September 12
Speaker: Khiem Nguyen
Title: On Khinchin's Constant
The goal of this talk is to give a tour of ergodic theory and given an application in the form of Khinchin's constant. For almost all real numbers, the geometric mean of the coefficients in the simple continued fraction expansion tend towards Khinchin's constant.
September 19
Speaker: Micah Coats
Title: The Isoperimetric Inequality for Sets with Sufficiently Smooth Boundaries
The Isoperimetric inequality for sets in Euclidean space states that the ball uniquely minimizes the surface area for sets of some fixed measure. Proving this in general for measurable sets is challenging but an elegant proof for sets with sufficiently smooth boundaries is much easier and will be presented in this talk.
October 3
Speaker: Khiem Nguyen
Title: The Gromov-Hausdorff Metric (Part 1)
The Gromov Hausdorff metric is a way to measure how far two compact metric spaces are away from being isometric. This is a two part talk. This week we will focus on introducing the notion of the Gromov Hausdorff metric and constructing the Gromov Hausdorff metric space.
October 10
Speaker: Khiem Nguyen
Title: The Gromov-Hausdorff Metric (Part 2)
The Gromov Hausdorff metric is a way to measure how far two compact metric spaces are away from being isometric. This week, the second part will focus on showing that the Gromov Hausdorff metric space is separable and complete as well as giving conditions on when sequences converge in the Gromov Hausdorff metric space.
October 17
Speaker: Chase Giles
Title: 'On the Number of Prime Numbers Less than a Given Magnitude' and its Relation to the Prime Number Theorem
Bernhard Riemann's titanic paper "On the Number of Prime Numbers Less than a Given Magnitude", despite hardly comprising 8 pages, has undoubtedly been the bedrock of modern analytic number theory since its publication. We explore the work in this paper (in greater explanatory depth than Riemann offered his readers), a brief history of the prime number theorem, and how Riemann's results lead to its first proof. We will show the connection between the zeros of the zeta function and the prime numbers, deriving the famous functional equation and other results along the way.
October 24
Speaker: Billy Duckworth
Title: Categories for Analysts (Part 1.2)
In this talk we will use the basic concepts from category theory we setup last time to go through a slick proof of the Riesz-Markov representation theorem.
October 31
Speaker: Billy Duckworth
Title: Monad Amenability
There are a few different kinds of "amenable groups" that are important in all sorts of dynamics. Their definitions may not have anything obvious in common, but there is a secret theme that runs through them. Amenable groups are groups whose actions have "virtual fixed points". We will use the categorical concept of monads to formalize what is meant by "virtual points" to get a whole family of amenabilities.
November 7
Speaker: Agil Muradov
Title: An Ultrafilter Proof of Hahn-Banach
The Hahn–Banach Theorem is a central result in functional analysis, usually proved with Zorn’s Lemma. In this talk, I’ll explore an alternative proof that uses the Ultrafilter Lemma instead. This approach, developed by Łoś, Ryll-Nardzewski, and Luxemburg, replaces the usual maximal-chain argument with a compactness argument based on ultrafilters. It provides a different perspective on the logical principles behind the Hahn–Banach Theorem and shows how ultrafilters can be used as a constructive tool in analysis.
December 12
Speaker: Micah Coats
Title: Intro to Schwarz Symmetrization on Euclidean and Hyperbolic Spaces
The Schwarz symmetrization of a function f:R^n->R is a rearrangement of f such that the measures of level sets are unchanged and where the rearrangement f* is radially symmetric and decreasing. Despite not being linear, the rearrangement map (*) has a number of very nice properties such as decreasing the modulus of continuity, increasing the L^p norm, and preserving limits of functions under mild conditions. It is particularly useful as a tool for some optimization problems and classically was important for proving the isoperimetric inequality.
January 31
Speaker: Billy Duckworth
Title: Density of Partial Geometric Sums
Based on a conversation I had with Micah, we will examine the question of whether certain sums of powers of a single complex number can be dense in the plane. Curiously, there will be an appearance of the golden ratio and Niven's Theorem on the rationality of the values of cosine.
February 21
Speaker: Chad Berner
Title: Framlike Fourier Expansions for Finite Borel Measures
The exponential functions form an orthonormal basis for L^2[0,1), and therefore, elements of L^2[0,1) are approximated by Fourier series in a nice stable way. This talk explores the question about when other finite Borel measures on the torus can have elements of their L^2 space approximated by Fourier series in a somewhat stable or "frame-like" way. We also provide some classifying results for measures with these properties.
