Mathematics

Semi-Coherent Multi-Population Mortality Modeling: The Impact on Longevity Risk Securitization

Wai-Sum Chan (The Chinese University of Hong Kong)

Abstract: Multi-population mortality models play an important role in longevity risk transfers involving more than one population. Most of the existing multi-population mortality models are built upon the hypothesis of coherence, which stipulates the differential between the mortality rates of two related populations at any given age to revert to its corresponding long-term equilibrium, thereby preventing diverging long-term forecasts that do not seem to be biologically reasonable. However, the coherence assumption may be perceived by market participants as too strong and is in fact not always supported by empirical observations. In this paper, we introduce a new concept called 'semi-coherence', which is less stringent in the sense that it permits the mortality trajectories of two related populations to diverge, as long as the divergence does not exceed a specific tolerance corridor, beyond which mean-reversion will come into effect. We further propose to produce semi-coherent mortality forecasts by using a vector threshold autoregression. The proposed modeling approach is illustrated with mortality data from US and English and Welsh male populations, and is applied to several pricing and hedging scenarios.

Investment, Consumption, and Pricing: A Forward-Thinking Approach

Wing Fung Chong (University of Illinois at Urbana-Champaign)

Abstract: Under the expected utility paradigm, the optimal investment and consumption problem was solved by Merton in 1969. His pioneering work adopting the expected utility framework inspired numerous optimality problems in various topics of actuarial and financial mathematics. The theory of indifference pricing is one of the representative instances. However, these elegant theories were built upon, first, fixing an investment and consumption horizon and, second, a priori assuming future utilities. In this talk, a recently developed notion called forward investment performance process by Musiela and Zariphopoulou will be introduced. With the element of consumption, SPDE and BSDE representations of forward investment and consumption performance processes will be presented. Properties and the large maturity behavior of forward entropic risk measures will be explored via BSDE representations.

Garden of Eden Theorems for dynamical systems: old and new

Tullio Ceccherini-Silberstein (Universita' del Sannio (Benevento))

Abstract: In this talk, completely self-contained, I would like to survey the classical Garden of Eden Theorem for cellular automata with finite alphabet over the group Z^d, originally establisehd by E.F. Moore and J. Myhill in the early 1960', and its more recent generalizations (with other alphabet sets and over larger classes of groups) as well as some extensions to other dynamical systems of algebraic origin, obtained in collaboration with Michel Coornaert.

Conformal Tilings of the Plane: Theory and Practice

Kenneth Stephenson (University of Tennessee)

Abstract: The Penrose Tiling of the plane is the most famous traditional "tiling" of the type which motivated this work. With just two tile types, a "kite" shape and a "dart" shape, and a corresponding subdivision rule, it provides fascinating tilings, ones in which the eye finds endless repetitions, but which in fact have no periodicities. Here we will discuss a new, even richer class of tilings, one in which conformal shapes replace the traditional euclidean shapes. The talk will intermingle the discrete world of circle packing with the continuous world of analytic maps and will convey --- I hope --- the pleasing blend of theory, computation, experimentation, and visualization that I have so enjoyed in this new topic. It will be a largely visual tour, so no background in tiling, circle packing, or analytic function theory is needed. (Much of this is joint work with Phil Bowers of Florida State University.)

Relaxations of Hadwiger's conjecture

Sergey Norin (McGill University)

Abstract: Hadwiger's conjecture from 1943 states that every simple graph with no $K_t$ minor can be properly colored using t-1 colors. This is a far-reaching strengthening of the four-color theorem and appears to be currently out of reach in its full generality. In the last three years, however, several relaxations have been proven. In these relaxations one considers colorings such that every color class induces a subgraph with bounded maximum degree or with bounded component size. We will survey recent results on such improper colorings of minor-closed classes of graphs. Based on joint work with Zdenek Dvorak and with Alex Scott, Paul Seymour and David Wood.

