Margaret Regan : Using homotopy continuation to solve parametrized polynomial systems in applications
- Graduate/Faculty Seminar,Uploaded Videos ( 1637 Views )Many problems that arise in mathematics, science, and engineering can be formulated as solving a parameterized system of polynomial equations which must be solved for given instances of the parameters. One way to solve these systems is to use a common technique within numerical algebraic geometry called homotopy continuation. My talk will start with background on homotopy continuation and parametrized polynomial systems, followed by applications to problems in computer vision and kinematics. Of these, I will first present a new approach which uses locally adaptive methods and sparse matrix calculations to solve parameterized overdetermined systems in projective space. Examples will be provided in 2D image reconstruction to compare the new methods with traditional approaches in numerical algebraic geometry. Second, I will discuss a new definition of monodromy action over the real numbers which encodes tiered characteristics regarding real solutions. Examples will be given to show the benefits of this definition over a naive extension of the monodromy group (over the complex numbers). In addition, an application in kinematics will be discussed to highlight the computational method and impact on calibration.
Rafah Hajjar Munoz : On the residually indistinguishable case of Ribet’s lemma
- Number Theory ( 217 Views )Ribet’s method describes a way to construct a certain extension of fields from the existence of a suitable modular form. To do so, we consider the Galois representation of an appropriate cuspform, which gives rise to a cohomology class that cuts out our desired extension. The process of obtaining a cohomology class from such a representation is usually known as Ribet’s lemma. Several generalizations of this lemma have been stated and proved during the last decades, but the vast majority of them makes the assumption that the representation is residually distinguishable, meaning that the characters of its residual decomposition are non-congruent modulo the maximal ideal. However, recent applications of Ribet’s method, such as for the proof of the 2-part of the Brumer-Stark conjecture, have encountered the challenge that the representation we obtain does not satisfy this assumption. In my talk, I describe the limitations of the residually indistinguishable case and conjecture a new general version of Ribet’s lemma in this context, giving a proof in some particular cases.
Dave Rose : Cartans theorem on maximal tori
- Graduate/Faculty Seminar ( 121 Views )Cartan's theorem on maximal tori in compact Lie groups can be thought of as a generalization of the spectral theorem for unitary matrices. The goal of this talk will be to sketch the `topological' proof of this theorem, based on the Lefschetz fixed point theorem. Along the way, we'll encounter the flag variety, an interesting object whose geometry encodes the representation theory of the Lie group. Those who don't specialize in geometry or topology fear not, we will give examples showing that these concepts are very concrete objects familiar from linear algebra.
Jill Pipher : Geometric discrepancy theory: directional discrepancy in 2-D
- Applied Math and Analysis ( 90 Views )Discrepancy theory originated with some apparently simple questions about
sequences of numbers. The discrepancy of an infinite sequence is a
quantitative measure of how far it is from being uniformly distributed.
Precisely, an infinite sequence { a1,a2, ...} is said to be uniformly
distributed in [0, 1] if
lim_{n\to\infty} (1/n|{a1, a2,...an} intersect [s,t]|) = t-s.
If a sequence {ak} is uniformly distributed, then it is also the case
that for all (Riemann) integrable functions f on [0, 1],
lim_{n\to\infty} (1/n\sum_{k=1}^n f(ak))=\int_0^1 f(x)dx.
Thus, uniformly distributed sequences provide good numerical schemes
for approximating integrals. For example, if alpha is any irrational
number in [0, 1], then the fractional part {alphak}:=ak is uniformly
distributed. Classical Fourier analysis enters here, in the form of
Weyl's criterion.
The discrepancy of a sequence with respect to its first n entries is
D({ak},n) := sup_{s
See PDF.