February 28
Speaker: Billy Duckworth
Title: Infinite Dimensional Banach Spaces have at least Continuum Hamel Dimension
It is a typical first semester exercise in functional analysis to show that an infinite dimensional Banach space cannot have a countable Hamel basis. The typical way to show this uses the Baire category theorem, but we will showcase an elementary proof of a stronger statement: 'In any infinite dimensional Banach space, one can find continuum many linearly independent vectors.'
March 14
Speaker: Chad Berner
Title: Operator Orbit Frames and Framelike Expansions
Frames in a Hilbert space that are generated by operator orbits are vastly studied because of the applications in dynamic sampling and signal recovery. Using the Kaczmarz algorithm, we demonstrate in this talk a representation for frames generated by operator orbits when the operators are not surjective. After this, we describe a large class of operator orbit frames that arise from Fourier expansions for singular measures.
March 28
Speaker: Khiem Nguyen
Title: The Function Guessing Game
It is possible to almost surely predict the future using Choice. Namely, there is a strategy such that, for any fixed f : ℝ → ℝ and for almost all t ∈ ℝ, given f|_{(-∞, t)} one selects g : ℝ → ℝ such that there is some ε > 0 satisfying g|_{(-∞, t + ε)} = f|_{(-∞, t + ε)}.
April 4
Speaker: Micah Coats
Title: The Nikodym Metric Space and Vitali-Hahn-Saks
If (X, M, µ) is a finite measure space, then a metric on sets can be defined by ρ(A,B)= µ(AΔB) (where AΔB is the set difference of A and B). This so called Nikodym Metric is complete and it turns finite absolutely continuous measures with respect to µ into uniformly continuous functions with respect to the Nikodym metric. An application of the Baire category theorem gives the Vitali-Hahn-Saks theorem, a result remarkably similar to the continuity of pointwise limits of uniformly bounded linear operators.
April 11
Speaker: Billy Duckworth
Title: Choquet's Theorem
A theorem of Minkowski is that a compact, convex set in finite dimensional Euclidean space can be represented as both a convex hull of and the set of convex combinations of its extreme points. The Krein-Milman theorem is a generalization of the first part to locally convex Hausdorff spaces, but the second part does not hold in that level of generality. Choquet's theorem is a way to fill this gap; we can represent points as integrals with respect to a probability measure supported on the extreme points.
April 18
Speaker: Billy Duckworth
Title: Orbit Almost Periodicity
We will introduce amenable groups and their almost periodic functions. Then, we will consider the ergodic action of such a group on a standard probability space. There is a very close connection between almost periodicity with respect to the action and almost periodicity of the function restricted to orbits.
May 2
Speaker: Micah Coats
Title: Equivalence of Compactness and Sequential Compactness in Metric Spaces
A fundamental result from topology is that the abstract definition of compactness and sequential compactness are equivalent in metric spaces. A proof for this result is provided in today’s talk.
October 4
Speaker: Yifan Hu
Title: Adaptive Mesh Refinement Algorithms and Applications to Vlasov Simulations
In numerical simulations for Vlasov systems, uniformly high order discretization is often not practical due to high dimensionality of the solution. Adaptive mesh refinement (AMR) algorithms are especially suitable for Vlasov simulations, since the plasma distribution decays exponentially fast as the particle velocity increases, and the interesting plasma dynamics are often confined in certain regions. These features provide physics-informed insights for AMR algorithms. With such insights, various AMR algorithms aim to produce high order solutions in some selected regions where interesting plasma dynamics happen, which will greatly reduce the overall computation cost. In this talk, we will investigate the algorithms proposed by Hittinger and Banks in the journal article "Block-structured adaptive mesh refinement algorithms for Vlasov simulation" (2013).
October 25
Speaker: Micah Coats
Title: Subharmonic in the Distribution Sense Implies Mean Value Subharmonic
Subharmonic functions can be equivalently defined in a mean value sense and using the usual definition that -Δu≤0. However, these definitions can be extended by integrating functions against the Laplacians of non-negative test functions and defining the result as the distributional Laplacian. In this proof by Caffarelli, it is shown that a Subharmonic function in the distributional sense is also equivalent to subharmonic functions in the mean value sense.
November 1
Speaker: Khiem Nguyen
Title: Complete Ordered Fields are Unique up to Isomorphism
The real numbers are defined as a complete ordered field. In this talk, we will show this uniquely characterize the real numbers up to isomorphism, and discuss the construction of the ordered field isomorphism between any two construction of the real numbers.
