**Spring 2017**

**Tuesdays at 4:10 p.m. in Carver 150
**

The ISU Mathematics Department Colloquium is
co-organized by

David P. Herzog (dherzog@iastate.edu), and

Pablo Raúl Stinga (stinga@iastate.edu).

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**February 28**

**Speaker: Andrew Suk**

University of Illinois at Chicago

**Title: ****
On the Erdos-Szekeres convex polygon problem**

**Abstract:**The classic 1935 paper of Erdos and Szekeres entitled "A combinatorial problem in geometry" was a starting point of a very rich discipline within combinatorics: Ramsey theory. In that paper, Erdos and Szekeres studied the following geometric problem. For every integer n \geq 3, determine the smallest integer ES(n) such that any set of ES(n) points in the plane in general position contains n members in convex position, that is, n points that form the vertex set of a convex polygon. Their main result showed that ES(n) \leq {2n - 4\choose n-2} + 1 = 4^{n -o(n)}. In 1960, they showed that ES(n) \geq 2^{n-2} + 1 and conjectured this to be optimal. Despite the efforts of many researchers, no improvement in the order of magnitude has been made on the upper bound over the last 81 years. In this talk, we will sketch a proof showing that ES(n) =2^{n +o(n)}.

**March 7**

**Speaker: Mark Allen**

Brigham Young University

**Title: A Free Boundary Problem on Cones**

**Abstract:**The one phase free boundary problem shares a well-known connection to area-minimizing surfaces. In this talk we review this connection and then discuss the one-phase problem on rough surfaces, and in particular cones. After reviewing results of the author with Chang Lara for the one-phase problem on two-dimensional cones, we revisit the connection to area-minimizing surfaces to gain insight into the problem on higher dimensional cones. We then present new results on when the free boundary is allowed to pass through the vertex of a three-dimensional cone as well as results for higher dimensional cones.

**March 14**

**Spring Break**

**March 28**

**Speaker: Giles Auchmuty**

University of Houston

**Title: Real Hilbert Spaces and Laplace's
equation**

**Abstract:**This talk will describe the use of elementary Hilbert space methods to prove results about solutions of boundary value problems for Laplace's equation. Results about the subspaces of real harmonic functions as subspaces of the Hilbert spaces L^2(\Omega) and H^m(\Omega) will be described. Boundary value problems for Laplace’s equation may be viewed as studying the linear mapping of some space of allowable boundary data to these Hilbert spaces.

Under natural conditions on the boundary, these are compact linear transformations that have a singular value decomposition (SVD). This SVD will be described in terms of Steklov eigenvalues and eigenfunctions and their use for efficient approximations of solutions of different boundary value problems will be illustrated. These representations of the solutions are related to applications such as pipe flow and electrostatic fields.

**March 30 (Thursday, 4:10pm, Carver 268)
**

**Speaker: Heather Smith**

Georgia Tech

**Title: Sampling and Counting Genome
Rearrangement Scenarios **

**Abstract:**Genome rearrangement is a common mode of molecular evolution. Representing genomes with edge-labelled, directed graphs, we explore three models for genome rearrangement - reversal, single cut-or-join (SCJ), and double cut-and-join (DCJ). Even for moderate size genomes and regardless of the model, there are a tremendous number of optimal rearrangement scenarios. When hypothesizing, giving one optimal solution might be misleading and cannot be used for statistical inference. With a focus on the SCJ model, we summarize the state-of-the-art in computational complexity and uniform sampling questions surrounding optimal scenarios and phylogenetic trees.

**April 11
**

**Speaker: Tadele Mengesha **

The University of Tennessee, Knoxville

**Title:**
**Regularity
estimates for solutions of a class of elliptic
equations with rough coefficients**

**Abstract:**In this talk I will present recent results on the regularity theory of a class of elliptic equations on weighted Sobolev spaces. Our focus will be on equations with coefficient matrix that is symmetric, nonnegative definite, and both its smallest and largest eigenvalues are proportional to a weight with certain properties. The weighted estimates are obtained under a smallness condition on the mean oscillation of the coefficients. We demonstrate via a counterexample that this condition is necessary. The motivation for this work comes from an effort to obtain a fractional Sobolev regularity theory for fractional nonlocal problems that are defined via local degenerate elliptic problems. Connections between our weighted Sobolev estimates for degenerate problems and estimates for solutions of fractional elliptic problems will be established. [This is based on a joint work with D. Cao and T. Phan.]