Lydia Bilinsky : A Mathematical Model of Glutamate and Glutamine Metabolism in the Rat: Implications for Glutathione Production
- Mathematical Biology ( 100 Views )Glutathione (GSH), a tripeptide formed from glutamate, cysteine, and
glycine, is arguably the most important antioxidant in the body. NAPQI, a
byproduct of acetaminophen (APAP) metabolism which is toxic to liver
cells, is neutralized by GSH. Although produced in great quantity by the
liver, in cases of APAP overdose demand for GSH can outstrip supply,
causing liver failure. Currently, patients presenting to the ER with APAP
overdose are given an infusion of cysteine since it is believed to be the
rate-limiting amino acid in GSH synthesis, however, there is evidence that
under some circumstances glutamate can become rate-limiting. Complicating
the issue is that in most hepatocytes, glutamate is not absorbable from
blood plasma but is formed from glutamine, which is produced in large
amounts by the skeletal muscle. In order to develop better rescue
protocols for APAP overdose, we have developed a mathematical model of
glutamate and glutamine metabolism in the rat. We have also investigated
how model parameters should change in the case of increased cortisol
production, such as occurs during sepsis, trauma, burns, and other
pathological states; the cortisol-stressed state has been studied in rats
by giving them dexamethasone. We compare model predictions with
experimental data for the normal, healthy rat and dexamethasone-stressed
rat. Biological parameters are taken from the literature wherever possible.
Emanuele Macri : MMP for moduli spaces of sheaves on K3 surfaces and Cone Conjectures
- Algebraic Geometry ( 133 Views )We report on joint work with A. Bayer on how one can use wall-crossing techniques to study the birational geometry of a moduli space M of Gieseker-stable sheaves on a K3 surface X. In particular: (--) We will give a "modular interpretation" for all minimal models of M. (--) We will describe the nef cone, the movable cone, and the effective cone of M in terms of the algebraic Mukai lattice of X. (--) We will establish the so called Tyurin/Bogomolov/Hassett-Tschinkel/Huybrechts/Sawon Conjecture on the existence of Lagrangian fibrations on M.
Demetre Kazaras:The geometry and topology of positive scalar curvature
- Graduate/Faculty Seminar,Uploaded Videos ( 1585 Views )I will give an informal overview of the history and status of my field. Local invariants of Riemannian metrics are called curvature, the weakest of which is known as "scalar curvature." The study of metrics with positive scalar curvature is very rich with >100 year old connections to General Relativity and smooth topology. Does this geometric condition have topological implications? The answer turns out to be "yes," but mathematicians continue to search for the true heart of the positive scalar curvature conditions.
Oguz Savk : Bridging the gaps between homology planes and Mazur manifolds.
- Geometry and Topology,Uploaded Videos ( 249 Views )We call a non-trivial homology 3-sphere a Kirby-Ramanujam sphere if it bounds a homology plane, an algebraic complex smooth surface with the same homology groups of the complex plane. In this talk, we present several infinite families of Kirby-Ramanujam spheres bounding Mazur type 4-manifolds, compact contractible smooth 4-manifolds built with only 0-, 1-, and 2-handles. Such an interplay between complex surfaces and 4-manifolds was first observed by Ramanujam and Kirby around nineteen-eighties. This is upcoming joint work with Rodolfo Aguilar Aguilar.
John McGreevy : Linear sigma models for open strings, with applications
- String Theory ( 11 Views )I will couple two-dimensional $N=(2,2)$ gauge theories to boundary matter in order to make them flow to worldsheet CFTs for open strings. In particular, I will focus on models for strings ending on BPS D-branes on even-dimensional cycles of Calabi-Yau threefolds. This framework provides an effective way of thinking about the role of the derived category of coherent sheaves in D-brane physics. Applications include a study of degenerations and singularities of the CFT, monodromy in closed string moduli space, local models for marginal stability behavior, and a possible matrix definition of these backgrounds.
Peter Miller : Integrable Nonlinear Waves and Singular Asymptotics
- Applied Math and Analysis ( 139 Views )This talk will be concerned with nonlinear analogues of the classical methods of analysis for exponential integrals that one uses to study singular limits for linear wave propagation problems solved by Fourier transforms. These analogues apply to nonlinear wave problems that may be treated by a nonlinear analogue of the Fourier transform, the "inverse-scattering transform". We will discuss the use of these techniques to study the semiclassical limit for the focusing nonlinear Schr\"odinger (NLS) equation, and we will also mention some recent work on the modified focusing NLS equation (an equation that tries to make up for shortcomings of the focusing NLS equation arising from modulational instability) and the sine-Gordon equation. The work on sine-Gordon is joint with Robert Buckingham, a recent Duke PhD.