November 8
Speaker: Billy Duckworth
Title: First Introduction to Filters in Analysis
We will define filters and how sequences converge along them. After exploring some elementary properties of this new type of convergence, we will find another way to 'take a limit' of arbitrary bounded sequences.
November 22
Speaker: Alex Heitzman (University of Nebraska Lincoln)
Title: One-Sided Nonlocal Antiderivatives
We will investigate antiderivatives of nonlocal operators of the form D_k(f) = \int_0^x [f(x)-f(t)]k(x,t)dt. Such operators can approximate the widely known Caputo derivative or classical derivatives. To solve an equation like D_k(f) = g for f, we partition the interval into subintervals and construct a piecewise-constant approximation for f. We present a generalization of the Arzela-Ascoli theorem to establish convergence of the approximate solutions to a solution of the nonlocal equation.
December 13
Speaker: Billy Duckworth
Title: Computability for Analysis
This talk will be a crash course in computability theory with an eye towards analysis. We will discuss (primitive) recursive functions, the Church-Turing thesis, computable sets, computably enumerable sets, and computable approximations. Then we will see how these ideas can be brought into the continuous world of analysis.
January 26
Speaker: Joe Miller
Title: Multiplication Operators on Weighted Banach Spaces (Part 1)
In 2016, Dr. Robert Allen and one of his students, Isaac Craig, studied multiplication operators on weighted Banach spaces of infinite trees. They characterized the bounded and compact multiplication operators in such a context and determined which were isometries. In this talk, we will define all these terms and understand these characterizations.
February 2
Speaker: Joe Miller
Title: Multiplication Operators on Weighted Banach Spaces (Part 2)
As a continuation of last week's talk, we will discuss Dr. Robert Allen and Isaac Craig's paper on multiplication operators on weighted Banach spaces of infinite trees. Last week we characterized the bounded and compact multiplication operators in such a context. This week, we will discuss the spectra of these operators and when isometries arise.
February 16
Speaker: Chad Berner
Title: Backward Shift Invariant Subspaces of the Hardy Space
We define the Hardy space on the disk, and we discuss some of its properties. Next, we proof a representation theorem for analytic functions on the disk with non-negative real part, which gives a correspondence between finite Borel measures and bounded analytic functions. We further explore this correspondence with singular measures and the normalized Cauchy transformation, which encodes information about backward invariant subspaces of the Hardy space. Finally, we discuss a normalized Cauchy transformation in two dimensions and how this relates to another approach of backward invariant subspaces.
March 1
Speaker: Billy Duckworth
Title: The Weird and Wonderful World of Ultrametric Analysis
In this talk we will take a quick tour through a branch of analysis in which the integers are bounded: non-Archimedean analysis. We will see familiar and extremely unfamiliar results alike.
March 8
Speaker: Micah Coats
Title: Using Riesz-Schauder Theory to Solve an Applied Math 2 Qual Problem
Integral equations of the 2nd kind are an example of an operator equation of the form T+I=f, where T is a compact operator on a Hilbert space. Riesz-Schauder Theory allows T+I to be written as a Fredholm operator where the dimension of the range is the same as the dimension of the null space, thus characterizing the solution set of the operator equation T+I=f. In this talk, we prove Riesz-Schauder Theory and then use it to solve a qualifying exam problem.
March 22
Speaker: Mitch Haeuser
Title: A Kaczmarz Algorithm for Solving a Distributed System of Equations on a Lattice
In this work we develop a modified Kaczmarz algorithm to solve a system of linear equations distributed over the vertices of a lattice. We prove convergence of the modified algorithm. In particular, we prove that the algorithm converges to the solution, or solution of minimal norm, provided that the system is consistent. In the case that the system is inconsistent, we introduce a relaxed algorithm which converges to a weighted least squares solution as the relaxation parameter tends to 0. Finally, we consider the case of ‘node failure’, that is, when one of the vertices and equations are eliminated from the lattice. We prove a stability result that relates the convergence of the algorithm on the amended lattice to that of the original lattice.
March 29
Speaker: Sydney Miyasaki
Title: Introduction to Analytic Combinatorics
Counting combinatorial objects is often hard, with many such problems lacking closed-form formulas. However, an approximate answer to such a problem often suffices. Such answers can often be obtained via complex analysis and lead to useful theorem's such as Stirling's Formula. We will go through a few general techniques to tackle such approximation formulas, including a wonderful theorem of Hayman on entire functions.
