Mathematics Department Colloquium

 
 


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.