**April 13 (Thursday, 4:10pm, Carver 268)
**

**Speaker: Jason McCullough **

Rider University

**Title:** **Rees-like Algebras and
the Eisenbud-Goto Conjecture
**

**Abstract:
**Regularity is a measure of the computational
complexity of a homogeneous ideal in a polynomial
ring, and thereby its associated projective
variety. There are examples in which the
regularity growth is doubly exponential in terms of
the degrees of the generators but better bounds were
conjectured for "nice" ideals. Together with
Irena Peeva, I discovered a construction that
overturned some of the conjectured bounds for "nice"
ideals - including the long-standing Eisenbud-Goto conjecture.
Our construction involves two new ideas that we
believe will be of independent interest: Rees-like algebras and
step-by-step homogenization. I'll explain the
construction and some of its consequences.

**April 18**

**Speaker: Diego Maldonado**

Kansas State University

**Title: On the elliptic Harnack inequality **

**Abstract:**This talk can be regarded as a leisurely tour of the ideas involved in proving the celebrated Harnack inequality for nonnegative solutions to elliptic PDEs in various contexts. We will start with the ever-present Laplacian and then move on to looking at the Harnack properties for divergence and non-divergence form operators (revisiting the former's role in the solution to Hilbert's 19th problem) and describing the corresponding DeGiorgi-Nash-Moser and Caffarelli-Krylov-Safonov techniques. The talk is intended for a broad audience and graduate students are specially encouraged to attend.

**April 19 (Wednesday, 4:10pm, Carver 294)
**

**Speaker: Sunder Sethuraman
**

University of Arizona

**Title:
Modularity clustering, random
geometric graphs, and Kelvin's tiling problem**

**Abstract:**Given a graph, the popular `modularity' clustering method specifices a partition of the vertex set as the solution of a certain optimization problem. In this talk, we will discuss scaling limits, or `consistency' properties, of this method with respect to random geometric graphs constructed from n points, X_1, X_2, . . . ,X_n, drawn independently according to a probability measure supported on a bounded domain in R^d, edges being placed between vertices X_i and X_j only if they are within \epsilon distance of each other.

A main result is the following: Suppose the number of clusters, or partitioning sets of V_n, is bounded above by a fixed level, then we show that the discrete optimal modularity partitions converge, as n grows, in a specific sense to a continuum partition of the underlying domain, characterized as the solution of a `soap bubble', or `Kelvin'-type shape optimization problem.

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**Fall 2016**

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**August 30**

**Speaker: László Székely**

University of South Carolina

**Title: Using the Lovász Local Lemma in
asymptotic enumeration**

**Abstract:** Since its introduction
in 1975, the Lovász Local Lemma (LLL) has been the
tool of probabilistic combinatorics to find the
proverbial needle in the haystack. The conclusion of
LLL is that "none of n events occur" has positive
probability, and the key concept for the LLL is the
dependency graph. In 1991, Erdős and Spencer observed
that LLL still works (with minimal changes in the
proof), if the dependency graph is relaxed to a
negative dependency graph. The difficulty of using
their observation is that independence usually has
clear combinatorial reasons and is easy to see, if it
is there. Showing the correlation required for the
negative dependency graph is less straightforward,
hence the Erdős-Spencer result found few applications.
In joint papers with Lincoln Lu (and in part with
Austin Mohr) we defined some generic types of events
in the space of random perfect matchings of complete
graphs (and in analogous situations), where negative
dependency graphs arise. For these types of events we
managed to provide a good upper bound for the
probability that "none of the n events occur". These
results have many applications to graph and hypergraph
enumeration (even derangements can be counted using
LLL!) though often we do not obtain the strongest
known results. A classic existential result of Erdős
(1961) showed that for any given g and k, there are
graphs with girth at least g and chromatic number at
least k. Using the methods described above, one can
obtain a universal result: almost all graphs with
prescribed degree sequence and girth at least g are
not k-colorable under suitable conditions on the
degree sequence and k.