April 5
Speaker: Chase Foxen
Title: Calculus of Variations
This presentation discusses the foundational concepts from the calculus of variations. Preliminary concepts regarding normed linear spaces are covered and proofs of a few important lemmas and theorems for working with functionals are shown. The Euler-Lagrange Equation, a fundamental result, is derived and applied to the famous brachistochrone problem, the shortest path a ball rolling under gravity without friction can take between two points.
April 12
Speaker: Mitch Haeuser
Title: Regularity Theory for Equations with Nonlocal Boundary Conditions
We will discuss regularity for a problem involving a fractional Dirichlet-to-Neumann operator associated to harmonic functions. In particular, we will define fractional powers of the normal derivative, compatible Sobolev spaces, and consider various examples. We will further look at the extension problem characterization to obtain various Schauder estimates.
September 8
Speaker: Billy Duckworth
Title of the talk: Eudoxus Reals
I will present an unusual construction of the real numbers that goes by the name ‘Eudoxus Reals’. Its most striking feature is that it constructs the reals from the integers, skipping the rationals altogether.
September 15
Speaker: Agil Muradov
Title: An Alternate Way for Constructing the Reals
I will present an alternative way of constructing the set of real numbers, different from the Dedekind cuts and from the Eudoxus reals. Effectively I will be constructing the set of real numbers from just two integer numbers {0,1} but skipping explicitly constructing N, Z and Q.
September 22
Speaker: Chad Berner
Title: A Fourier Expansion for Singular Measures on the Real Line and Fourier Dextroduals
We will discuss a Fourier expansion result for L^2(R,μ) where mu is a singular Borel probability measure on the real line using theory from the Kaczmarz algorithm. We also discuss examples, non-examples, and properties of measures on the torus that are not absolutely continuous or singular that admit Fourier frame-like expansions.
October 6
Speaker: Mitch Haeuser
Title: Nonlocal Equations on the Boundary
We will discuss regularity for a problem involving a fractional Dirichlet-to-Neumann operator associated to harmonic functions. In particular, we will define a fractional powers of the normal derivative, compatible Sobolev spaces, and consider various examples. We will further look at the extension problem characterization to obtain various estimates. This is joint work with Luis Caffarelli (UT Austin) and Pablo Raúl Stinga (Iowa State University)
October 13
Speaker: Mitchell Ashburn
Title: Lie Groups and Sub-Riemannian Geodesics
A Lie group is a group that is also a manifold, this gives it a bit more structure than an arbitrary manifold. We will utilize this additional structure to consider a sub-Riemannian problem SL(2). We call a metric sub-Riemannian if it is positive definite only on a subspace of the tangent bundle, this leads to the notion of a sub-Riemannian geodesic. This talk will start with an introduction to Lie groups followed by a summary of a part of the paper "Sub-Riemannian Geodesics on SL(2,R)" by Domenico D'Alessandro and Gunhee Cho.
October 20
Speaker: Brian Zilli
Title of the talk: Limit Sets of Computable Sequences – Definitions & Summary of Results
We define computability of a sequence of reals and a way of measuring the computational complexity of closed sets of reals. We do this toward answering the problem: How computationally complex is the set of limit points of a computable sequence of reals? We prove only one short lemma giving an upper bound on complexity in preparation for a later talk where we will show the bound is saturated and other results (that will be stated without proof today).
March 10
Speaker: Mitchell Haeuser
Title: Nonlocal Equations on the Boundary
We will discuss regularity for a problem involving a fractional Neumann boundary condition with harmonic interior. In particular, we will define a fractional normal derivative, compatible Sobolev spaces, and consider various examples. We will further look at the extension problem to obtain various estimates.
April 21
Speaker: Billy Duckworth
Title: All Topologies Come from Generalized Metrics
Most of us started our education in topology by first studying metric spaces. We quickly learn that there are many interesting and useful topological spaces that cannot have come from a metric. Various generalizations of the concept of metrics (such as are psuedo-metrics, quasi-metrics, etc...) are used in various areas, but none of them are general enough to describe arbitrary spaces. This paper discuses a way to generalize the concept of a metric in a broad enough way to make all topological spaces "metric" spaces. Further, some basic separation properties are described in terms of additional properties of this new distance function.
September 9
Speaker: Chad Berner
Title: Fourier Analysis from Singular Measures
Any square integrable function on the torus is a norm limit of its Fourier series, but what if you change the measure from Lebesgue measure to a singular measure? It turns out you will lose orthogonality of the exponentials, but by the Kaczmarz algorithm, any function that is square integrable in this new measure space has a Fourier series converging in norm. We discuss further results in higher dimensions as well as analytic operators and their relation to the Hardy space and Fourier series.