Victoria Booth : Dynamics of sleep-wake regulation
- CGTP Group Meeting Seminar ( 159 Views )Sleep and wake states are regulated by the interactions among a number of
brainstem and hypothalamic neuronal populations and the expression of
their neurotransmitters. Based on experimental studies, several different
structures have been proposed for this sleep-wake regulatory network with
particular debate over components involved in rapid-eye movement (REM)
sleep regulation. We have developed a mathematical modeling framework
that is uniquely suited for investigating the structure and dynamics of
proposed sleep-wake regulatory networks. Using this framework, we are
analyzing the competing proposed network structures for the regulation of
REM sleep to determine how the structure of the sleep-wake regulatory
network determines sleep-wake behavior and the dynamics of behavioral
state transitions.
Laura DeMarco : Complex dynamics and potential theory
- Presentations ( 148 Views )I will begin with the basics of the two subjects, with the goal of explaining how each has been used as a method to obtain results in the other. The first half will be devoted to foundational results, dating to the 1930s for potential theory and the 1980s for complex dynamics. The second half will be devoted to more recent developments.
Christoph Ortner : Multi-scale simulation of crystal defects
- Applied Math and Analysis ( 110 Views )PART 1: I will construct a mathematical model of a defect embedded in an infinite homogeneous crystal. I will then establish a regularity result for minimisers, which given the crucial information on which approximation schemes are based. As an elementary application of this framework I will prove convergence rates for two computational schemes: (1) clamped far-field and (2) coupling to harmonic far-field model.
PART 2: The conditions under which the theory of Part 1 holds are separability and locality of the total energy. In Part 2 I will show how for a tight-binding model (a minimalistic electronic structure model) these two condition arise. This analysis raises some interesting (open) questions.
PART 3: Finally, I will use the theory developed in PART 1 and PART 2 to construct and analyse a new family of QM/MM embedding schemes with rigorous error estimates.
Kate Stange : Visualising the arithmetic of imaginary quadratic fields
- Number Theory ( 96 Views )Let $K$ be an imaginary quadratic field with ring of integers $\mathcal{O}_K$. The Schmidt arrangement of $K$ is the orbit of the extended real line in the extended complex plane under the Mobius transformation action of the Bianchi group $\operatorname{PSL}(2,\mathcal{O}_K)$. The arrangement takes the form of a dense collection of intricately nested circles. Aspects of the number theory of $\mathcal{O}_K$ can be characterised by properties of this picture: for example, the arrangement is connected if and only if $\mathcal{O}_K$ is Euclidean. I'll explore this structure and its connection to Apollonian circle packings. Specifically, the Schmidt arrangement for the Gaussian integers is a disjoint union of all primitive integral Apollonian circle packings. Generalizing this relationship to all imaginary quadratic $K$, the geometry naturally defines some new circle packings and thin groups of arithmetic interest.
Erin Beckman : A Look at Branching Processes
- Graduate/Faculty Seminar ( 122 Views )In 1873, a man named Francis Galton posed a question in Educational Times, calling for the mathematical study of the extinction of family surnames over time. Within a year, mathematician Henry Watson replied with a solution. But instead of ending there, this question opened up a new direction of mathematics: the study of branching processes. A branching process is a particle system in which the particles undergo splitting or branching events dictated by particular rules. This talk will introduce some examples of these systems (from the basic Galton-Watson model to more general branching-selection models), interesting questions people ask about branching processes, and some recent research done in this area.
Ilarion Melnikov : The Coulomb Branch in Gauged Linear Sigma Models
- String Theory ( 14 Views )I will present recent work with M.R. Plesser on the GLSM perspective on closed string tachyon condensation in non-supersymmetric orbifolds. These models are distinguished from their supersymmetric relatives by the appearance of weakly coupled Coulomb vacua in certain phases of the GLSM. We study the form of A-twisted correlators in these theories. Using standard techniques, we are able to compute these correlators in a pure Higgs phase (one without Coulomb vacua). These correlators are independent of which phase is used to compute them, and thus we can learn about the properties of the Coulomb branch, where direct computations are difficult.
Frank Thorne : Secondary Terms in Counting Functions for Cubic Fields
- Number Theory ( 122 Views )I will speak about recent progress on the enumeration of number fields, with particular attention to joint work with Taniguchi, which proved the existence of a negative secondary term in the counting function for cubic fields by discriminant. Among other results, we also found surprising biases in arithmetic progressions -- e.g., cubic field discriminants are more likely to be 5 (mod 7) than 3 (mod 7). Our work applies the analytic theory of the Shintani zeta function, which I will describe briefly. I will also discuss other approaches to related questions (and in particular an independent, and different, proof of the secondary term due to Bhargava, Shankar, and Tsimerman), using approaches as diverse as the geometry of numbers, algebraic geometry, and class field theory.
