Dr. Andrew Barnes : Risk Measurement and Capital Allocation for large loan portfolios
- Presentations ( 199 Views )Calculation of portfolio loss distributions is an important part of credit risk management in all large banking institutions. Mathematically, this calculation is tantamount to efficiently computing the probability distribution of the sum of a very large number of correlated random variables. Typical Monte Carlo aggregation models apply brute force computation to this problem and suffer from two main drawbacks: lack of speed and lack of transparency for further credit risk analysis. I will describe an attempt to ameliorate these drawbacks via an asymptotic probabilistic method based on the Central Limit Theorem. I will next describe capital allocation, a process of attributing risk to individual transactions or subportfolios of a given portfolio. In so doing, I will state axioms for coherent risk measures. These axioms place the notion of risk measurement and diversification on a firm mathematical foundation. I will then describe axioms for capital allocation via coherent risk measures, and illustrate the ideas with efficient computational formulae for allocating capital based on a couple of commonly used risk measures. In the course of this talk, which will be geared towards graduate students, I will attempt to give a flavor of industrial research and role of applied mathematics in industry.
Ellen Eischen : L-functions, congruences, and applications
- Presentations ( 168 Views )L-functions, certain meromorphic functions that include the Riemann zeta-function, encode important number-theoretic information. The first part of this talk will focus on some striking properties of special values of the Riemann zeta-function and certain other L-functions (namely, congruences modulo powers of a prime number). In the second part of the talk, I will introduce tools that are useful for studying these congruences. These tools have applications not only to number theory, but also to homotopy theory. This will be a colloquium style talk, intended for a broad audience.
Florian Naef : A real description of brackets and cobrackets in string topology
- Presentations ( 268 Views )Let M be a manifold with non-vanishing vectorfield. The homology of the space of loops in M carries a natural Lie bialgebra structure described by Sullivan as string topology operations. If M is a surface, these operations where originally defined by Goldman and Turaev. We study formal descriptions of these Lie bialgebras. More precisely, for surfaces these Lie bialgebras are formal in the sense that they are isomorphic (after completion) to their algebraic analogues (Schedler's necklace Lie bialgebras) built from the homology of the surface. For higher dimensional manifolds we give a similar description that turns out to depend on the Chern-Simons partition function.
This talk is based on joint work with A. Alekseev, N. Kawazumi, Y. Kuno and T. Willwacher.
Ben Weinkove : Symplectic forms, Kahler metrics and the Calabi-Yau equation
- Presentations ( 158 Views )Yau's theorem on Kahler manifolds states that there exists a unique Kahler metric in every Kahler class with prescribed volume form. This has many applications in complex geometry. I will discuss symplectic manifolds. In a different direction, I will talk about the problem of existence of constant scalar curvature Kahler metrics, which can also be considered a generalization of Yau's theorem.
Ravi Vakil : Murphys Law in algebraic geometry: Badly-behaved moduli spaces
- Presentations ( 167 Views )We consider the question: ``How bad can the deformation space of an object be?'' (Alternatively: ``What singularities can appear on a moduli space?'') The answer seems to be: ``Unless there is some a priori reason otherwise, the deformation space can be arbitrarily bad.'' We show this for a number of important moduli spaces. More precisely, up to smooth parameters, every singularity that can be described by equations with integer coefficients appears on moduli spaces parameterizing: smooth projective surfaces (or higher-dimensional manifolds); smooth curves in projective space (the space of stable maps, or the Hilbert scheme); plane curves with nodes and cusps; stable sheaves; isolated threefold singularities; and more. The objects themselves are not pathological, and are in fact as nice as can be. This justifies Mumford's philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. I will begin by telling you what ``moduli spaces'' and ``deformation spaces'' are. The complex-minded listener can work in the holomorphic category; the arithmetic listener can think in mixed or positive characteristic. This talk is intended to be (mostly) comprehensible to a broad audience.
Kirsten Wickelgren : An arithmetic count of the lines on a cubic surface
- Presentations ( 280 Views )A celebrated 19th century result of Cayley and Salmon is that a smooth cubic surface over the complex numbers contains exactly 27 lines. By contrast, over the real numbers, the number of real lines depends on the surface. A classification was obtained by Segre, but it is a recent observation of Benedetti-Silhol, Finashin-Kharlamov, Horev-Solomon and Okonek-Teleman that a certain signed count of lines is always 3. We extend this count to an arbitrary field k using an Euler number in A1-homotopy theory. The resulting count is valued in the Grothendieck-Witt group of non-degenerate symmetric bilinear forms. (No knowledge of A1-homotopy theory will be assumed in the talk.) This is joint work with Jesse Kass.
Margaret Beck : Using global invariant manifolds to understand metastability in Burgers equation with small viscosity.
