I will explain the basic notions and methods of an algebro-geometric theory over semirings and over somewhat exotic objects called `hyperrings'. Both developments reveal previously unseen links to other theories which include tropical geometry. In particular, I will focus on the following: 1. Cech cohomology on semiring schemes, 2. Construction of hyperring schemes, 3. Hyperstructure of the underlying space of an affine algebraic group scheme.

In this talk, we will look at how congruences between Hecke eigensystems of modular forms affect the Iwasawa invariants of their anticyclotomic p-adic L-functions. It can be regarded as an application of the ideas of Greenberg-Vatsal and Emerton-Pollack-Weston on the variation of Iwasawa invariants to the anticyclotomic setting. As an application, we establish new examples of the anticyclotomic main conjecture for modular forms. At the end, we discuss a higher weight generalization of the result (joint work with F. Castella and M. Longo) and give an explicit example.

Bounds on the complex dielectric constant of a two-component material at fixed 

frequency were derived about 35 years ago independently by Milton and Bergman 

using the analytic representation formula for the effective dielectric constant as a 

function of the component dielectric constant. These bounds become tighter the more 

information is incorporated about the composite geometry, such as the volume 

fractions of the constituents and whether it is isotropic or not. These bounds were 

subsequently generalized to elasticity in works of Berryman, Gibiansky, Lakes and 

Milton, using the variational principles of Cherkaev and Gibiansky. All these bounds 

are applicable when the applied fields are time harmonic. But what happens when the 

applied fields are not time harmonic? One would like to bound for each moment in time,

 the transient response of the induced average displacement field given the applied time

 varying electric field. We obtain such bounds using the analytic method, and we find 

that they can be very tight, tighter the more information is known about the composite. 

The bounds are also applicable to the mathematically equivalent problem of antiplane 

elasticity, where one is interested in bounding the stress relaxation and creep of 

composites of two viscoelastic phases.

Here we show how the analytic properties reviewed in Lecture 1 can be used to derive bounds on the effective moduli of composites, in particular the "Bergman-Milton" bounds that were derived independently by David Bergman and myself way back in 1979. (Chapter 27 of book "Theory of Composites” by Graeme Milton). 

Lecture 1: A Landscape of Graph Polynomials.

We introduce the most prominent graph polynomials (characteristic, Laplacian, chromatic, matching, Tutte) and discuss how to compare them.

Lecture 2: Why is the Chromatic Polynomial a Polynomial?

We give an alternative proof for the fact that the chromatic polynomial is indeed a polynomial. From this we introduce generalized chromatic polynomials, and show that this actually represents the most general case; Every (reasonably defined) graph polynomial can be represented as a generalized chromatic polynomial.

Lecture 3: Hankel matrices and Graph Polynomials.

We introduce Hankel matrices of graph paramaters, which generalize Lovasz’ connection matrices. We show that many (but not all) graph polynomials have Hankel matrices of finite rank. We show how to use the Finite Rank Property to show definability/non-definability of graph parameters/polynomials in Monadic Second Order Logic.

Network and graph theory has proven useful for modelling, analysis, and solving of problems arising in mathematics, theoretical computer science, natural sciences, social sciences, and even in finance. The connectivity, interdependence, and complexity in financial markets and systems are increasing. The analysis of networks and graphs will help us understand issues and problems arising in finance and provide appropriate models. This talk is a gentle introduction to network and graph theory.

Photonic devices are emerging for an increasing variety of technological applications, ranging from sensors to solar cells. I will show how large-scale computational optimization and rigorous analytical frameworks enable rapid search through large design spaces, and spur discovery of fundamental limits to interactions between light and matter. Our simple analysis of solar-cell emissivity showed that solar cells should be good LEDs, a counterintuitive idea leveraged by a start-up company that recently set a record for single-junction photovoltaic efficiency. I will then pivot to reviewing large-scale adjoint-based optimization methods, which we used to design new solar-cell textures and super-scattering nanoparticles. Finally, our computational nanoparticle designs led to new analytical limits to the response of metals, which have applications ranging from smoke-grenade obscurance to the near-field radiative transfer of heat.

In this lecture we will review and discuss several aspects of linear (time) translation-invariant (LTI) systems. We will begin by focusing our attention on causal and passive LTI systems, their fundamental properties, and the relation- ship between causality, passivity, and energy dissipation. After we have given a discussion of such systems in the time domain, we will discuss their properties in the frequency domain (dispersion). This leads naturally to positive (real) functions and Herglotz functions. We will then review their analytic properties and how they are related to causality, passivity, dissipation, and the Kramers-Kronig relations (i.e., dispersion relations). Finally, we will introduce the notion of a transparency window for a passive LTI system and describe its consequences. Simple examples from mathematics (e.g., the resolvent of a self-adjoint opera- tor), physics, and engineering (e.g., a spring-mass-damper system or an RLC circuit with one degree-of-freedom; constitutive relations in electromagnetism) will be used to illustrate how ubiquitous such passive LTI systems are in many areas of science.

Lecture 1: A Landscape of Graph Polynomials.

