CANCELLED

### Left-orderability and Three-manifolds

Friday, October 7, 2016, 4:00pm
Ungar Room 402

Abstract: A group is called left-orderable if it can be given a left-invariant total order. We will discuss the question of when the fundamental group of a three-manifold Y is left-orderable. Orderability is known to be related to certain topological aspects of Y, such as the surfaces which sit inside it. We will discuss a conjectural relationship between left-orderability and the solutions to a certain nonlinear PDE on Y.

CANCELLED

### On the Topology of Steel

Thursday, October 6, 2016, 5:00pm
Ungar Room 402

Abstract: Polycrystalline materials, such as metals, are composed of crystal grains of varying size and shape. Typically, the occurring grain cells have the combinatorial types of 3-dimensional simple polytopes, and together they tile 3-dimensional space.

We will see that some of the occurring grain types are substantially more frequent than others - where the frequent types turn out to be "combinatorially round". Here, the classification of grain types gives us, as an application of combinatorial low-dimensional topology, a new starting point for a topological microstructure analysis of steel.

### Partial Differential Equations in Finance

Thursday, August 25, 2016, 5:00pm
Ungar Room 402

Abstract: Starting with the Black & Scholes equation, I will give a tour of partial differential equations arising in various financial models. My goal is to emphasize what is mathematically interesting about each equation and how solutions are used in applications.

### The Importance of Synchrony in Mass Drug Administration

Friday, July 8, 2016, 4:30pm
Ungar Room 402

Abstract: Mass drug administration (MDA), a strategy in which all individuals in a population are subject to treatment without individual diagnosis, has been recommended by the World Health Organization for controlling and eliminating several neglected tropical diseases. In this talk, I will present some results arising from mass treatment of trachoma with azithromycin. In the first part, we compare three typical drug distribution strategies (regardless of health status): constant treatment, impulsive synchronized MDA, and impulsive non-synchronized treatment. We show that synchronized and constant strategies are respectively the most and least effective treatments in disease control. Elimination through synchronized treatment is always possible when adequate drug efficacy and coverage is fulfilled and sustained. In the second part, the optimal seasonal timing of mass administration of azithromycin for maximum antimalarial benefit has been established. This is joint work with Thomas M. Lietman and Travis C. Porco.

### Nonlocal Effects and Nonlocal Dispersal

Thursday, July 7, 2016, 4:30pm
Ungar Room 402

Abstract: This talk is concerned with some aspects of nonlocal dispersal equations. It consists of three parts. In part one, I will present some relations between local (random) and nonlocal dispersal problems. In part two, I will report our recent results on traveling waves and entire solutions of nonlocal dispersal equations. Part III is devoted to some problems on traveling waves and entire solutions of nonlocal dispersal equations.

### Traveling Wave Solutions of Evolutionary Models without Monotonicity

Monday, May 9, 2016, 4:00pm
Ungar Room 402

Abstract: This talk is concerned with the traveling wave solutions of evolutionary systems including delayed reaction-diffusion systems and integrodifference equations. Even if the general monotone conditions fail, the existence of traveling wave solutions is studied by generalized upper and lower solutions. The asymptotic behavior is established by the idea of contracting rectangles. In the study of classical Lotka-Volterra competitive systems, we obtain the existence of nonmonotone traveling wave solutions, which weakly confirms the conjecture by Tang and Fife [ARMA, 1980].

### Spectral Asymptotics for First Order Systems

Thursday, April 14, 2016, 5:00pm
Ungar Room 402

Abstract: In layman's terms a typical problem in this subject area is formulated as follows. Suppose that our universe has finite size but does not have a boundary. An example of such a situation would be a universe in the shape of a 3-dimensional sphere embedded in 4-dimensional Euclidean space. And imagine now that there is only one particle living in this universe, say, a massless neutrino. Then one can address a number of mathematical questions. How does the neutrino field (solution of the massless Dirac equation) propagate as a function of time? What are the eigenvalues (energy levels) of the particle? Are there nontrivial (i.e. without obvious symmetries) special cases when the eigenvalues can be evaluated explicitly? What is the difference between the neutrino (positive energy) and the antineutrino (negative energy)? What is the nature of spin? Why do neutrinos propagate with the speed of light? Why are neutrinos and photons (solutions of the Maxwell system) so different and, yet, so similar?

The speaker will approach the study of first order systems of partial differential equations from the perspective of a spectral theorist using techniques of microlocal analysis and without involving geometry or physics. However, a fascinating feature of the subject is that this purely analytic approach inevitably leads to differential geometric constructions with a strong theoretical physics flavour.

