Miles Crosskey : Spectral bounds on empirical operators
- Graduate/Faculty Seminar ( 99 Views )Many machine learning algorithms are based upon estimating eigenvalues and eigenfunctions of certain integral operators. In practice, we have only finitely many randomly drawn points. How close are the eigenvalues and eigenfunctions of the finite dimensional matrix we construct in comparison to the infinite dimensional integral operator? In what way can we say these two are close if they do not even operate on the same spaces? To answer these questions, I will be showing some results from a paper "On Learning with Integral Operators" by Rosasco, Belkin, and De Vito.
Hugh Bray : Update on Dark Matter, Spiral Galaxies, and the Axioms of General Relativity
- Graduate/Faculty Seminar ( 115 Views )We will give an update on our last talk on a new connection between differential geometry and astrophysics which involves a model for dark matter and a possible explanation for barred spiral patterns in galaxies. We will also briefly discuss the Tully-Fisher relation, a mysterious experimental fact relating the visible mass of a galaxy to the speed of the stars in the galaxy, which to this point defies a convincing theoretical explanation.
Spencer Leslie : Intro to crystal graphs and their connections with number theory
- Graduate/Faculty Seminar ( 183 Views )I will review some basics of crystal bases for highest-weight representations for a semisimple Lie algebra. I will also point to some connections with number theory through Fourier coefficients of Eisenstein series, mostly in type A.
Shishi Luo : Getting a job after your PhD (and you thought graduating was hard...)
- Graduate/Faculty Seminar ( 118 Views )According to the NSF Survey of Earned Doctorates, of the Mathematics PhDs earned in 2011, 37% had an offer for postdoctoral research, 33% had definite employment in either academic, government, or industry positions, and 28% were seeking employment/postdoc. This means that although 100% of you are doing research right now, most likely only a third of you will continue to do research in a university setting. That means (a) getting a postdoc will be competitive and/or (b) you need to familiarize yourself with non-postdoc opportunities. To help you in this process, we've assembled a panel of local experts who have recently been through the job application process: Christine Berkesch (research faculty), Emily Braley (teaching faculty), Liz Munch (postdoc and non-academic), as well as senior faculty who can talk about qualities that are sought in both university (Mike Reed) and government research (Bill Pardon) settings. Come and learn what you can do now and in the future to make your job search more successful!
Rick Durrett : Random graphs as models of social networks.
- Graduate/Faculty Seminar ( 112 Views )We will describe the configuration model and discuss what happens when the people's opinions and the connections in the network coevolve. Despite the combined efforts of James Gleeson, Peter Mucha, Bill Shi, David Sivakoff, Josh Socolar, Chris Varghese and myself, we cannot prove any rigorous results so the talk should be accessible to almost anyone.
Jayce Getz : Trace formulae
- Graduate/Faculty Seminar ( 110 Views )All right, brain. You don't like me and I don't like you, but let's just do this and I can get back to killing you with trace formulae. -Homer Simpson (misquoted) We will discuss trace formulae starting with the Poisson summation formula and working towards the case of compact locally symmetric spaces. No background is assumed. Oh, and I'll bring beverages (both the big kid and little kid kind).
Chris O'Neill : An Introduction to Ehrhart Theory and Lattice Point Enumeration
- Graduate/Faculty Seminar ( 110 Views )A polytope is a subset of R^d which is the convex hull of a finite set of vertices. Given a polytope P, we can consider integer dilations of P, and ask how many integer points are contained in each dilation, as a function of the dilation factor. A theorem by Eugene Ehrhart tells us that, under the right conditions, this counting function is a polynomial, with some very interesting and unexpected properties. To demonstrate the usefulness of these results, we will give alternative proofs to some well known results from far outside the realm of geometry, including some basic facts about the chromatic polynomial of a graph. This talk will contain a little geometry, a little analysis, a little algebra, and a little combinatorics, and will be accessible to anyone who enjoys at least one of these topics.
Brian Fitzpatrick : A Gentle Introduction to Categories and Categorification
- Graduate/Faculty Seminar ( 114 Views )Category theory is a language used to describe mathematical structures. Its focus is on the relationship between mathematical objects rather than the objects themselves. Categorification is then an attempt to find category-theoretic analogues of classical set-theoretic results. This talk will present the basic notions needed to develop a theory of categorification and will show how many familiar mathematical results are, in fact, "shadows" of a hidden categorical structure.
