Michael Jenista : Global dynamics of switching networks in biology
- Graduate/Faculty Seminar ( 111 Views )The study of biological networks is an increasingly popular area of mathematical research. Many different approaches are applied to answer many different kinds of questions. We ask, "what kinds of behavior are observed in biological switching networks, and how can we produce this behavior?" This is therefore a question of modelling. We start with two different frameworks: boolean and continuous. Both are frequently used to model genetic transcription networks which are examples of switching networks. We then explore several principles of global dynamics that are true in both frameworks. We finish with some current research conjectures and sketches of proposed proofs.
Kevin Kordek : Geography of Mapping Class Groups and Moduli Spaces
- Graduate/Faculty Seminar ( 177 Views )Mapping class groups are topological objects which can be used to describe the continuous symmetries of a surface. On the other hand, every compact orientable surface has a moduli space, a complex variety whose points parametrize all of its inequivalent complex structures. These concepts turn out to be closely related. In this talk, we'll cover the basics of both mapping class groups and moduli of Riemann surfaces, as well as explore their relationship.
Richard Hain : Scissors Congruence
- Graduate/Faculty Seminar ( 127 Views )Is it true that two polygons in the plane have the same area if and only if they can be decomposed into congruent polygons? What about in three and higher dimensions? And what about the analogous questions for polygons in the hyperbolic plane and polyhedra in higher dimensional hyperbolic spaces? Some aspects of the subject are elementary, while others involve Abel's dilogarithm and arcane subjects, such as algebraic K-theory. This talk will be largely elementary, and fun.
Chad Schoen : Algebraic geometry and complex analytic geometry
- Graduate/Faculty Seminar ( 117 Views )This talk will introduce parts of algebraic geometry and complex analytic geometry which are closely related to each other. These are important areas of pure mathematics. The presentation will start from scratch and hopefully reach the statement of a famous conjecture by the end. The first half of the talk should be accessible to incoming graduate students who have worked an exercise or two on projective space. Familiarity with complex analysis at the undergraduate level would be helpful. The second half of the talk will make use of cohomology. Students who have taken a semester of algebraic topology and are now taking a second semester together with a course on Riemann surfaces should be able to follow many of the details. Those who have completed a course on Riemann Surfaces and on Algebraic Geometry should be able to follow all the details. Professional mathematicians working in this general area will likely find new insights few and far between if present at all.
Robert Bryant : Curves, Surfaces, and Webs: An Episode in 19th Century Geometry
- Graduate/Faculty Seminar ( 172 Views )An old question about surfaces in 3-space is: When can a surface be written as a sum of two curves? For example, the elliptic paraboloid z = x^2 + y^2 can be thought of as the sum of the two space curves (x,0,x^2) and (0,y,y^2). However, a little thought shows that most surfaces in space should not be expressible parametrically as X(s) + Y(t) where X and Y are space curves. Surfaces for which this can be done are called `surfaces of translation'. This raises the question of determining whether or not this is possible for a given surface and in how many ways. This simple question leads to some surprisingly deep mathematics, involving complex analysis and overdetermined systems of PDE, and to other questions that are still open today. I will explain some of these developments (and what they have to do with my own work). There will even be a few pictures.
Aubrey HB : Persistent Homology
- Graduate/Faculty Seminar ( 171 Views )Persistent Homology is an emerging field of Computational Topology that is developing tools to discover the underlying structure in high-dimensional data sets. I will discuss the origins and main concepts involved in Persistent Homology in an accessible way, with illustrations and comprehensive examples. If time allows, I will also describe some current, as well as, future applications of Persistent Homology.
Leonardo Mihalcea : What is Schubert calculus?
- Graduate/Faculty Seminar ( 150 Views )Do you ever wanted to know how many lines in 3−space intersect 4 given random lines ? (Answer: 2.) One way to prove this is to do explicit computations in the cohomology of the Grassmannian of lines in the projective space. But interestingly enough, one can also use Representation Theory, or symmetric functions (Schur polynomials), to answer this question. The aim of this talk is to present the basics of Schubert Calculus, as seen from the cohomological point of view. I will define Schubert varieties in Grassmannians, and discuss about how they intersect. The final goal is to show that 2 = 1+ 1 (and I may also use Knutsons puzzles for another proof of this).
