant sectional curvature or to a metric of a compact rank one symmetric space?
In other words, are there any other models of Einstein manifolds with pinched positive sectional curvature in each dimension n ≥ 4?
In this talk, I shall discuss some old and new results in this area for n=4.Abstract: Dengue is a mosquito-borne viral pathogen that causes large amounts of disease in the tropics and sub-tropics. Dengue viruses are divided into four large clades, called serotypes: infection with a virus produces complete immunity to viruses within that same serotype, but increases the risk of severe disease upon infection with a virus from a different serotype. Multiple mechanisms have been hypothesized for this interaction between serotypes in the human immune system, which, combined with seasonal oscillations in mosquito abundances, lead to complex behavior in mathematical models. In addition, two new interventions for dengue are currently in intense development: a vaccine that protects against all four serotypes and transgenic mosquitoes that are less-suitable vectors. In this talk, I will discuss a model for evolution of dengue viruses in response to these new interventions and work in progress on the best population groups to target with vaccine to minimize disease burden.
Dr. Jan Medlock
Clemson University
will present
Issues in the Ecology and Evolution of Dengue
Friday, April 8, 2011, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: Macdonald polynomials are symmetric functions in a set of variables X which also depend on two parameters q,t. In this talk we describe how a formula of Haiman for the Hilbert series of the quotient ring of diagonal coinvariants in terms of Macdonald polynomials implies a much simpler expression for the Hilbert series involving matrices satisfying certain constraints.
Dr. Jim Haglund
University of Pennsylvania
will present
Macdonald Polynomials and the Hilbert Series of the Quotient Ring of Diagonal Coinvariants
Thursday, April 7, 2011, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: In the early 80's Hawking et al. generalized the positive mass theorem to include charge. It was conjectured that the case of equality should occur only for the extremal black hole solutions known as Majumdar-Papapetrou spacetimes. Chrusciel et al. confirmed this under extra assumptions. In this talk we will show how these extra hypotheses may be removed. This is joint work with Gilbert Weinstein.
Dr. Marcus Khuri
Stony Brook University
will present
The Positive Mass Theorem with Charge Revisited
Thursday, March 31, 2011, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: The volume entropy is a very interesting invariant of a Riemannian manifold. When the Ricci curvature has a negative lower bound, there is a sharp lower bound for the volume entropy. I will discuss why the equality case characterizes hyperbolic manifolds. In certain cases, we can also prove that the manifold is close to a hyperbolic manifold in the Gromov-Hausdorff sense if the volume entropy is close to the sharp lower bound. The method involves the Busemann compactification and Patterson-Sullivan measure. This is a joint work with Francois Ledrappier.
Richard P. Stanley
Norman Levinson Professor of Applied Mathematics
M.I.T.
will present
A Survey of Alternating Permutations
Tuesday, March 22, 2011, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Professor Xiaodong Wang
Michigan State University
will present
Volume Entropy and Ricci Curvature
Thursday, March 10, 2011, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: Mathematical modeling for mosquito-borne diseases has been used to develop theory and guide disease control for more than a century, but the demands on models have been changing. Analysis of a comprehensive review of mosquito-borne transmission models demonstrated that mosquito-borne disease models follow the conventions of the Ross-Macdonald model and that there has been little innovation in modeling transmission. A new mathematical description of transmission was based on mosquito movement, aquatic ecology and blood feeding behavior. Mosquito movement can be described concisely as a random walk on a bipartite graph. Transmission also depends on the ways that mosquitos allocate bites on humans and the way humans allocate their time at risk. This framework provides a starting point for reformulating a new theory of transmission that captures other aspects of mosquito behavior that are important for transmission but absent from the Ross-Macdonald model.
Dr. David Smith
University of Florida
and
Center for Disease Dynamics Economics and Policy Washington, DC
will present
Recasting the Theory of Transmission by Mosquitoes
Thursday, February 24, 2011, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: Epidemic models are vital for implementing, evaluating, and optimizing control schemes in order to eradicate a disease. These mathematical models may be oversimplified, but they are useful for gaining knowledge of the underlying mechanics driving the spread of a disease, and for estimating the number of vaccinations required to eradicate a disease. This talk discusses some epidemic models with switching parameters. Both constant control and pulse control schemes are examined, and, in doing so, we hope to gain insight into the effects of a time-varying contact rate on critical control levels required for eradication.