### Rectifiability of Harmonic Measure

Wednesday, April 13, 2016, 4:00pm
Ungar Room 402

Abstract: In a recent multi-authored paper by J. Azzam, S. Hofmann, J.-M. Martell, S. Mayboroda, M. Mourgoglou, X. Tolsa and myself the following result is proved:

### Souls of Manifolds via Curvature and Surgery

Thursday, January 27, 2011, 5:00pm
Ungar Room 402

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.

### A Mathematical Model of Chronic Wounds

Thursday, January 20, 2011, 5:00pm
Ungar Room 402

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.

### A Shape-based Method for Determining Protein Binding Sites in a Genome

Tuesday, January 18, 2011, 5:00pm
Ungar Room 402

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.

### Imatinib Dynamics and Cancer Vaccines: From Agent-Based Models to PDEs

Thursday, January 13, 2011, 5:00pm
Ungar Room 402

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.

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.

### Understanding Random Surfaces

Monday, December 20, 2010, 4:30pm
Ungar Room 402

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.

### Bifurcation Problems for Structured Population Dynamics Models

Monday, December 6, 2010, 4:30pm
Ungar Room 402

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.

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.

### Controlling Mosquitoes by Classical or Transgenic Sterile Insect Techniques

Monday, November 22, 2010, 4:00pm
Ungar Room 402

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.

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.

### The Euclidean Algorithm and Primitive Roots

Thursday, November 18, 2010, 5:00pm
Ungar Room 402

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.

### Bounded and Unbounded Motions for Asymmetric Oscillators at Resonance

Monday, November 8, 2010, 4:30pm
Ungar Room 402

Abstract: In this talk, we consider the boundedness and unboundedness of solutions for the asymmetric oscillator

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.

### Evolution Problem in General Relativity

Wednesday, November 3, 2010, 5:00pm
Ungar Room 402

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.

### The Black Hole Stability Problem

Friday, October 22, 2010, 4:00pm
Ungar Room 402

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.

### Persistence of a Single Phytoplankton Species

Thursday, October 21, 2010, 4:30pm
Ungar Room 402

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.

### Macdonald Polynomials in Representation Theory and Combinatorics

Friday, October 8, 2010, 4:30pm
Ungar Room 402

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.

### Modeling Approaches for Influenza and HIV

Wednesday, September 8, 2010, 4:00pm
Ungar Room 402

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.

### A Non-neutral Theory of Dispersal-limited Community Dynamics

Thursday, April 22, 2010, 4:30pm
Ungar Room 402

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.

### The Role of Light and Nutrients in Aquatic Trophic Interactions

Friday, April 16, 2010, 4:30pm
Ungar Room 402

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.

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.

### Numerical Solutions to Noisy Systems

Thursday, April 15, 2010, 5:00pm
Ungar Room 402

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.

### Ecological and Evolutionary Consequences of Dispersal in Multi-trophic Communities

Friday, March 12, 2010, 4:30pm
Ungar Room 402

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.

### How to Lose as Little as Possible

Thursday, March 11, 2010, 5:00pm
Ungar Room 402

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.

### Sobolev Inequalities on Probability Metric Spaces

Friday, March 5, 2010, 5:00pm
Ungar Room 402

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.

CANCELLED

### Problems in Probability

Friday, February 26, 2010, 5:00pm
Ungar Room 402

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.

### Riemannian Manifolds of Constant Scalar Curvature

Tuesday, February 23, 2010, 5:00pm
Ungar Room 402

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.

### Critical Metrics for the Volume Functional on Compact Manifolds with Boundary

Thursday, February 4, 2010, 5:00pm
Ungar Room 402

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.

### Einstein Metrics, the Bach Tensor, and Metric Degenerations

Monday, February 1, 2010, 5:00pm
Ungar Room 402

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 $CP^2 # 2\overline 2CP^2$.

### Metabelian SL(n,C) Representations of Knot Groups

Thursday, January 28, 2010, 5:00pm
Ungar Room 402

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.

### Backwards Uniqueness for the Ricci Flow and the Non-expansion of the Isometry Group

Monday, January 25, 2010, 5:00pm
Ungar Room 402

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.

### Impulsive Delay Equation Models for the Control of Vector-borne Diseases

Friday, January 15, 2010, 4:00pm
Ungar Room 402

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.

### Einstein Spacetimes with Bounded Curvature

Thursday, December 10, 2009, 4:30pm
Ungar Room 402

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.

CANCELLED

### Evolutionary Implications of Influenza Medication Strategies

Tuesday, November 24, 2009, 5:00pm
Ungar Room 402

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)

### Gauge Theory of Faddeev-Skyrme Functionals

Friday, November 20, 2009, 3:30pm
Ungar Room 402

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.

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.

### Graph Theoretic Methods in Algebraic Statistics

Thursday, November 12, 2009, 5:00pm
Ungar Room 402

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.

### Slice Knots and the Alexander Polynomial

Thursday, November 5, 2009, 5:00pm
Ungar Room 402

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.