We introduce the most prominent graph polynomials (characteristic, Laplacian, chromatic, matching, Tutte) and discuss how to compare them.

Lecture 2: Why is the Chromatic Polynomial a Polynomial?

We give an alternative proof for the fact that the chromatic polynomial is indeed a polynomial. From this we introduce generalized chromatic polynomials, and show that this actually represents the most general case; Every (reasonably defined) graph polynomial can be represented as a generalized chromatic polynomial.

Lecture 3: Hankel matrices and Graph Polynomials.

We introduce Hankel matrices of graph paramaters, which generalize Lovasz’ connection matrices. We show that many (but not all) graph polynomials have Hankel matrices of finite rank. We show how to use the Finite Rank Property to show definability/non-definability of graph parameters/polynomials in Monadic Second Order Logic.

Here we discuss the analytic properties of the effective (conductivity, elastic, piezoelectric, etc.) tensor of composite materials as a function of the moduli of the component moduli, and present the associated representation formulas for these functions. (Chapter 18 of book "Theory of Composites” by Graeme Milton).

Lecture 1: A Landscape of Graph Polynomials.

We introduce the most prominent graph polynomials (characteristic, Laplacian, chromatic, matching, Tutte) and discuss how to compare them.

Lecture 2: Why is the Chromatic Polynomial a Polynomial?

We give an alternative proof for the fact that the chromatic polynomial is indeed a polynomial. From this we introduce generalized chromatic polynomials, and show that this actually represents the most general case; Every (reasonably defined) graph polynomial can be represented as a generalized chromatic polynomial.

Lecture 3: Hankel matrices and Graph Polynomials.

We introduce Hankel matrices of graph paramaters, which generalize Lovasz’ connection matrices. We show that many (but not all) graph polynomials have Hankel matrices of finite rank. We show how to use the Finite Rank Property to show definability/non-definability of graph parameters/polynomials in Monadic Second Order Logic.

Let A be a commutative ring. A subset X of A^n is a polynomial

family with d parameters if it is the range of a polynomial map from A^d to

A^n. It is an old question of Skolem (1938) whether SL_2(A) is a polynomial

family, where A is the ring of integers. Only recently Vaserstein (2010)

answered Skolem's question in the affirmative. In this talk, I will discuss

my recent result proving that SL_2(A) is a polynomial family, where A is

the ring of polynomials over a finite field of q elements. This is a

function field analogue of Vaserstein's theorem.

The Siegel-Ramachandra invariants, as special values of Siegel functions of one variable, generate ray class fields over imaginary quadratic fields. Generalizing these invariants we shall introduce ray class invariants of certain CM-fields obtained from classical theta constants of multi-variables. And we will determine the action of the Galois group on these invariants in a concrete way by making use of Shimura's reciprocity law.

This is a joint work with Koo and Shin.

In this talk, we develop an equivalent condition for a primitive Fricke family of level $N$ to be totally primitive when $N$ is different from $4$. Furthermore, we present generators of the function field of the modular curve of level $N$ in terms of Fricke and Siegel functions. By using the functions belonging to Fricke families, we shall construct generators of the ray class fields over imaginary quadratic fields as an application of class field theory.

This is a joint work with Koo and Shin.

The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock theta function ω(q) (resp. ν(-q)). Similar results for partitions with the corresponding restriction on each even part are also obtained, one of which involves the third order mock theta function φ(q). Congruences for the smallest parts functions associated to such partitions are obtained. Two analogues of the partition-theoretic interpretation of Euler’s pentagonal theorem are also obtained. This is joint work with George Andrews and Atul Dixit.

We will highlight two examples of the interplay of combinatorics and orthogonal polynomials by considering two recent one-parameter extensions of Hermite polynomials: a curious q-analog in connection with q-Weyl algebra and the 2D-Hermite polynomials. As application we derive a generalization of Touchard-Riordan formula for crossings of chords joining pairs of 2n points on a circle and a new Kibble-Slepian type formula for the 2D-Hermite polynomials, which extends the Poisson kernel for these polynomials.

The Riemann zeta-function, which encodes information about the integers and the prime numbers, has been studied extensively. Its values at 2,4,... are well-known, but much less is known about its values at 3,5,... . This difference can be explained to an extent by the different behaviour of certain groups (algebraic K-groups) of the rationals.
In this talk, we discuss some basic examples of such K-groups, and some links between them and arithmetic.

Approximate random k-colouring of a graph G=(V,E) is a very well

 Combining information from different source is an important practical problem. Using hierarchical area level models, we establish a frequentist framework for combining information from different source to get improved prediction for small or large area estimation. The best prediction is obtained by the conditional expectation of the observable latent variable given all available observation. The model parameters are estimated by two-level EM algorithm. Estimation of the mean squared prediction error is discussed.

  Sponsored by National Agricultural Statistical Agency (NASS) of US department of Agriculture, the proposed method was applied to the crop acreage prediction problem combining information from three sources: The first source is the June Area Survey (JAS), which is obtained by the probability sampling. The second source is from the Farm Service Agency (FSA) data, which is obtained from a voluntary participation of certain programs. The third source is from the classification of the satellite image data, called Cropland Data Layer (CDL).