Dr. Xinzhi Liu
Department of Applied Mathematics
University of Waterloo
Waterloo, Canada
will present
Epidemic Models with Switching Parameters
Wednesday, February 23, 2011, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: A central question in Potential Theory is the extent to which the geometry of a domain influences the boundary regularity of solutions to divergence form elliptic operators. To answer this question one studies the properties of the corresponding elliptic measure. On the other hand one of the central questions in Geometric Measure Theory (GMT) is the extent to which the regularity of a measure determines the geometry of its support. The goal of this talk is to present a few instances in which techniques from GMT and Harmonic Analysis come together to produce new results in both of these areas.
Professor Tatiana Toro
University of Washington
will present
Potential Theory Meets Geometric Measure Theory
Tuesday, February 22, 2011, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: In this talk I will describe my work in progress with Laurent Meersseman and Alberto Verjovvsky on the moduli space of Toric Manifolds. Using specific families of foliations and the Gale transform we describe some basic geometric and topological properties of this moduli space.
Professor Ernesto Lupercio
Research and Advanced Studies Center of the National Polytechnic Institute of Mexico (Cinvestav - IPN)
Winner of the 2009 Srinivasa Ramanujan Prize
will present
The Moduli Space of (Non-commutative) Toric Varieties
Friday, February 18, 2011, 4:30pm
Ungar Room 402
Refreshments at 4:00pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: Those who are skeptical of the Darwinian view of evolution often argue that since there are K^n possible n letter words over a K letter alphabet, it must take an exponentially long time before random mutations of the letters will produce "the right word." We show that if the effects of natural selection are taken into account in a reasonable way, the K^n time estimate can be replaced by Kn log n. As a byproduct we obtain the mean of the largest of many geometrically distributed random variables. This is joint work with Warren Ewens.
Professor Herbert S. Wilf
Thomas A. Scott Emeritus Professor of Mathematics
University of Pennsylvania
will present
There's Plenty of Time for Evolution
Thursday, February 17, 2011, 2:00pm
Ungar Room 402
Refreshments at 1:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: In this talk I will discuss a problem in quantum topology called the Volume Conjecture. This conjecture relates the colored Jones polynomials with the hyperbolic volume of the knot complement. As I will explain the Volume Conjecture links together elements of topology, geometry and algebra. I will begin with a gentle introduction to knot theory and the definition of the Jones polynomial. Then I will show how to compute the colored Jones polynomial using algebra. Finally after stating the conjecture, I will discuss some related topological invariants.
Dr. Nathan Geer
Utah State University
will present
The Colored Jones Polynomial and Some of Its Relatives
Tuesday, February 15, 2011, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: I will first update the situation discussed last year re the artificial pancreas project. The problem is now to upgrade the sensor by getting a better algorithm for recalibration. This is a really important and beautiful problem. I will tell you about my recent work on it; it's still very open.
Professor Larry Shepp
Wharton School, University of Pennsylvania
and
Rutgers University
will present
Some New Probability Problems; Some New Solutions
Thursday, February 10, 2011, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
The second update is a beautiful theoretical problem recently posed by Mike Steele. The problem is this: consider a sequence of n iid uniform variables on [0,1]. Call a subsequence "upsy-downsy", if no three successive terms of the subsequence are monotonic. It is known (Houdre-Stanley-Widom) that if one can search all subsequences, then the expected length of the longest upsy-downsy subsequence is asymptotic to n times 2/3. Steele asked a question of interest to people in stochastic optimization: what is the length of the upsy-downsy subsequence if each term is optimally chosen, {\em without knowing the rest of the sequence, as in the famous secretary problem. We showed the answer is asymptotic to n times c, where c = 2 - \sqrt{2} = .586\ldots < 2/3$. I will indicate two approaches to this problem, each of which gives the right answer, but only one of which I regard as mathematically legitimate.Abstract: Deep connections between topology and geometry will be discussed in the case of manifolds with non-negative (sectional) curvature. Historical perspective of these connections and new developments will be presented.
Dr. Slawomir Kwasik
Tulane University
will present
Souls of Manifolds via Curvature and Surgery
Thursday, January 27, 2011, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: Chronic wound healing is a staggering public health problem, affecting 6.5 million individuals annually in the U.S. Ischemia, caused primarily by peripheral artery diseases, represents a major complicating factor in the healing process. In this talk, I will present a mathematical model of chronic wounds that represents the wounded tissue as a quasi-stationary Maxwell material, and incorporates the major biological processes involved in the wound closure. The model was formulated in terms of a system of partial differential equations with the surface of the open wound as a free boundary. Simulations of the model demonstrate how oxygen deficiency caused by ischemia limit macrophage recruitment to the wound-site and impair wound closure. The results are in tight agreement with recent experimental findings in a porcine model. I will also show analytical results of the model on the large-time asymptotic behavior of the free boundary under different ischemic conditions of the wound.