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Andrea Bertozzi (UCLA)

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Andrea Bertozzi (UCLA)

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Andrea Bertozzi (UCLA)

Ideals in L(L_p)

William B. Johnson (Texas A&M University)

Abstract: I'll discuss the Banach algebra structure of the spaces of bounded linear operators on l_p and L_p := L_p(0, 1). The main new results are
1. The only non trivial closed ideal in L(L_p), 1 ≤  p < ∞ , that has a left approximate identity is the ideal of compact operators (joint with N. C. Phillips and G. Schechtman).
2. There are innitely many; in fact, a continuum; of closed ideals in L(L_1) (joint with G. Pisier and G. Schechtman).
The second result answers a question from the 1978 book of A. Pietsch, "Operator ideals".

Puzzles, flag manifolds, and Gromov-Witten invariants

Anders Skovsted Buch (Rutgers University)

Abstract: The development of algebraic geometry has been motivated by enumerative geometric questions where one asks for the number of geometric figures of some type that satisfy a list of conditions. For example, the Gromov-Witten invariants of a flag manifold counts the number of curves that meet a list of Schubert varieties in general position. Many enumerative problems can be reduced to understanding the Schubert structure constants of flag manifolds. Standard conjectures about #P-functions indicate that these structure constants are best expressed as the number of objects in some combinatorially defined set. The classical Littlewood-Richardson rule for the structure constants of Grassmannians is an example of this, but it is not known if all Schubert structure constants can be (reasonably) expressed in this way. I will speak about recent results that express Schubert structure constants as the number of puzzles that can be created from a given list of puzzle pieces, as well as relations to the Gromov-Witten invariants of Grassmannians.

Random Matrices, Heat Flow, and Lie Groups

Todd Kemp (UCSD)

Abstract: Random matrix theory studies the behavior of the eigenvalues and eigenvectors of random matrices as the dimension grows. In the age of data science, it has become one of the hottest fields in probability theory and many parts of applied science, from material deposition to wireless communication. Initiated by Wigner in the 1950s (with some key results going back further to Wishart and other statisticians in the 1920s), there is now a rich and well-developed theory of the universal behavior of random spectral statistics in models that are natural generalizations of the Gaussian case. In this talk, I will discuss a generalization of these kinds of results in a new direction. A Gaussian random matrix can be thought of as an instance of Brownian motion on a Lie algebra; this opens the door to studying the eigenvalues of Brownian motion on Lie groups. I will present recent progress understanding the asymptotic spectral distribution of Brownian motion on unitary groups and general linear groups. The tools needed include probability theory, functional analysis, combinatorics, and representation theory. No technical background is required; only an interest in trying to understand some cool and mysterious pictures.

Associahedra, cluster algebras, and scattering amplitudes

Hugh Thomas (University of Quebec at Montreal)

Abstract: The past several years have seen a flurry of activity in the physics of scattering amplitudes, in part motivated by a new geometric approach to the problem, which finds the solution encoded in a geometrical object, most famously in the amplituhedron of Arkani-Hamed and Trnka for N=4 super Yang-Mills. I will discuss a version of this approach for a simpler quantum field theory (biadjoint scalar $\varphi^3$ theory), where the geometrical object encoding the answer at tree level has recently been shown by Arkani-Hamed et al. to be an associahedron, a polytope originally defined by Jim Stasheff in the context of homotopy theory, and now well-known thanks to its connection to type $A_n$ cluster algebras. In recent work with my students Bazier-Matte, Chapelier, Douville, Mousavand, and former student Yıldırım, we showed that the construction of the associahedron developed by Arkani-Hamed et al. for their purposes is also applicable to other finite type cluster algebras, yielding simple constructions both of generalized associahedra, and, unexpectedly, of the Newton polytopes of the cluster variables. Time permitting, I will discuss the possibility (which we are investigating with Arkani-Hamed) that this construction in other types also has a physical interpretation.

Model theory and ultraproducts

Maryanthe Malliaris (University of Chicago)

Abstract: The ultraproduct construction gives a way of averaging an infinite sequence of mathematical structures, such as fields, graphs, or linear orders. The talk will be about the strength of such a construction.