Andrew Goetz : The Einstein-Klein-Gordon Equations, Wave Dark Matter, and the Tully-Fisher Relation
- Graduate/Faculty Seminar ( 123 Views )We describe a geometric theory of dark matter called "wave dark matter" whose underlying equations are the Einstein-Klein-Gordon system of PDEs. In spherical symmetry this system has simple static state solutions which we use to model dark galactic halos. We outline some scaling properties of these states including two new boundary conditions which might account for the existence of an astrophysical scaling relation called the baryonic Tully-Fisher relation.
Nadav Dym : Linear computation of angle preserving mappings
- Graduate/Faculty Seminar ( 205 Views )We will discuss recent work on computing angle preserving mappings (a.k.a. conformal mappings) using linear methods. We will begin with an intro/reminder on what these mappings are, and why would one to compute them. Then we will discuss the results themselves which show that when choosing a good target domain, computation of angle preserving mappings can be made linear in the sense that (i) They are a solution of a linear PDE (ii) They can be approximated by solving a finite dimensional linear system and (iii) the approximates are themselves homeomorphisms and "discrete conformal".
Erin Beckman : The frog model on trees with drift
- Graduate/Faculty Seminar ( 177 Views )In this talk, I will introduce a version of the frog model interacting particle system. The system initially consists of a single active particle at the root of a d-ary tree and an inactive particle at every other node on the tree. Active particles move according to a biased random walk and when an active particle encounters an inactive particle, the inactive particle becomes active and begins its own biased random walk. I will begin with an introduction and history of the model before moving on to talk about recent results, giving bounds on the drift such that the model is recurrent. I will go briefly into the techniques of proving such bounds, which involve a subprocess of the frog model that can be coupled across trees of different degrees. This is based on joint work with Frank, Jiang, Junge, and Tang.
Aaron Pollack : Modular forms on exceptional groups
- Graduate/Faculty Seminar ( 214 Views )Classically, a modular form for a reductive group G is an automorphic form that gives rise to a holomorphic function on the symmetric space G/K, when this symmetric space has complex structure. However, there are very interesting groups G, such as those of type G_2 and E_8, for which G/K does not have complex structure. Nevertheless, there is a theory of modular forms on these exceptional groups, whose study was initiated by Gross-Wallach and Gan-Gross-Savin. I will define these objects and describe what is known about them.
Heekyoung Hahn : Distribution of integer valued sequences associated to elliptic curves
- Graduate/Faculty Seminar ( 91 Views )Let $E$ be a non-CM elliptic curve defined over $\mathbb{Q}$. For each prime $p$ of good reduction, $E$ reduces to a curve $E_p$ over the finite field $\mathbb{F}_p$. In this talk, we are particularly interested in ssquare-free values of two sequences: $f_p(E) =p + 1 - a_p(E)$ and $f_p(E) = a_p(E)^2 - 4p$, where $a_p(E)=p+1-|E(\mathbb{F}_p)|$. More precisely for any fixed curve $E$, we first give an upper bound for the number of primes $p$ up to $X$ for which $f_p(E)$ is square-free. Second, we show that the average results on this prime counting function are compatible with the corresponding conjectures at the level of the constants, i.e., whether the average of the conjectured constants is equivalent to the constant obtained via the average conjecture. This is joint work with S. Akhtari, C. David and L. Thompson.
Phillip Andreae : Spectral geometry and topology; Euler characteristic and analytic torsion
- Graduate/Faculty Seminar ( 201 Views )What do eigenvalues have to do with geometry and topology? The first part of the talk will provide a few answers to that very broad question, including a discussion of the Euler characteristic from a spectral theory perspective. The second part of the talk will be a brief introduction to my research in analytic torsion, a topological invariant defined in terms of eigenvalues. In particular I'll explain some similarities and differences between analytic torsion and Euler characteristic.