Dr. Chuan Xue
Mathematical Biosciences Institute at Ohio State
will present
A Mathematical Model of Chronic Wounds
Thursday, January 20, 2011, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: We present a new algorithm for the identification of bound regions from ChIP-Seq experiments. ChIP-Seq is a relatively new assay for measuring the interactions of proteins with DNA. The binding sites for a given protein in a genome are "peaks" in the data, which is given by an integer-valued height function defined on the genome. Our method for identifying statistically significant peaks is inspired by the notion of persistence in topological data analysis and provides a non-parametric approach that is robust to noise in experiments. Specifically, our method reduces the peak calling problem to the study of tree-based statistics derived from the data. The software T-PIC (Tree shape Peak Identification for ChIP-Seq) is available at http://math.berkeley.edu/~vhower/tpic.html and provides a fast and accurate solution for ChIP-Seq peak finding.
Dr. Valerie Hower
University of California, Berkeley
will present
A Shape-based Method for Determining Protein Binding Sites in a Genome
Tuesday, January 18, 2011, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: Various models exist for the interaction between the drug imatinib and chronic myelogenous leukemia. However, the role of the immune response during imatinib treatment remains unclear. Based on experimental data, we hypothesize that imatinib gives rise to a brief anti-leukemia immune response as patients enter remission.
Dr. Peter Kim
University of Utah
will present
Imatinib Dynamics and Cancer Vaccines:
From Agent-Based Models to PDEs
Thursday, January 13, 2011, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
We propose that cancer vaccinations during imatinib treatment can boost the existing immune response and lead to a sustained remission or a potential cure. To examine this hypothesis, we take a model by Michor et al. and extend it to a delay differential equation (DDE) model by incorporating an anti-leukemia immune response. We show that properly-timed vaccines can sustain the immune response to potentially prolong remission or eliminate cancer.
For comparison, we analyze an agent-based model developed independently by Roeder et al. We develop a partial differential equation (PDE) model that captures the same behavior as the Roeder agent-based model and extend it by incorporating an immune response. We conclude that both the DDE and PDE models exhibit similar behaviors with regard to cancer remission, implying that anti-leukemia immune responses may play a role in leukemia treatment.Abstract: There is a bijection between a class of piece-wise linear surfaces and dimer configurations on planar graphs. A dimer configuration on a graph is a perfect matching on vertices connected by edges. Dimers are well known in biology, chemistry and statistical mechanics. For certain very natural probability measures on dimer configurations, important correlation functions can be computed as Pfaffians of N\times N matrices. This reduces the statistics of such special random surfaces to a reasonable problem in linear algebra. This allows to study such random surfaces corresponding tolarge graphs. The talk will outline this story and at the end the discussion will focus on the "continuum limit" of such random surfaces.
Professor Nicolai Reshetikhin
University of California, Berkeley
will present
Understanding Random Surfaces
Monday, December 20, 2010, 4:30pm
Ungar Room 402
Refreshments at 4:00pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: This presentation is devoted to bifurcation problems for some classes of PDE arising in the context of population dynamics. The main difficulty in such a context is to understand the dynamical properties of a PDE with non-linear and non-local boundary conditions. A typical class of examples is the so called age structured models. Age structured models have been well understood in terms of existence, uniqueness, and stability of equilibria since the 80's. Nevertheless, up to recently, the bifurcation properties of the semiflow generated by such a system has been only poorly understood.
Professor Pierre Magal
University of Bordeaux, France
will present
Bifurcation Problems for Structured Population Dynamics Models
Monday, December 6, 2010, 4:30pm
Ungar Room 402
Refreshments at 4:00pm in Ungar Room 521
All interested persons are welcome to attend.
In this presentation, we will start with some results about existence and smoothness of the center mainfold, and we will present some general Hopf bifurcation results applying to age structured models. Then we will turn to normal theory in such a context. The point here is to obtain formula to compute the first order terms of the Taylor expansion of the reduced system.Abstract: For centuries, humans have attempted to control insect populations. This is in part because of the significant mortality and morbidity burden associated with insect vector-borne diseases, but also due to the huge economic impact of insect pests leading to losses in global food production. The development of transgenic technologies, coupled with sterile insect techniques (SIT), is being explored in relation to new approaches for the biological control of insect pests.
Dr. Steven White
Centre for Ecology & Hydrology
Wallingford, Oxon, UK
will present
Controlling Mosquitoes by Classical or Transgenic Sterile Insect Techniques
Monday, November 22, 2010, 4:00pm
Ungar Room 402
Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.