Computer Driven Questions, Theorems and Pre-theorems in Low Dimensional Topology

Moira Chas (Stony Brook University)

Abstract: Consider an orientable surface S with negative Euler characteristic, a minimal set of generators of the fundamental group of S, and a hyperbolic metric on S. Then each unbased homotopy class C of closed oriented curves on S determines three numbers: the word length (that is, the minimal number of letters needed to express C as a cyclic word in the generators and their inverses), the minimal geometric self-¬intersection number, and finally the geometric length. Also, the set of free homotopy classes of closed directed curves on S (as a set) is the vector space basis of a Lie algebra discovered by Goldman. This Lie algebra is closely related to the intersection structure of curves on S. These three numbers, as well as the Goldman Lie bracket of two classes, can be explicitly computed (or approximated) using a computer. These computations led us to counterexamples to existing conjectures, to formulate new conjectures and (sometimes) to subsequent theorems.

Resonance rigidity for Schrödinger operators

Tanya Christiansen (University of Missouri)

Abstract: From a mathematical point of view, resonances may provide a replacement for discrete spectral data for a class of operators with continuous spectrum. Physically, resonances may correspond to decaying waves. This talk will introduce the notion of resonances for Schrödinger operators. We discuss results, both by the speaker and others, related to the rigidity of the set of resonances of a Schrödinger operator on ${\mathbb R}^d$ with potential $V\in L^\infty_c({\mathbb R}^d)$. For example, within this class of operators, is the Schrödinger operator with $0$ potential determined by its resonances? What can we say about other sets of isoresonant potentials?

Rigidity of uniform Roe algebras and coronas

Ilijas Farah (York University)

Abstract: Coarse geometry is the study of large-scale properties of metric spaces. Two metric spaces are coarsely equivalent if their "large-scale geometries" agree. The uniform Roe algebra $C^*_u(X)$ is a norm-closed algebra of bounded linear operators on the Hilbert space $\ell_2(X)$. It is the algebra of all bounded linear operators on $\ell_2(X)$ that can be uniformly approximated by operators of "finite propagation". The uniform Roe algebra is a coarse invariant of the space $X$. It includes $\ell_\infty(X)$ (as the algebra of all operators of zero propagation) and the algebra of compact operators. The uniform Roe corona $Q^*_u(X)$ is obtained by modding out the compact operators from $C^*_u(X)$. After introducing the basics of coarse spaces and uniform Roe algebras, we will study implications (or lack thereof) between the following three assertions and their variants:

What we still don't know about addition and multiplication

Carl Pomerance (Dartmouth College)

Abstract: How could there be something we don’t know about arithmetic? It would seem that subject was sewn up in third grade. But here’s a problem: What is the most efficient method for multiplication? No one knows. And here is another: How many different numbers appear in a large multiplication table? There are many more such problems, largely unsolved, and for which we could use some help!

A reception will be held from 5-6 pm in 239 Altgeld Hall following this first lecture.

Random number theory

Carl Pomerance (Dartmouth College)

Abstract: No, this is not a talk about random numbers! Rather, we discuss the role of randomness in number theory, from Euler to the present. We'll visit the probabilistic background of Fermat's Last Theorem, the ABC conjecture, the Prime Number Theorem, the Riemann Hypothesis, Goldbach's Conjecture, and the Twin-Prime Conjecture, among other famous problems and results. Randomness has had a profound influence in computational number theory. In combinatorial number theory, the Probabilistic Method (best known in graph theory and combinatorics) is used to prove the existence of strange structures. In algebraic number theory the probability-based Cohen-Lenstra heuristics lead us to conjectures (and theorems too) about the distribution of algebraic number fields. We close with the famous Covering Congruences problem of Paul Erdös, which was recently settled with probabilistic tools.

Primality testing: then and now

Carl Pomerance (Dartmouth College)

Abstract: The task is simply stated. Given a large integer, decide if it is prime or composite. Gauss wrote of this algorithmic problem (and the twin task of factoring composites) in 1801: "the dignity of science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated." Though progress with factoring composites has been steady and substantial, I think Gauss would be especially pleased with the enormous progress in primality testing, both in practice and in theory. In fact, one of the latest developments strangely and aptly employs a construct Gauss used to deal with ruler and compass constructions of regular polygons! This talk will present a survey of some of the principal ideas used in the prime recognition problem starting with the 19th century work of Lucas, to the 21st century work of Agrawal, Kayal, and Saxena, and beyond.