**September 6**

**Speaker: Mark Huber**

Claremont McKenna College

**Title: Estimates with user-specified relative
error **

**Abstract:** Monte Carlo algorithms
are often used in problems involving high dimensional
integration. Application include finding maximum
likelihood estimators, Bayes Factors for model
selection, and approximation of #P complete problems.
Often, the output of these algorithms is treated as
data that comes from regular experiments, and
classical statistical estimators are used. In this
talk, I will present ways to take advantage of the
fact that this data comes from Monte Carlo experiments
to give estimates for the mean of Bernoulli and
Poisson random variables with the remarkable property
that the relative error in these estimates does not
depend on the quantity being estimated, but is instead
user-specified. That is, the user can decide ahead of
time on the distribution of the relative error of
these estimates. This gives a simple framework for
exact confidence intervals for these estimates that
are highly accurate even out in the tails where
central limit approximations are inappropriate.

**September 13**

**Speaker: Christopher Hoffman**

University of Washington

**Title: First passage percolation and KPZ
universality **

**Abstract: ** First passage
percolation is a classic model of a random metric
space. This model has contributed to several powerful
tools in probability, including the subadditive
ergodic theorem and KPZ universality, but to few
theorems. I will describe this model and the
conjectures for the model as well as the latest
results in the field.

**September 20**

**Speaker: Jay Newby**

University of North Carolina

**Title: How first passage time problems can
help us understand transport of biomolecules in
crowded environments. **

**Abstract:** My talk will explore how
first passage time problems are used to model
molecular transport in biology. Cellular environments
are typically crowded and highly heterogeneous. Even
if large molecular species are not directly involved
in a given reaction, they can influence it through
steric interactions. By modeling the random motion of
individual molecules in heterogeneous environments,
first passage time statistics can be used to
understand the dynamics of complex physiological
processes. I will discuss two examples. (i) First, I
will show how antibodies are dynamically tuned to
anchor large nanoparticles, such as viruses, to
constitutive elements of a mucin polymer gel. Mucus is
a vital component of our immune system and provides a
first line of defense against infection. Large
nanoparticles such as bacteria are trapped within the
tangled polymer network, preventing contact with the
mucus membrane and subsequent infection. However, some
nanoparticles, such as certain viruses, are small
enough that they can freely diffuse through the
polymer matrix. One hypothesis for how smaller
nanoparticles could be trapped is that they are
crosslinked to the mucin network by antibodies.
Indeed, antibodies are present in large quantities
within mucus. However, the hypothesis was previously
discounted because antibodies typically have very weak
affinity for mucin. Counter to the prevailing theory
that antibodies are only effective if they have strong
affinity to mucin, I will show how weak affinity and
rapid binding kinetics substantially improves their
ability to trap large nanoparticles. (ii) In the
second half of my talk I will present theoretical
support for a hypothesis about cell-cell contact,
which plays a critical role in immune function. A
fundamental question for all cell-cell interfaces is
how receptors and ligands come into contact, despite
being separated by large molecules, the extracellular
fluid, and other structures in the glycocalyx. The
cell membrane is a crowded domain filled with large
glycoproteins that impair interactions between smaller
pairs of molecules, such as the T cell receptor and
its ligand, which is a key step in immunological
information processing and decision-making. A first
passage time problem allows us to gauge whether a
reaction zone can be cleared of large molecules
through passive diffusion on biologically relevant
timescales. I combine numerical and asymptotic
approaches to obtain a complete picture of the first
passage time, which shows that passive diffusion alone
would take far too long to account for experimentally
observed cell-cell contact formation times. The result
suggests that cell-cell contact formation may involve
previously unknown active mechanical processes.