In this talk, I explore the impact of two control strategies (classical SIT and transgenic late-acting bisex lethality) using a stage-structured mathematical model, which is parameterized for the mosquito Aedes aegypti, which can spread yellow fever, dengue fever and Chikungunya disease. Counter to the majority of studies, I use realistic pulsed release strategies and incorporate a fitness cost, which is manifested as a reduction in male mating competitiveness.
I will explore the timing of control release in constant and cyclic wild-type mosquito populations, and demonstrate that this timing is critical for effective pest management. Furthermore, I will incorporate these control strategies into an integrated pest management program (IPM) and find the optimal release strategy. Finally, I will extend the models to a spatial context, determining conditions for the prevention of mosquito invasion by the use of a barrier wall.Abstract: Artin's famous primitive root conjecture states that if n is an integer other than -1 or a square, then there are infinitely many primes p such that n is a primitive root modulo p. Although this conjecture is not known to hold for any value of n, Hooley proved it to be true under the assumption of the generalized Riemann hypothesis (GRH). We will discuss a number field version of this conjecture and its connection to the following Euclidean algorithm problem. Let O be the ring of integers of a number field K. It is well-known that if O is a Euclidean domain, then O is a unique factorization domain. With the exception of the imaginary quadratic number fields, it is conjectured that the reverse implication is true. This was proven by Weinberger under the assumption of the GRH. We will discuss recent progress towards the unconditional resolution of the Euclidean algorithm problem and the related primitive root problem. This is joint work with M. Ram Murty.
Dr. Kate Petersen
Florida State University
will present
The Euclidean Algorithm and Primitive Roots
Thursday, November 18, 2010, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: In this talk, we consider the boundedness and unboundedness of solutions for the asymmetric oscillator
Dr. Shiwang Ma
Nankai University, China
will present
Bounded and Unbounded Motions for Asymmetric Oscillators at Resonance
Monday, November 8, 2010, 4:30pm
Ungar Room 402
Refreshments at 4:00pm in Ungar Room 521
All interested persons are welcome to attend.
x" + ax+ - bx- + g(x) = p(t), where x+ = max{x,0},x- = max{-x,0}, a and b are two positive constants, p(t) is a 2π-periodic smooth function and g(x) satisfies lim|x|→+∞x-1g(x) = 0. We establish some sharp sufficient conditions concerning the boundedness of all the solutions and the existence of unbounded solutions. Unlike many existing results in the literature where the function g(x) is required to be a bounded function with asymptotic limits, here we allow g(x) be unbounded or oscillatory without asymptotic limits. Some critical cases will also be considered.Abstract: The talk will introduce basic mathematical concepts of General Relativity and review the progress, main challenges and open problems, viewed through the prism of the evolution problem. I will illustrate interaction of Geometry and PDE methods in the context of General Relativity on examples ranging from incompleteness theorems and formation of trapped surfaces to stability problems.
Dr. Igor Rodnianski
Princeton University
will present
Evolution Problem in General Relativity
Wednesday, November 3, 2010, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: The problem of nonlinear stability for the Kerr model of a rotating black hole is one of the central problems in general relativity. The analysis of linear fields on the Kerr spacetime is an important model problem for full nonlinear stability. In this talk, I will present recent work with Pieter Blue which makes use of the hidden symmetry related to the Carter constant to circumvent these difficulties and give a "physical space" approach to estimates for the wave equation, including energy bounds, trapping, and dispersive estimates. I will also discuss the field equations for higher spin fields including linearized gravity.
Dr. Lars Andersson
Albert Einstein Institute
will present
The Black Hole Stability Problem
Friday, October 22, 2010, 4:00pm
Ungar Room 402
Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: Phytoplankton need light to grow. However, most of phytoplankton are heavier than water, so they sink. How can phytoplankton persist? We investigate a nonlocal reaction-diffusion-advection equation which models the growth of a single phytoplankton species in a water column where the species depends solely on light for its metabolism. We study the effect of sinking rate, water column depth and vertical turbulent diffusion rate on the persistence of a single phytoplankton species. This is based upon a joint work with Sze-Bi Hsu, National Tsing-Hua University.
Dr. Yuan Lou
Ohio State University
will present
Persistence of a Single Phytoplankton Species
Thursday, October 21, 2010, 4:30pm
Ungar Room 402
Refreshments at 4:00pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: This talk surveys some recent work in algebraic combinatorics that illustrates surprising connections between representation theory and enumerative combinatorics. We describe how to calculate the Hilbert series of various spaces of polynomials (harmonics, diagonal harmonics, and Garsia-Haiman modules) using combinatorial statistics on permutations and parking functions. This leads to a discussion of the algebraic and combinatorial significance of the Macdonald polynomials, which have played a central role in the theory of symmetric functions for the past two decades.