Kingshuk Roy Choudury : Statistical modelling and comparison of tumor growth
- CGTP Group Meeting Seminar ( 169 Views )"I will talk about tumor growth modelling in xenograft experiments. It's a statistical approach (see e.g. the ref. below), but I'd like to incorporate more mathematical modelling." Roy Choudhury, K., Kasman, I., Plowman, G., 2010, Analysis of multi-arm tumor growth trials in xenograft animals using phase change adaptive piecewise quadratic models, Statistics in Medicine, 29, 2399-2409
Anette Hosoi : Small Swimming Lessons: Optimizing Low Reynolds Number Locomotion
- Mathematical Biology ( 89 Views )ABSTRACT: The past decade has seen a number of engineering innovations that make construction of devices of micro- and even nanometric dimensions feasible. Hence, there is a growing interest in exploring new and efficient ways to generate propulsion at these small scales. Here we explore optimization of one particular type of low Reynolds number propulsion mechanism flagella. Beyond the general challenges associated with optimization, there are a number of issues that are unique to swimming at low Reynolds numbers. At small scales, the fluid equations of motion are linear and time-reversible, hence reciprocal motion i.e., strokes that are symmetric with respect to time reversal cannot generate any net translation (a limitation commonly referred to as the Scallop Theorem). One possible way to break this symmetry is through carefully chosen morphologies and kinematics. One symmetry-breaking solution commonly employed by eukaryotic microorganisms is to select nonreciprocal stroke patterns by actively generating torques at fixed intervals along the organism. Hence, we will address the question: For a given morphology, what are the optimal kinematics? In this talk we present optimal stroke patterns using biologically inspired geometries such as single-tailed spermatozoa and the double-tail morphology of Chlamydomonas, a genus of green alga widely considered to be a model system in molecular biology.
Caroline Turnage-Butterbaugh : The Distribution of the Primes and Moments of Products of Automorphic $L$-functions
- Number Theory ( 121 Views )The prime numbers are the multiplicative building blocks of the integers, and much thought has been given towards understanding their behavior. In this talk, we will examine prime numbers from two points of view. We will first consider questions on the distribution of the primes. In particular, we will illustrate how the breakthrough work of Maynard and Tao on bounded gaps between primes settles an old problem of Erdos and Turan. Secondly, we will explore the relationship between prime numbers and zeros of the Riemann zeta-function, as a way to motivate the study of the moments of the Riemann zeta function and more general L-functions. In particular, we consider arbitrary products of L-functions attached to irreducible cuspidal automorphic representations of GL(m) over the rationals. The Langlands program suggests essentially all L-functions are of this form. Assuming some standard conjectures, I will discuss how to estimate two types of moments: the continuous moment of an arbitrary product of primitive automorphic L-functions and the discrete moment (taken over fundamental discriminants) of an arbitrary product of primitive automorphic L-functions twisted by quadratic Dirichlet characters.
Kash Balachandran : The Kakeya Conjecture
- Graduate/Faculty Seminar ( 131 Views )In 1917, Soichi Kakeya posed the question: What is the smallest amount of area required to continuously rotate a unit line segment in the plane by a full rotation? Inpsired by this, what is the smallest measure of a set in $\mathbb{R}^n$ that contains a unit line segment in every direction? Such sets are called Kakeya sets, and can be shown to have arbitrarily small measure w.r.t. n-dimensional Lebesgue measure [and in fact, measure zero]. The Kakeya conjecture asserts that the Hausdorff and Minkowski dimension of these sets in $\mathbb{R}^n$ is $n$. In this talk, I will introduce at a very elementary level the machinery necessary to understand what the Kakeya conjecture is asking, and how the Kakeya conjecture has consequences for fields diverse as multidimensional Fourier summation, wave equations, Dirichlet series in analytic number theory, and random number generation. I'll also touch on how tools from various mathematical disciplines from additive combinatorics and algebraic geometry to multiscale analysis and heat flow can be used to obtain partial results to this problem. The talk will be geared towards a general audience.