**September 27**

**Speaker: Anna Romanowska**

Warsaw University of Technology

**Title: An algebraist's view of convexity and
duality. **

**Abstract: Click
here**

**September 29**

**Speaker: Mireille Boutin**

Purdue University

**Title: Invariant Representations for Object
Recognition and Symmetry Detection **

**Abstract: **In many engineering
applications, the objects of interest are either
invariant under certain groups of transformations or
are characterized by certain types symmetries. In this
talk, we will present object representation methods
that exploit such invariances or symmetries in order
to be more application friendly (e.g., less
computation or more robustness.) To illustrate how
invariances can be exploited in applications, we will
begin with the case of an object represented by a set
of points in a vector space. We will assume that the
object is unchanged by a (simultaneous) rigid motion
of all the points, as well as a reordering of the
points. (For example, the points could represent the
minutia of a fingerprint image.) In this case, we
showed that any generic such object can be represented
without any loss of information by the multi-set of
(unordered) pairwise distances between the points.
This "bag of pairwise distances" representation can
also be viewed as the distribution of an invariant
(the distance). More generally, invariant statistics
can be used to represent complex (generic) objects
without any information loss. For example, we will
describe how complete weighted graphs can be
represented by two invariant statistics; for a generic
choice of weights, these statistics are a lossless
representation of the original graph. This allows for
quick comparison (polynomial time) of two generic
graphs modulo isomorphism. To illustrate how
symmetries can be exploited in applications, we will
look at the problem of recognizing an image
representing an object with a known symmetry. We will
present a new representation called the Pascal
Triangle of the image, which is written in terms of
complex moments of the image and has a direct
connection with the radon transform. We will show how
different types of symmetries manifest themselves in
the Pascal triangle, and show how we applied these
observation to quickly recognize HAZMAT signs (which
are characterized by a 4-fold rotational symmetry)
using a smart phone equipped with a camera. We will
finish the talk by briefly discussing other (current
and future) signal processing and machine learning
problems where geometry plays an important role.

**October 11**

**Speaker: Tim McNicholl**

Iowa State University

**Title: Structural Computable Analysis **

**Abstract:** Abstract: Computability
theory is the mathematical study of the limits and
potentialities of discrete computing devices.
Computable analysis is the theory of computing with
continuous data such as real numbers. Computable
structure theory examines which
computability-theoretic properties are possessed by
the structures in various classes such as partial
orders, Abelian groups, and Boolean algebras. Until
recently computable structure theory has focused on
classes of countable algebraic structures and has
neglected the uncountable structures that occur in
analysis such as metric spaces and Banach spaces.
However, a program has now emerged to use computable
analysis to broaden the purview of computable
structure theory so as to include analytic structures.
The solutions of some of the resulting problems have
involved a blend of methods from functional analysis
and classical computability theory. We will discuss
progress so far on metric spaces and Banach spaces, in
particular $\ell^p$ spaces, as well as open problems
and new areas for investigation.

**October 18**

**Speaker: Palle Jorgensen**

University of Iowa

**Title: Harmonic analysis on fractals **

**Abstract:** We study spectral
duality for singular measures \mu. Complex Hadamard
matrices is one source of such examples, and there are
others. The main question is to decide when L^2( \mu)
will have an orthogonal Fourier basis; i.e., when is
there a fractal Fourier transform? Not so for the
middle-third Cantor! Nonetheless, Jorgensen and
Pedersen [JoPe98] showed that spectral duality does
hold for the middle-1/4 Cantor measure. Higher
dimensional L^2 fractals are associated to certain
Complex Hadamard matrices. For affine fractals, the
distribution of Fourier frequencies satisfies very
definite lacunary properties, in the form of geometric
almost gaps; the size of the gaps grows exponentially,
with sparsity between partitions. Motivated by wavelet
analysis (on fractals), R. Strichartz showed (shortly
after [JoPe98] ) that these lacunary Fourier series
offer better convergence properties than the classical
counterparts; one reason is that, like wavelets, they
are better localized. Another family of Cantor spaces
we study arise as limits of infinite discrete models;
e.g., infinite weighted graphs from electrical
networks with resistors. Then the Cantor spaces arise
as boundaries; for example, Poisson boundaries, Shilov
boundaries, Martin boundaries, path-space boundaries,
and metric boundaries.