Professor Nick Loehr
Virginia Tech
will present
Macdonald Polynomials in Representation Theory and Combinatorics
Friday, October 8, 2010, 4:30pm
Ungar Room 402
Refreshments at 4:00pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: In this talk, I will present a survey of research projects on different mathematical models for influenza and HIV. For influenza, I will discuss two different modeling approaches. In the first approach, I will present a multi-strain/multi-host (MSMH) model that tracks the spread of inter-species strains between birds, pigs and humans. In the MSMH model, pigs are "mixing vessels" between avian and human strains and are capable of producing super-strains as a consequence of genetic recombination of these strains. I will show how specific subtypes can cause an epidemic then virtually disappear for years or even decades before reemerging (e.g., the case of H1N1). In the second approach, I will present a model that tracks the spread of influenza within flight transmission. A plane flight is much shorter scale than influenza's infectious duration; hence, we use methods from microbial risk management to assess the number of potential infections. We show that the flight duration along with the compartment will ultimately determine the passenger's risk. For HIV, I will present cross-sectional data on HIV prevalence in Lesotho, a small sub-Saharan African nation with HIV prevalence at approximately 23%. I will present our current progress on data analysis from the Health and Demographic Survey (DHS) to develop risk maps by district based on prevalence and treatment, feasibility analysis of a clinical trial, and efficacy of male circumcision as prevention for HIV.
Brian J. Coburn, Ph.D.
Center for Biomedical Modeling
Semel Institute of Neuroscience and Human Behavior
David Geffen School of Medicine
University of California, Los Angeles
will present
Modeling Approaches for Influenza and HIV
Wednesday, September 8, 2010, 4:00pm
Ungar Room 402
Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: We introduce the first analytical model of a dispersal-limited, niche-structured community to yield Hubbell's neutral theory in the limit of functional equivalence among all species. Dynamics of the multivariate species abundance distribution (SAD) for an asymmetric local community are modeled explicitly as a dispersal-limited sampling of the surrounding metacommunity. Coexistence may arise either from approximate functional equivalence or a competition-colonization tradeoff. At equilibrium, these symmetric and asymmetric mechanisms both generate unimodal SADs. Multiple modes only arise in asymmetric communities and provide a strong indication of non-neutral dynamics. Although these stationary distributions must be calculated numerically in the general theory, we derive the first analytical sampling distribution for a nearly neutral community where symmetry is broken by a single species distinct in ecological fitness and dispersal ability. Novel asymptotic expansions of hypergeometric functions are developed to make evaluations of the sampling distribution tractable for large communities. In this regime, population fluctuations become negligible. A calculation of the macroscopic limits for the symmetric and asymmetric theories yields a new class of deterministic competition models for communities of fixed-size where rescue effects facilitate coexistence. For nearly neutral communities where the asymmetric species experiences linear density-dependence in ecological fitness, strong Allee-type effects emerge from a saddle-node bifurcation at a critical point in dispersal limitation. The bistable dynamics governing a canonical Allee effect are modified by a constant influx of migrants, which raises the lower stable fixed point above zero. In the stochastic theory, a saddle-node bifurcation corresponds to the development of bimodal stationary distributions and the emergence of inflection points in plots of mean first-time to extirpation as a function of abundance.
Dr. Andrew Noble
University of Maryland
will present
A Non-neutral Theory of Dispersal-limited Community Dynamics
Thursday, April 22, 2010, 4:30pm
Ungar Room 402
Refreshments at 4:00pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: Carbon (C), nitrogen (N), and phosphorus (P) are vital constituents in biomass: C supplies energy to cells, N is essential to build proteins, and P is an essential component of nucleic acids. The scarcity of any of these elements can severely restrict organism and population growth. Thus in nutrient deficient environments, the consideration of nutrient cycling, or stoichiometry, may be essential for population models. To show this idea, I will present two case studies in this talk.
Dr. Hao Wang
University of Alberta
will present
The Role of Light and Nutrients in Aquatic Trophic Interactions
Friday, April 16, 2010, 4:30pm
Ungar Room 402
Refreshments at 4:00pm in Ungar Room 521
All interested persons are welcome to attend.
We carried out a microcosm experiment evaluating competition of an invasive species Daphnia lumholtzi with a widespread native species, Daphnia pulex. We applied two light treatments to these two different microcosms and found strong context-dependent competitive exclusion in both treatments. To better understand these results we developed and tested a mechanistically formulated stoichiometric model. This model exhibits chaotic coexistence of the competing species of Daphnia. The rich dynamics of this model as well as the experiment allow us to suggest some plausible strategies to control the invasive species D. lumholtzi.
