2016-2017

Given Banach spaces X and Y , a single-valued (possibly non-smooth) mapping f: X → Y and a set-valued mapping F: X ⇉ Y, we study local convergence properties of the (in)exact Newton-type iterative schemes for solving the so-called generalized equation:

Find x ∈ X such that 0 ∈ f(x) + F(x). (1)

This model covers various problems such as equations, inequalities, variational inequalities, and, in particular, optimality conditions. The mapping f is approximated by a "generalized set-valued derivative" which in finite dimensions may be represented by Clarke's generalized Jacobian while in Banach spaces it may be identified with Ioffe's strict prederivative.

Based on various kinds of metric regularity, we intend to present Newton, Kantorovich, and Dennis-Moré theorems within the framework (1). As corollaries, we obtain results on convergence of inexact quasi-Newton type methods for semismooth equations. The presentation is based on the papers [1, 2, 3].

[1] S. Adly, R. Cibulka, and H. Van Ngai, Newton’s method for solving inclusions using set-valued approximations, SIAM J. Optim. 25 (2015) 159–184.

[2] R. Cibulka, A.L. Dontchev, and M.H. Geoffroy, Inexact Newtonmethods and Dennis–Moré theorems for nonsmooth generalized equations, SIAM J. Control Optim. 53 (2015) 1003–1019.

[3] R. Cibulka, A.L. Dontchev, J. Preininger, T. Roubal, and V.Veliov, Kantorovich-type theorems for generalized equations, to appear in J. Convex Anal., available at http://orcos.tuwien.ac.at/fileadmin/t/orcos/Research Reports/2015-16.pdf.

Starting at classical Ramsey numbers, we will take a journey through the study of monochromatic subgraphs of edge colored graphs. Many wonderful properties hold for 2-colorings but fail, sometimes badly, when more colors are introduced. One of the main topics of this talk will be considering those properties of 2-colorings that are preserved when more colors are allowed but certain colored substructures are forbidden.

Speaker's bio: Colton Magnant received his Ph.D. in 2008 from Emory University under the direction of Ron Gould. After a post-doc at Lehigh University, he joined Georgia Southern University. In addition to over fifty publications, he has founded two research journals, one as Editor-In-Chief called Theory and Applications of Graphs.

We will discuss the solvability of a class of Dirichlet Problem associated with nonlinear integro-differential operator. The main ingredient is the probabilistic construction of continuous supersolution via the identification of the continuity set of the exit time operators under Skorohod topology. We will demonstrate the main idea using some examples with fractional PDEs. Most of this talk would be accessible to a graduate student in math department.

Recent developments in the literature of partial identification have significant implications for the econometric estimation of important policy effects. In empirical economics, it is often of interest to estimate the effect of a binary policy treatment variable on a binary outcome variable where the treatment allocation is not random and both may be driven by common observable and unobservable factors. A common approach is to assume a parametric model, such as a bivariate probit, to achieve point identification. However such approach is often termed “identification by functional form” and there is often confusion regarding the identification without instrumental variables. Partial identification analysis of such problems allows for less restrictive assumptions for the underlying data generating process (DGP) in empirical applications, and the estimated bounds offer more robust measures for policy impacts. This talk presents Monte Carlo results on finite sample performance of average treatment effect (ATE) estimates from correct and mis-specified models, and the role played by the strength of the instrumental variables. We also graphically illustrate how both the correct- and mis-specified model estimates can all be sitting within the ATE bounds. Finally we apply the partial identification approach to a health economics application. We estimate the bounds for ATE of private health insurance status on dental service utilisation, using data from the Australian National Health Survey. Four sets of bounds from the literature under varying DGP assumptions and their 95% confidence regions are estimated. The resulted ATE confidence bounds are much wider than the confidence intervals using a conventional bivariate probit. We found that two of the bounds have reasonably narrow widths to be informative. We also estimate bounds for different sub-populations with varying widths. Performances of global parametric, local parametric, and smoothed and raw non-parametric estimators for bounds are studied using generated data.

The master knapsack polytope is the convex hull of non-negative integer solutions to a single knapsack equation with right-hand side n and variables with coefficients ranging from 1 to n. The knapsack facets are the inequalities necessary for the description of this polytope. Single row relaxations of many integer programming problems can be described as projections of the master knapsack polytope; thus a good understanding of the structural properties of the knapsack facets has wide ranging applications in discrete optimization. In this work, we start by describing a series of computational experiments that shed light on the properties of the strongest, or most important, knapsack facets. Based on the experiments we identify a particularly important category of these facets, the 1/k-facets, and derive characterizations of their structural properties, leading us to a concise and compact representation. We then show how our compact representation can be useful for efficiently enumerating and separating these knapsack facets. Our new techniques are particularly efficient when applied to the sparse knapsack subproblems, which correspond to the typically sparse integer programming relaxations motivating the study. This presentation is based on joint work with Sunil Chopra and Sangho Shim.

Since the 19th century the theory of Riemann surfaces has had a central place in mathematics putting together different branches of mathematics such as complex analysis, algebraic and hyperbolic geometry, group theory and combinatorial methods. From Riemann, Klein and Poincare among others, we know that a compact Riemann surface is a complex curve, and also the quotient of the hyperbolic plane by a Fuchsian group.