Henry Adams : Evasion Paths in Mobile Sensor Networks
- Graduate/Faculty Seminar ( 140 Views )Suppose ball-shaped sensors wander in a bounded domain. A sensor doesn't know its location but does know when it overlaps a nearby sensor. We say that an evasion path exists in this sensor network if a moving intruder can avoid detection. Vin de Silva and Robert Ghrist give a necessary condition, depending only on the time-varying connectivity data of the sensors, for an evasion path to exist. Using zigzag persistent homology, we provide an equivalent condition that moreover can be computed in a streaming fashion. However, no method with time-varying connectivity data (i.e. Cech complexes) as input can give necessary and sufficient conditions for the existence of an evasion path. Indeed, we show that the existence of an evasion path depends on more than just the fibrewise homotopy type of the region covered by sensors. In the setting of planar sensors that also measure weak rotation information, we provide necessary and sufficient conditions for the existence of an evasion path, and we pose an open question concerning Cech and alpha complexes. Joint with Gunnar Carlsson.
Mark Stern : In Search of the Second Derivative
- Graduate/Faculty Seminar ( 121 Views )The goal of this talk is to show how the search for a proper definition of the second derivative of a function leads us to Yang Mills theory, the mathematics underlying modern particle physics and the source of powerful tools in topology and geometry. Along the way, we will introduce vector bundles, connections, and curvature. We assume a knowledge of multivariable calculus (and the first derivative).
Ioannis Sgouralis : A numerical method for solving the advection-diffusion equation in moving domains
- Graduate/Faculty Seminar ( 106 Views )The advection-diffusion equation commonly arises in the tracking of substances that are carried by and diffuse in a fluid. My interest lies in models of biological flows, where the equation is defined on domains representing the lumen of blood vessels, tubules, or ducts, which owing to its complex geometry and deformation, has a significant impact on the underlying substance that cannot be ignored. In the talk, I will go through the basic steps involved in the development of a numerical method for the solution of the advection-diffusion equation on such domains with an emphasis on domains with moving boundaries. To handle the difficulties introduced by the boundary, the proposed method will follow the finite volume formulation.
Austin Baird : MOVING FLUID IN TUBES: HOW DO WE USE MATHEMATICS TO HELP UNDERSTAND CIRCULATORY SYSTEMS IN SMALL ORGANISMS?
- Graduate/Faculty Seminar ( 98 Views )The human heart is a multi chambered organ using valves and compression to create pressure heads, which drive blood flow. Some organisms don't share this type of heart structure, namely, very small invertebrates and embryonic vertebrates. We begin by discussing scaling in fluid dynamics and why merely existing at such a small scales can cause difficulty in effectively transporting fluid. We then move to describing effective pumping mechanisms in valveless tubular hearts and how numerical modeling can help understand how these organisms are able to transport fluid, despite their limitations. Once we have a numerical framework to investigate these organisms we can then begin to add more biological structure in our models and begin to investigate optimal regimes of transport.
Shahed Sharif : Class field theory and cyclotomic fields
- Graduate/Faculty Seminar ( 191 Views )We'll undertake a gentle introduction to class field theory by investigating cyclotomic fields, including a proof of quadratic reciprocity. The results we'll discuss complement Les Saper's Grad Faculty seminar talk, though by no means is the latter a prerequisite. As a special treat, I will reveal a completely new, elementary proof of Fermat's Last Theorem.
Rachel Rujie Yin : Machine Learning in Art
- Graduate/Faculty Seminar ( 98 Views )I will talk about machine learning ideas and techniques used in my art-related projects, in particular the platypus project of cradle-removal in digitized X-ray images of paintings. Using machine learning techniques, we can extract and remove wood-grain patterns from the cradle in X-ray images which cannot be done using any existing Photoshop tools. I will also talk about interesting machine learning problems involved in other projects if time permits.
Lillian Pierce : Class numbers of quadratic number fields: a few highlights on the timeline from Gauss to today
- Graduate/Faculty Seminar ( 129 Views )Each number field (finite extension of the rational numbers) has an invariant associated to it called the class number (the cardinality of the class group of the field). Class numbers pop up throughout number theory, and over the last two hundred years people have been considering questions about the growth and divisibility properties of class numbers. Well focus on class numbers of quadratic extensions of the rationals, surveying some key results in the two centuries since the pioneering work of Gauss, and then turning to very recent joint work of the speaker with Roger Heath-Brown on averages and moments associated to class numbers of imaginary quadratic fields.