We modeled bacteria-algae interactions in the epilimnion with the explicit consideration of carbon (energy) and phosphorus (nutrient). We hypothesized that there are three dynamical scenarios determined by the basic reproductive numbers of bacteria and algae. Effects of key environmental conditions were examined through these scenarios. Competition of bacterial strains were modeled to examine Nishimura's hypothesis that in severely P-limited environments such as Lake Biwa, P-limitation exerts more severe constraints on the growth of bacterial groups with higher nucleic acid contents, which allows low nucleic acid bacteria to be competitive.Abstract: We study input-affine systems where input represents some bounded noise. The system can be rewritten as differential inclusion describing the evolution. Differential inclusions are a generalization of differential equations with multivalued right-hand side. They have applications in many areas of science, such as mechanics, electrical engineering, the theory of automatic control, economical, biological, and social macrosystems. A numerical method for rigorous over-approximation of a solution set of input-affine system will be presented. The method gives high order error for a single time step and a uniform bound on the error over the finite time interval. The approach is based on the approximations of inputs by piecewise linear functions.
Dr. Sanja Zivanovic
Centrum Wiskunde en Informatica (CWI), Amsterdam, Netherlands
will present
Numerical Solutions to Noisy Systems
Thursday, April 15, 2010, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: I investigate the effects of non-random dispersal strategies on coexistence and species distributions in multi-trophic communities with competition and predation. I conduct a comparative analysis of dispersal strategies with random and fitness-dependent dispersal at the extremes and two intermediate strategies that rely on cues (density and habitat quality) that serve as proxies for fitness. The most important finding is an asymmetry between consumer species in their dispersal effects. The dispersal strategy of inferior resource competitors that are less susceptible to predation have a large effect on both coexistence and species distributions, but the dispersal strategy of the superior resource competitor that is more susceptible to predation has little or no effect on dispersal. I explore the consequences of this asymmetry for the evolution of dispersal.
Dr. Priyanga Amarasekare
University of California Los Angeles
will present
Ecological and Evolutionary Consequences of Dispersal in Multi-trophic Communities
Friday, March 12, 2010, 4:30pm
Ungar Room 402
Refreshments at 4:00pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: Suppose Alice has a coin with heads probability q and Bob has one with heads probability p > q. Now each of them will toss their coin n times, and Alice will win iff she gets more heads than Bob does. Of course, the game favors Bob, but for the given p, q, what is the choice, N(q,p), of n that maximizes Alice's chances of winning? The analysis uses the multivariate form of Zeilberger's algorithm, so a portion of the talk will be a review of the ideas underlying symbolic summation.
Dr. Herbert Wilf
Thomas A. Scott Emeritus Professor of Mathematics
University of Pennsylvania
will present
How to Lose as Little as Possible
Thursday, March 11, 2010, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: To formulate new Sobolev inequalities one needsto answer questions like: what is the role of dimension? What norms are appropriate to measure the integrabilitygains? Just to name a few...For example, in contrast to the Euclidean case, the integrability gains in Gaussian measure are logarithmic but dimension free (log Sobolev inequalities). So it is easy to understand the difficulties to derive a general theory. I will discuss some new methods to prove general Sobolev inequalities that unify the Euclidean and the Gaussian cases, as well as several important model manifolds.
Dr. Mario Milman
Florida Atlantic University
will present
Sobolev Inequalities on Probability Metric Spaces
Friday, March 5, 2010, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: Several problems will be discussed: 1) What is the distribution of the empirical correlation coefficient of two (actually independent) Wiener processes? It is far from zero - correlation is induced by the arc sine law property of the sample paths. This is used by (bad) statisticians to show correlation between time series when none exists. It is a non-trivial calculation to find the actual distribution. 2) What is the relationship between the coefficients of a polynomial of degree n and the number of its real zeros? Descartes had something to say about it, but Mark Kac showed that probability theory can add a lot of insight. 3) An update on the situation discussed last year re the artificial pancreas project.
CANCELLED
Dr. Larry Shepp
Rutgers University
Member, National Academy of Science (NAS)
Member, Institute of Medicine (IOM)
will present
Problems in Probability
Friday, February 26, 2010, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: The problem of constructing Riemannian metrics of constant scalar curvature is called the Yamabe problem. It is an important variational problem in conformal geometry, and also relates directly to the Einstein equations of general relativity. We will give a brief history and introduction to this problem and describe some new phenomena which have been discovered recently concerning issues of singular behavior and blow up of such metrics.