**October 25**

**Speaker: Michael Damron**

Georgia Tech

**Title: The shortest crossing of a box in
percolation **

**Abstract:** On the two-dimensional
square lattice, call each nearest-neighbor bond
``open'' with probability 1/2 and ``closed'' with
probability 1/2, each independently. Conditioned on
the existence of an open left-right crossing path of a
box of side-length n, it was shown by
Aizenman-Burchard that, with high probability, the
shortest crossing has at least n^{1+\epsilon} edges,
for some \epsilon>0. It was also shown by
Morrow-Zhang that the *lowest* crossing has order
n^{4/3} edges. In 1992, Kesten and Zhang asked if the
shortest crossing has the same length as the lowest
crossing. Specifically, conditioned on the existence
of an open crossing, does the ratio of the length of
the shortest crossing to the length of the lowest
crossing go to zero in probability as n tends to
infinity? I will talk about these questions, and joint
work with J. Hanson and P. Sosoe in which we show that
the answer to the Kesten-Zhang question is yes.

**November 1**

**Speaker: William Rundell **

Texas A&M University

**Title:Fractional confusion: some of the
things we have discovered about fractional order
differential equations; some we haven't and some
that are still a complete enigma. **

**Abstract:** The idea of a fractional
derivative dates back to the final years of the
seventeenth century. It had its rigorous mathematical
foundations in the mid-nineteenth century and was a
nearly complete theory by 1970. Work on physical
models of diffusion over the last 50 years has
indicated that traditional assumptions based on
Einstein's formulation of Brownian motion may not hold
in general and instead of a Gaussian process and the
heat equation being the basic building block, current
thinking has taken this into the realm of fractional
differential operators. We will explore this avenue
and show how fractional calculus plays a role. We will
also ask questions about the qualitative behaviour of
fractional PDEs and how/why this may differ from the
traditional elliptic/parabolic equation cases.
Finally, we ask about inverse problems: can we recover
unknown coefficients and initial/boundary conditions
from these models and will the answers be very similar
to the classical case? We will show the answer to the
first question is a qualified ``yes'' and for the
second question the answer is ``sometimes, but
certainly not always.''

**November 8**

**Speaker: Justin Webster**

College of Charlston

**Title: Flutter Dynamics: The Waltz of the
Wave and Plate Equations **

**Abstract:** When a thin elastic
structure is immersed in a fluid flow, certain
conditions may bring about excitations in the
structure. That is, the ``dynamic loading" of the
fluid couples with ``natural oscillatory modes" of the
structure. In this case we have a bounded-response
feedback, and the oscillatory behavior persists until
the flow velocity changes or energy is dissipated from
the structure. This interactive phenomenon is referred
to as flutter. Beyond the obvious applications in
aeroscience (projectile paneling and flaps, flags, and
airfoils), the flutter phenomenon arises in: (i) the
biomedical realm (in treating sleep apnea), and (ii)
sustainable energies (in providing a low-cost power
generating mechanisms). Modeling flutter, and
predicting, controlling, and preventing (or bringing
about) its emergence has been a foremost problem in
engineering for nearly 70 years. In this talk we
describe the basics of modeling flutter in the
simplest configuration (an aircraft panel) using
differential equations and dynamical systems. After
discussing the partial differential equations model,
we will discuss theorems that can be proved about
solutions to these equations using modern analysis
(e.g., nonlinear functional analysis, semigroups,
monotone operator theory, the theory of global
attractors, elliptic theory). We will relate these
results back to (experimental) results in engineering
and recent numerical work. We will also describe very
recent (and very open) problems in the analysis of
wing and flag configurations, where a portion of the
structure is unsupported.

**December 6**

**Speaker: Hongjie Dong **

Brown University

**Title: Schauder estimates for nonlocal fully
nonlinear elliptic and parabolic equations **

**Abstract:** In 1934, J. Schauder
first established by now well-known Schauder estimates
for linear elliptic equations, which became an
indispensable tool in the theory of partial
differential equations. For fully nonlinear concave
elliptic equations, such result was obtained by M. V.
Safonov in 1988, following the seminal work of L. C.
Evans and N. V. Krylov in early 1980s. I will present
a recent work on Schauder estimates for a class of
concave fully nonlinear nonlocal elliptic and
parabolic equations with rough and non-symmetric
kernels, where the data are allowed to be bounded and
measurable. A further Dini type estimate will also be
discussed.