In this talk we study the connectivity of the moduli spaces of Riemann surfaces (i.e in spaces of Fuchsian groups). Spaces of Fuchsian groups are orbifolds where the singular locus is formed by Riemann surfaces with automorphisms: the branch loci. We will show that with a few exceptions the branch loci is disconnected and consists of several connected components. This talk is a survey of the different methods and topics playing together in the theory of Riemann surfaces and intended for a general audience.

Speaker's biography: Milagros Izquierdo was born in Burgo de Osma, Spain. She studied mathematics at University of Zaragoza, where she received a Master’s Degree in 1986 and a PhD degree in 1990 under the direction of M. T. Lozano and J. M. Montesinos. Her PhD thesis was on NEC groups. During 1991-93, Milagros was a post-doctorate fellow at the University of Southampton, where she worked with D. Singerman. She was an Associate Professor at the University of Västerås during 1993-2002 and since 2007 is a Professor of Mathematics at Linköping University in Sweden.

Professor’s Izquierdo’s interest lies on Riemann and Klein surfaces, i.e. complex and real curves and moduli spaces of curves, using combinatorial methods. She studies loci of surfaces with prescribed automorphisms, and uses Fuchsian and NEC groups and automorphisms of surfaces to obtained results on their moduli spaces. She is one of the leading experts in the area. Among her collaborators are E. Bujalance, A. Broughton, M. Conder, A. Costa, R. Hidalgo, H. Parlier, A. Rojas, T. Shaska, and D. Singerman. She is currently the president of the Swedish Mathematical Society.

Cybersecurity dynamics is promising in serving as the foundation of the emerging Science of Cybersecurity. Recent progresses on complex networks, which model viruses and malware codes via networked systems, offer a multidisciplinary platform where mathematics (broadly defined, including stochastic processes, dynamical systems, control theory, game theory, etc), statistics, statistical physics, complexity science, and network science can be applied. In this talk, I would like to briefly review some recent results towards the understanding and measuring of cybersecurity and the mathematics that has been used in this area.

Generating functions for the number of solutions of systems of polynomial equations over finite fields are called local zeta functions. In 19th century, Weil formulated a few conjectures about these functions. He was able to prove them in the case of a single polynomial of two variables - a curve. The part of this theorem that is relevant for our talk states that the local zeta function of a curve over a finite field with q elements is a rational function of one variable T, and squares of its complex roots lie on a circle of radius 1/q. If we substitute the -s power of q for T, the theorem states that the roots of the zeta function lie on the critical line Re(s)=1/2, which is a classical Riemann Hypothesis type of result.

In this talk, I will review some basics of classical coding theory and then introduce zeta functions of linear codes. An analogue of Riemann Hypothesis for linear codes and a discussion on the state of the conjecture will follow.

The talk will be self-contained, so most of it should be easily accessible for undergraduate students.

On April 20, 2010, a dramatic explosion on the Deepwater Horizon “Macondo" oil well drilling rig in the Gulf of Mexico caused the largest in U.S. history marine oil spill. The ecological disaster required a massive coordinated response from National Oceanic and Atmospheric Administration (NOAA), and the research community. In this talk I will explore a potential novel way to track oil, especially submerged oil in shallow waters, based on well-known clustering data mining methodology, satellite and surface ships observations, and the widely available relevant software to analyze aggregate observations of temperature, salinity and depth acquired by NOAA. This approach promises precise results produced fast and at low cost.

I will discuss elliptic curves from the classical number theoretic point of view of trying to solve Diophantine equations. The aim will be both to explain how we think about these creatures and to give an overview of what we can (and sometimes can't) prove about them, and to illustrate it with explicit examples. I will not try to describe the huge modern technical machine that has been developed to study elliptic curves, so most of the results will come as black boxes. I will end by outlining my joint work with Tim Dokchitser on ranks of elliptic curves, and its applications to the Birch and Swinnerton-Dyer conjecture and the distribution of ranks of Bhargava et al.

Short Biography: Vladimir Dokchitser is a Reader in Number Theory and a Royal Society University Research Fellow at King’s College London, UK. He was an undergraduate and a graduate student at Trinity College, Cambridge. He then held a 1-year postdoctoral position at the Max Planck Institute in Bonn before returning to Cambridge first as a Junior Research Fellow at Gonville & Caius College, and then as the Meggitt Fellow at Emmanuel College. He moved to Warwick in 2013, and finally came to King’s in 2016. His research interests include the arithmetic of elliptic curves, the Birch-Swinnerton-Dyer conjecture and other L-value conjectures, Galois representations and hyperelliptic curves.

The rational solutions to a Diophantine equation defining an elliptic curve form a finitely generated abelian group. The main arithmetic invariant of the elliptic curve is the rank of this group, that is the number of generators of infinite order. Controlling the rank is very difficult and finding the rank in general remains an unsolved problem. As observed by Selmer in mid 20th century, the parity of the rank is much more well behaved, which is summarized by the “parity conjecture”. We will discuss this conjecture, its origin, consequences and current known results, and illustrate them with examples. We will end by discussing new results on parity of ranks of abelian surfaces.

Short Biography: Celine Maistret is a Senior Research Associate at the University of Bristol, UK. She is a number theorist, specializing in arithmetic of hyperelliptic curves and ranks of abelian surfaces. She completed her PhD at the University of Warwick under the supervisions of John Cremona and Vladimir Dokchitser.