- Presentations ( 109 Views )Finding globally stable states can provide useful information about the behavior of solutions to PDEs: for any initial condition, the solution will eventually approach such a state. However, in some cases, the solution can exhibit long transients in its approach to the state. If the transient is long enough, it may be this behavior, rather than the limiting behavior, that is observed in numerical simulations or experiments. This is referred to as "metastability" and has been found, for example, in the 2D Navier-Stokes equations with small viscosity. A similar phenomenon has been seen in Burgers equation, which can be explained using global invariant manifolds. More precisely, it is shown that in terms of similarity, or scaling, variables there exists a one-dimensional global center manifold of stationary solutions. Through each of these fixed points there exists a one-dimensional, global, attractive, invariant manifold. Metastability corresponds to a fast transient in which solutions approach this 'metastable' manifold, followed by a slow decay along this manifold, and, finally, convergence to the globally stable state.
Matthias Heymann : Computing maximum likelihood paths of rare transition events, and applications to synthetic biology
- Presentations ( 163 Views )Dynamical systems with small noise (e.g. SDEs) allow for rare transitions from one stable state into another that would not be possible without the presence of noise. Large deviation theory provides the means to analyze both the frequency of these transitions and the maximum likelihood transition path. The key object for the determination of both is the quasipotential, V(x,y) = inf S_T(phi), where S_T(phi) is the action functional associated to the system, and where the infimum is taken over all T>0 and all paths phi:[0,T]->R^n leading from x to y. The numerical evaluation of V(x,y) however is made difficult by the fact that in most cases of interest no minimizer exists.
In my work I prove an alternative geometric formulation of V(x,y) that resolves this issue by introducing an action on the space of curves ( i.e. this action is independent of the parametrization of phi). In this formulation, a minimizer exists, and we use it to build a flexible algorithm (the geometric minimum action method, gMAM) for finding the maximum likelihood transition curve.
In one application I show how the gMAM can be useful in the newly emerging field of synthetic biology: We propose a method to identify the sources of instabilities in (genetic) networks.
This work was done in collaboration with my adviser Eric Vanden-Eijnden and is the core of my PhD thesis at NYU.
Robert Ghrist : Sheaves and Sensors
- Presentations ( 207 Views )This work is motivated by a fundamental problem in sensor networks -- the need to aggregate redundant sensor data across a network. We focus on a simple problem of enumerating targets with a network of sensors that can detect nearby targets, but cannot identify or localize them. We show a clear, clean relationship between this problem and the topology of constructable sheaves. In particular, an integration theory from sheaf theory that uses Euler characteristic as a measure provides a computable, robust, and powerful tool for data aggregation.
James Nolen : Reaction-Diffusion Fronts in Heterogeneous Media
- Presentations ( 145 Views )Reaction-diffusion equations are used in mathematical models of many physical and biological phenomena involving front propagation and pulse propagation. How do variations in the environment effect these phenomena? In this seminar, I will describe recent progress in understanding how fronts propagate in heterogeneous media. In particular, I will describe properties of generalized traveling waves for one-dimensional reaction-diffusion equations with variable excitation. I also will discuss multi-dimensional fronts in stationary random media, a model relevant to premixed-turbulent combustion. Along the way, I plan to describe interesting topics for future research.
Joseph Rabinoff : From Diophantine equations to p-adic analytic geometry
- Presentations ( 235 Views )A Diophantine equation is a polynomial equation in several variables, generally with integer coefficients, like x3 + y3 = z3. Provably finding all integer solutions of a Diophantine equation is a storied mathematical problem that is easy to state and notoriously difficult to solve. The method of Chabauty--Coleman is one particularly successful technique for ruling out extraneous solutions of a certain class of Diophantine equations. The method is p-adic in nature, and involves producing p-adic analytic functions that vanish on all integer-valued solutions. I will discuss work with Katz and Zureick-Brown on finding uniform bounds on the number of rational points on a curve of fixed genus, defined over a number field, subject to a (conjecturally weak) restriction on its Jacobian. The same technique also makes progress on the uniform Manin-Mumford conjecture on the size of torsion packets on curves of fixed genus.
Laura DeMarco : Complex dynamics and potential theory
- Presentations ( 154 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.
Patrick Brosnan : Essential dimension and algebraic stacks
- Presentations ( 132 Views )Essential dimension is an invariant introduced by Buhler and Reichstein to measure how many parameters are needed to define an algebraic object such as a field extension or an algebraic curve over a field. I will describe joint work with Vistoli and Reichstein which studies essential dimension in the case where the algebraic objects are represented by a stack. I will also give examples of applications in the theory of quadratic forms.
Jason Mireles-James : Adaptive Set-Oriented Algorithms for Conservative Systems
- Presentations ( 131 Views )We describe an automatic chaos verification scheme based on set oriented numerical methods, which is especially well suited to the study of area and volume preserving diffeomorphisms. The novel feature of the scheme is an iterative algorithm for approximating connecting orbits between collections of hyperbolic fixed and periodic points with greater and greater accuracy. The algorithm is geometric rather than graph theoretic in nature and, unlike existing methods, does not require the computation of chain recurrent sets. We give several example computations in dimension two and three.