Dr. Richard Schoen
Bass Professor of Humanities and Sciences
Stanford University
will present
Riemannian Manifolds of Constant Scalar Curvature
Tuesday, February 23, 2010, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: It is known that, on closed manifolds, Einstein metrics of negative scalar curvature are critical points of the usual volume functional constrained to the space of metrics of constant scalar curvature. In this talk, I will discuss how this variational characterization of Einstein metrics can be localized to compact manifolds with boundary. I will derive the critical point equation and focus on geometric properties of its solutions. In particular, if a solution has zero scalar curvature and the boundary of the manifold can be isometrically embedded into the Euclidean space as a convex hypersurface, I will show that the volume of such a critical metric is always greater than or equal to the Euclidean volume enclosed by the image of the isometric embedding, and two volumes are the same if and only if the critical metric is isometric to the Euclidean metric on a round ball. I will also give a classification of all conformally flat critical metrics. This is joint work with Luen-Fai Tam.
Dr. Pengzi Miao
School of Mathematical Sciences
Monash University
will present
Critical Metrics for the Volume Functional on Compact Manifolds with Boundary
Thursday, February 4, 2010, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: One might search for "canonical metrics," such as Einstein metrics, on a manifold by trying to prove the convergence of a sequence of metrics that minimize some functional, although such a direct approach usually fails. In this talk we present an indirect approach which has been successful in some cases. A local obstruction to finding an Einstein metric in a conformal class is the non-vanishing of the Bach tensor, defined to be the gradient of the Weyl curvature functional $\int |W|^2$. On a Kaehler manifold there are no other obstructions, and any Bach-flat Kaehler metric is locally conformally Einsteinian. Additionally, the conformal factor is geometrically interesting and sometimes controllable. This talk will describe the results of a 2008 paper with X. Chen and C. LeBrun, where circumstances under which a Kaehler manifold is Bach-flat were established, and where it was shown that these conditions hold for a certain Kaehler metric on $CP^2 # 2\overline CP^2$ with non-zero conformal factor, establishing for the first time an Einstein metric on $CP^2 # 2\overline 2CP^2$.
Dr. Brian J. Weber
RTG/Simons Center Postdoc
Stony Brook University
will present
Einstein Metrics, the Bach Tensor, and Metric Degenerations
Monday, February 1, 2010, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: In this talk, which represents joint work with Stefan Friedl, we will present a classification of irreducible metabelian SL(n,C) representations of knots groups. Under a mild hypothesis, we prove that such representations factor through a finite group, hence they are all conjugate to unitary representations, and we give a simple formula for the number of conjugacy classes. For knots with nontrivial Alexander polynomial, we discuss an existence result for irreducible metabelian representations. Given a knot group, its SL(n,C) character variety admits a natural action by the cyclic group of order n, and we show how to identify the fixed points of this action with characters of metabelian representations. If time permits, we will describe conditions under which such points are simple points in the character variety using a deformation argument of Abdelghani, Heusener, and Jebali.
Dr. Hans Boden
McMaster University
will present
Metabelian SL(n,C) Representations of Knot Groups
Thursday, January 28, 2010, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: One of the fundamental properties of the Ricci flow -- an evolution equation for Riemannian metrics -- is that of isometry preservation, namely, that an isometry of the initial metric remains an isometry of the solution, at least as long as the curvature remains bounded. In this talk, I will take up the complementary problem of isometry development under the flow. While the solution may acquire new isometries in the limit, one does not expect the flow to sponsor their generation within the lifetime of the solution. The impossibility of such a phenomenon is equivalent to a backwards uniqueness (or unique-continuation) property for the equation: two solutions which agree at some non-initial time must agree identically at all previous times. I will discuss recent work which establishes this property for complete solutions of bounded curvature, and prohibits, additionally, a solution from becoming Einstein or self-similar in finite time.
Dr. Brett L. Kotschwar
C.L.E. Moore Instructor
Massachusetts Institute of Technology
will present
Backwards Uniqueness for the Ricci Flow and the Non-expansion of the Isometry Group
Monday, January 25, 2010, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: Delay equation models for the control of a vector-borne disease such as West Nile virus will be presented. The models make it possible to compare the effectiveness of larvicides and adulticides in controlling mosquito populations. The models take the form of autonomous delay differential equations with impulses (if the adult insects are culled) or a system of nonautonomous delay differential equations where the time-varying coefficients are determined by the culling times and rates (in the case where the insect larvae are culled). Sufficient conditions can be derived which ensure eradication of the disease. Eradication of vector-borne diseases is possible by culling the vector at either the immature or the mature phase. Very infrequent culling can actually lead to the mean insect population being increased rather than decreased.
