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Series: School of Mathematics Colloquium

Populations, whether they be viral particles, bio-chemicals, plants or animals, are subject to intrinsic and extrinsic sources of stochasticity. This stochasticity in conjunction with nonlinear interactions between individuals determines to what extinct populations are able to persist in the long-term. Understanding the precise nature of these interactive effects is a central issue in population biology from theoretical, empirical, and applied perspectives.
For the first part of this talk, I will discuss, briefly, the relationship between attractors of deterministic models and quasi-stationary distributions of their stochastic, finite population counterpoints i.e. models accounting for demographic stochasticity. These results shed some insight into when persistence should be observed over long time frames despite extinction being inevitable.
For the second part of the talk, I will discuss results on stochastic persistence and boundedness for stochastic models accounting for environmental (but not demographic) noise. Stochastic boundedness
asserts that asymptotically the population process tends to remain in compact sets. In contrast, stochastic persistence requires that the population process tends to be "repelled" by some "extinction set." Using these results, I will illustrate how environmental noise can facilitate coexistence of competing species and how dispersal in stochastic environments can rescue locally extinction prone populations. Empirical work on Kansas prairies, acorn woodpecker populations, and microcosm experiments demonstrating these phenomena will be discussed.

Series: School of Mathematics Colloquium

Isoperimetric problems in Gaussian spaces have been studied since the 1970s. The study of these problems involve geometric measure theory, symmetrization techniques, spherical geometry and the study of diffusions associated with the heat equation. I will discuss some of the main ideas and results in this area along with some new results jointly with Joe Neeman.

Series: School of Mathematics Colloquium

Self-adjoint $n$-by-$n$ matrices have a natural partial ordering,
namely $ A \leq B $ if the matrix $ B - A$ is positive semi-definite.
In 1934 K. Loewner characterized functions that preserve this ordering;
these functions are called $n$-matrix monotone.
The condition depends on the dimension $n$, but if a function
is $n$-matrix monotone for all $n$, then it must extend analytically
to a function that maps the upper half-plane to itself.
I will describe Loewner's results, and then discuss what happens
if one wants to characterize functions $f$ of two (or more) variables that
are matrix monotone in the following sense:
If $ A = (A_1, A_2)$ and $B = (B_1,B_2)$ are pairs of commuting
self-adjoint
$n$-by-$n$ matrices, with $A_1 \leq B_1 $ and $A_2 \leq B_2$,
then $f(A) \leq f (B)$.
This talk is based on joint work with Jim Agler and Nicholas Young.

Series: School of Mathematics Colloquium

A basic strategy for linear optimization over a complicated convex set is
to try to express the set as the projection of a simpler convex set that
admits efficient algorithms. This philosophy underlies all
"lift-and-project" methods in
optimization which attempt to find polyhedral or spectrahedral lifts of
complicated sets. In this talk I will explain how the existence of a lift
is equivalent to the ability to factorize a certain operator associated to
the convex set through a cone.
This theorem extends a result of Yannakakis who showed that polyhedral
lifts of polytopes are controlled by the nonnegative factorizations of the
slack matrix of the polytope. The connection between cone lifts and cone
factorizations of convex sets yields a uniform framework within which to
view all lift-and-project methods, as well as offers new tools for
understanding convex sets. I will survey this evolving area and the main
results that have emerged thus far.

Series: School of Mathematics Colloquium

It is well-known that a deterministic dynamical system can exhibit
stochastic behavior that is due to the fact that instability along
typical trajectories of the system drives orbits apart, while
compactness of the phase space forces them back together. The consequent
unending dispersal and return of nearby trajectories is one of the
hallmarks of chaos.
The hyperbolic theory of dynamical systems provides a mathematical
foundation for the paradigm that is widely known as "deterministic
chaos" -- the appearance of irregular chaotic motions in purely
deterministic dynamical systems. This phenomenon is considered as one of
the most fundamental discoveries in the theory of dynamical systems in
the second part of the last century. The hyperbolic behavior can be
interpreted in various ways and the weakest one is associated with
dynamical systems with non-zero Lyapunov exponents.
I will discuss the still-open problem of whether dynamical systems with
non-zero Lyapunov exponents are typical. I will outline some recent
results in this direction. The genericity problem is closely related to
two other important problems in dynamics on whether systems with nonzero
Lyapunov exponents exist on any phase space and whether nonzero
exponents can coexist with zero exponents in a robust way.

Series: School of Mathematics Colloquium

The study of random Cayley graphs of finite groups is related to the investigation of Expanders and to problems in Combinatorial Number Theory and in Information Theory. I will discuss this topic, describing
the motivation and focusing on the question of estimating the chromatic number of a random Cayley graph of a given group with a prescribed number of generators. The investigation of this problem combines combinatorial, algebraic and probabilistic tools. Several intriguing questions that remain open will be mentioned as well.

Series: School of Mathematics Colloquium

Series: School of Mathematics Colloquium

There is an extensive applied mathematics literature developed for problems in the biological and physical sciences. Our understanding of social science problems from a mathematical standpoint is less developed, but also presents some very interesting problems. This lecture uses crime as a case study for using applied mathematical techniques in a social science application and covers a variety of mathematical methods that are applicable to such problems. We will review recent work on agent based models, methods in linear and nonlinear partial differential equations, variational methods for inverse problems and statistical point process models. From an application standpoint we will look at problems in residential burglaries and gang crimes. Examples will consider both "bottom up" and "top down" approaches to understanding the mathematics of crime, and how the two approaches could converge to a unifying theory.

Series: School of Mathematics Colloquium

We show that the LIA approximation of the incompressible Euler equation describes the skew-mean-curvature flow on vortex membranes in any dimension. This generalizes the classical binormal, or vortex filament, equation in 3D. We present a Hamiltonian framework for higher-dimensional vortex filaments and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively. This framework, in particular, allows one to define the symplectic structures on the spaces of vortex sheets.

Series: School of Mathematics Colloquium

Building on work of Jordan from 1870, in 1979 Harris
showed that a geometric monodromy group associated to
a problem in enumerative geometry is equal to the Galois
group of an associated field extension. Vakil gave a
geometric-combinatorial criterion that implies a Galois
group contains the alternating group. With Brooks and
Martin del Campo, we used Vakil's criterion to show that
all Schubert problems involving lines have at least
alternating Galois group.
My talk will describe this background and sketch a
current project to systematically determine Galois groups
of all Schubert problems of moderate size on all small
classical flag manifolds, investigating at least several
million problems. This will use supercomputers employing
several overlapping methods, including combinatorial
criteria, symbolic computation, and numerical homotopy
continuation, and require the development of new
algorithms and software.