Anita Layton : Unraveling Kidney Physiology, Pathophysiology and Therapeutics: A Modeling Approach
- Graduate/Faculty Seminar ( 114 Views )The kidney not only filters metabolic wastes and toxins from the body, but it also regulates the body's water balance, electrolyte balance, and acid-base balance, blood pressure, and blood flow. Despite intense research, aspects of kidney functions remain incompletely understood. I will discuss how our group use mathematical modeling techniques to address a host of previously unanswered questions in renal physiology and pathophysiology: Why is the mammalian kidney so susceptible to hypoxia, despite receiving ~25% of the cardiac output? What are the mechanisms underlying the development of acute kidney injury in a patient who has undergone cardiac surgery performed on cardiopulmonary bypass? What is the effect of inhibiting sodium-glucose transport, a novel treatment for reducing renal glucose update in diabetes, on renal NaCl transport and oxygen consumption?
Humberto Diaz : A Tour of Heights & Rational Points
- Graduate/Faculty Seminar ( 104 Views )A very important (and difficult) problem for number theorists is to determine all the rational solutions to a polynomial equation defined over the rational numbers. The oldest nontrivial case, which dates back to Pythagoras, is that of finding all the rational points on a unit circle. In this talk, we will consider the case of elliptic curves, where the rational points have the structure of an Abelian group G under a curiously defined addition law. We will develop some preliminaries and introduce the classical height machinery, a powerful tool which helps us understand the complexity of the points of G. We will look at some important results about the height and about G and see what still remains very elusive.
Greg Herschlag : Fluid Flow Through Permeable Channels
- Graduate/Faculty Seminar ( 110 Views )This talk will serve as a brief introduction to Newtonian fluids and focus on flow through channels. It will begin with a brief introduction to the Navier-Stokes and Stokes equations and continue with a derivation of fluid flow through an impermeable channel leading to classical Poiseuille flow. Next, I will present some results for flow through permeable channels. Finally, I will discuss a new result that has been found in collaboration with Jian-Guo Liu, Anita Layton and myself, in which we have determined an analytic solution for Stokes flow in an infinite permeable pipe.
Dave Rose : Why I love cats, and you should too
- Graduate/Faculty Seminar ( 107 Views )Category theory can be described as a general mathematical theory of structures and of systems of structures. Originally developed in the 40's by Saunders Mac Lane and Samuel Eilenberg in the context of algebraic topology, category theory has since grown to serve as both an organizational tool in many areas of mathematics and as a deep theory connecting these areas. The aims of this talk are 3-fold: first, to introduce the basic notions of category theory and to give a wide range of examples; second, to show how abstract results in category theory can influence the way we think about mathematics; finally, to show how a knowledge of some general results in category theory can save us time and effort in our day to day mathematical work. Since I will be starting with the basics, this talk should be accessible to a wide audience. Students who are considering working in algebra, geometry, or topology are particularly encouraged to attend, as are any students who have ever wondered why I love covering the chalkboards of 274F with crazy-looking diagrams or why the word `natural' is the fifth most used word in my vocabulary.
Andrew Goetz : General Relativity, Wave Dark Matter, and the Tully-Fisher relation
- Graduate/Faculty Seminar ( 196 Views )Abstract: In this talk I will give a quick overview of Einstein's theory of general relativity. I will then move on to discuss the mystery of dark matter: why astrophysicists think it's out there in the universe and what phenomena any successful theory of dark matter will have to explain. One such phenomenon is the Tully-Fisher relation, an intriguing correlation between the visible mass of galaxies and the rotational velocities of their stars. I will wrap up by describing a theory of wave dark matter and how it could possibly explain the Tully-Fisher relation.
Liz Munch : Applied Topology: Basic Ideas and a Mess of Applications
- Graduate/Faculty Seminar ( 108 Views )This talk will discuss some of the basic ideas pervasive in computational topology, especially persistent homology. Then, we will look at some currently active areas of application for these ideas, including large data sets, sensor networks, protein docking, plant root systems, and natural images. I assume no topology background, so this talk will be a good introduction to anyone interested in seeing what is going on in the field.