Dr. Stephen Gourley
University of Surrey, UK
will present
Impulsive Delay Equation Models for the Control of Vector-borne Diseases
Friday, January 15, 2010, 4:00pm
Ungar Room 402
Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: I will present recent results on Einstein spacetimes of general relativity when the curvature is solely assumed to be bounded and no assumption on its derivatives is made. One such result, in a joint work with B.-L. Chen, concerns the optimal regularity of pointed spacetimes in which, by definition, an "observer" has been specified. Under geometric bounds on the curvature and injectivity radius near the observer, there exist a CMC (constant mean curvature) foliation as well as CMC--harmonic coordinates, which are defined in geodesic balls with definite size depending only on the assumed bounds, so that the components of the Lorentzian metric has optimal regularity in these coordinates. The proof combines geometric estimates (Jacobi field, comparison theorems) and quantitative estimates for nonlinear elliptic equations with low regularity.
Professor Philippe LeFloch
University of Paris 6 and CNRS
will present
Einstein Spacetimes with Bounded Curvature
Thursday, December 10, 2009, 4:30pm
Ungar Room 402
Refreshments at 4:00pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: Patients at risk for complications of influenza are commonly treated with antiviral medications, which however also could be used to control outbreaks. The adamantanes and neuraminidase inhibitors are active against influenza A, but avian influenza (H5N1) is resistant to oseltamivir and swine influenza (H1N1) to the adamantanes (but see postscript). To explore influenza medication strategies (pre-exposure or prophylaxis, post-exposure/pre-symptom onset, and treatment at successive clinical stages) that may affect evolution of resistance (select for resistant strains within or facilitate their spread between hosts), we elaborated a published transmission model and chose parameters from the literature. Then we derived the reproduction numbers of sensitive and resistant strains, peak and final sizes, and time to peak. Finally, we made these results accessible via user-friendly Mathematica notebooks. (Joint work with Rongsong Liu, Dashun Xu, Yiding Yang, and John Glasser)
CANCELLED
Dr. Zhilan Feng
Purdue University
will present
Evolutionary Implications of Influenza Medication Strategies
Tuesday, November 24, 2009, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: We study geometric variational problems for a class of nonlinear sigma-models in quantum field theory. Mathematically, one needs to minimize an energy functional on homotopy classes of maps from closed 3-manifolds into compact homogeneous spaces G/H, similar to the case of harmonic maps. The minimizers are known as Hopfions and exhibit localized knot-like structure. Our main results include proving existence of Hopfions as finite energy Sobolev maps in each (generalized) homotopy class when the target space is a symmetric space. For more general spaces we obtain a weaker result on existence of minimizers in each 2-homotopy class.
Professor Sergiy Koshkin
University of Houston-Downtown
will present
Gauge Theory of Faddeev-Skyrme Functionals
Friday, November 20, 2009, 3:30pm
Ungar Room 402
Refreshments at 3:00pm in Ungar Room 521
All interested persons are welcome to attend.
Our approach is based on representing maps into G/H by equivalence classes of flat connections. The equivalence is given by gauge symmetry on pullbacks of G-->G/H bundles. We work out a gauge calculus for connections under this symmetry, and use it to eliminate non-compactness from the minimization problem by fixing the gauge.Abstract: First I will review how methods from commutative algebra, for example Gröbner bases and toric ideals, can be used in statistics. Then I will describe two applications of graph theoretic methods in this context: My proof of Sturmfels and Sullivant's conjecture on cut ideals; and the ideals of graph homomorphisms introduced together with Patrik Noren.
Dr. Alexander Engström
Miller Research Fellow
University of California, Berkeley
will present
Graph Theoretic Methods in Algebraic Statistics
Thursday, November 12, 2009, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
Abstract: A knot in the 3-sphere is slice if it bounds an embedded disk in the 4-ball. The disk may be topologically embedded, or we may require the stronger condition that it be smoothly embedded; the knot is said to be (respectively) topologically or smoothly slice. It has been known since the early 1980's that there are knots that are topologically slice, but not smoothly slice. These result from Freedman's proof that knots with trivial Alexander polynomial are topologically slice, combined with gauge-theory techniques originating with Donaldson. In joint work with C. Livingston and M. Hedden, we answer the natural question of whether Freedman's result is responsible for all topologically slice knots. We show that the group of topologically slice knots, modulo those with trivial Alexander polynomial, is infinitely generated. The proof uses Heegaard-Floer theory.
Dr. Daniel Ruberman
Brandeis University
will present
Slice Knots and the Alexander Polynomial
Thursday, November 5, 2009, 5:00pm
Ungar Room 402
Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.
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