Kyle Thicke : Applied math techniques in electronic structure calculations
- Graduate/Faculty Seminar ( 110 Views )In this talk, I will use my recent project (a fast algorithm for calculating the energy of a many-body quantum system in the random phase approximation) as an outline to present two cool techniques in applied math and show their actual applications in the project. First, we'll see that the trapezoid rule you teach in Calculus, when applied to periodic functions, is far more impressive than you thought. We'll also get a taste of the surprisingly nice properties that come from combining matrix decompositions with randomized algorithms. Finally, as an added bonus, we'll see how Cauchy's integral formula can be used (in this project) to sum N^2 things in O(N) time.
Mark Stern : Gauge theory : the geometry and physics of the ambiguity of acceleration
- Graduate/Faculty Seminar ( 147 Views )I will discuss the rich mathematical structures which arise when one asks how to define acceleration in the absence of a preferred coordinate system. I will introduce the Yang-Mills equations, which specialize to give electromagnetism and much of the physics of the standard model. I'll discuss aspects of the geometry, topology, and analysis of the Yang-Mills equations and how too much symmetry can actually make an analysis problem more difficult.
Yuriy Mileyko : Hierarchical ordering of reticular networks
- Graduate/Faculty Seminar ( 100 Views )Biological physical networks, especially those involved in resource delivery and distribution, often exhibit a hierarchical structure. Quantifying this structure is crucial to obtaining a better understanding of the processes underlying the network formation, and such a quantification has long been obtained using the Horton-Strahler ordering scheme. The scheme assigns an integer order to each edge in the network based on the topology of branching such that the order increases from distal parts of the network (e.g., mountain streams or capillaries) to the ``root'' of the network (e.g., the river outlet or the aorta). However, Horton-Strahler ordering cannot be applied to networks with loops because they they create a contradiction in the edge ordering in terms of which edge precedes another in the hierarchy. In this talk I will present a generalization of the Horton-Strahler order to weighted planar reticular networks, where weights are assumed to correlate with the importance of network edges, e.g., weights estimated from edge widths may correlate with flow capacity. The new method assigns hierarchical levels not only to edges of the network, but also to its loops, and classifies the edges into reticular edges, which are responsible for loop formation, and tree edges. I will show that the sensitivity of the hierarchical levels to weight perturbations can be analyzed in a rigorous way. I will also discuss applications of this generalized Horton-Strahler ordering to the study of leaf venation and other biological networks.
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.
Erin Beckman : The frog model on trees with drift
- Graduate/Faculty Seminar ( 178 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.
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.
Carla Cederbaum : The Newtonian Limit of General Relativity
- Graduate/Faculty Seminar ( 115 Views )Einstein's General Relativity is a geometric theory of space, time, and gravitation. In some sense, it is the successor of Newton's famous theory of gravitation -- the theory Newton is said to have come up with when an apple fell onto his head. But although Einstein's theory is much better at predicting gravitational effects in our universe, Newton's theory is not at all outdated or even obsolete. In fact, many astrophysical measurements and simulations still heavily rely on Newtonian intuitions, calculations, and concepts. In the talk, I will explain how and to what extent this usage of Newtonian theory in astrophysics and related fields is motivated and mathematically justified. This will lead us to the notion of Newtonian limit. We will also see some examples for the behavior of relativistic quantities like mass and center of mass under this Newtonian limit.
Michael Jenista : Dynamical Systems and the Conley Index
- Graduate/Faculty Seminar ( 164 Views )An introductory lecture to the Conley Index theory. We consider the flow case and introduce the key object of study: an index pair of an isolated invariant set. Index pairs are robust under perturbations and their homotopy type is invariant, making them an ideal tool for problems with error terms or even data-generated systems. The relevant tools are algebraic topology and some knowledge of continuous flows.
Joshua Cruz : An Introduction to the Riemann-Hilbert Correspondence
- Graduate/Faculty Seminar ( 135 Views )Early in the history of complex analysis, it was realized that there are no continuous versions of the square root or the logarithm on the entire complex plane; instead, analysts invented multi-valued functions to deal with these strange behaviors. The "graphs" of these multi-valued functions can get very interesting, and can be interpreted topologically. In general, the space of solutions to a "nice" system of holomorphic ordinary differential equations on the non-zero complex numbers will not be made up of functions, but of multi-functions. Studying these spaces of solutions have led to several ideas in algebraic topology, especially monodromy, and the relationship between systems of ODE and possible monodromies is called the Riemann-Hilbert Correspondence.