September 16
Speaker: Brian Zilli
Title: Boundary Spectra of Uniform Frostman Blaschke Products
In a 2006 paper, Alec Matheson proved that uniform Frostman Blaschke products have nowhere dense spectra and—conversely—that any closed, nowhere dense subset of the circle is the spectrum of some uniform Frostman Blaschke product. In this talk, we define these terms and outline the proof of Matheson’s first result.
September 23
Speaker: Mitch Haeuser
Title: Nonlocal Equations on the Boundary
We will discuss regularity for a problem involving a fractional Dirichlet-to-Neumann operator associated to harmonic functions. In particular, we will define a fractional power of the normal derivative, compatible Sobolev spaces, and consider various examples. We will further look at the extension problem characterization to obtain various estimates. This is joint work with Luis Caffarelli (UT Austin) and Pablo Raúl Stinga (Iowa State University)
October 7
Speaker: Joe Miller
Title: Multiplication Operators on Weighted Banach Spaces
In 2016, Dr. Robert Allen and one of his students, Isaac Craig, studied multiplication operators on weighted Banach spaces of infinite trees. They characterized the bounded and compact multiplication operators in such a context and determined which were isometries. In this talk, we will define all these terms and understand these characterizations.
October 14
Speaker: Sarah McCarty
Title: Piecewise Linear Functions Representable with Infinite-Width Neural Networks
A shallow neural network can be identified by the integral of the ReLU function with respect to a signed, finite measure on an appropriate parameter space. We map these measures on the parameter space with measures on the projective n-sphere cross R, allowing points in the parameter space to be bijectively mapped to hyperplanes in the domain of the function. This can in turn be mapped almost bijectively onto R^{n+1}. With these mappings and a large non-co-hyperplanar set, we can show that every continuous piecewise linear function expressible with an infinite width neural network is in fact expressible with a finite width network.
October 28
Speaker: Lillian Uhl
Title: Categorical Aspects of Integration Theory and Operator Algebras
One of the most influential results in modern operator theory is that of Gelfand and Naimark, which, in the form most often presented, demonstrates a deep and fundamental connection between compact topological spaces and commutative unital C*-algebras. Outside of its intrinsic significance, the result is also notable for catalyzing an extensive research program to the ends of illuminating relations between operator algebras relevant to mathematical analysis and different sorts of spaces primarily studied by topologists, order theorists, and geometers of all walks of mathematics. This talk is an exposition on one such connection, that between operator algebras known as "commutative von Neumann algebras" and spaces known as "localizable measurable spaces"; we start with a brief review of abstract measure theory, introducing and motivating less well known concepts as we go, then use the new machinery to motivate and relate the aforementioned fragment of the theory of von Neumann algebras.
November 11
Speaker: José David Beltran
Title: Applications of Young measures to scalar conservation laws
February 3
Speaker: Caleb Camrud
Title: Making Modal Logic Continuous
Modal language is used to model possibility, obligation, access to knowledge, and more. Logics of these modalities were then developed to reflect how we reason using modal language. But some modalities seem to exist in a continuum, rather than being discretely ordered. For a classic metaphysical example, it seems that the possible world in which I became a lawyer is more similar to the actual world than the possible world in which I am a frog. Indeed there intuitively seems to be an entire continuum of possible worlds "between" our actual world and the world in which I am a frog. As such, Dr. Ranpal Dosanjh (Iowa State University, Department of Philosophy) and I developed a method for making modal logic continuous. In this talk, I will present the syntax and semantics of basic modal logic, and discuss our method for constructing continuous modal logic.
February 10
Speaker: Jina Hyoungji Kim
Title: Kolmogorov diffusion with Infinite-dimensional Brownian motion
It is known that the Kolmogorov diffusion on a finite-dimensional space has the smooth density with respect to the Lebegue measure which is explicit. Now let us define the Kolmogorov diffusion on an infinite-dimensional space. Then we do not have a density of the heat kernel measure with respect to the Lebesgue measure anymore. But we can consider what is called quasi-invariance of a measure under some translation to obtain some type of smoothness of the heat kernel measure of an infinite-dimensional process.
February 17
Speaker: Kristina Moen
Title: Growth of the Ulam Sequence – An Open Problem
In his 1964 book “On Some Mathematical Problems Connected with Patterns of Growth Figures,” Stanislaw Ulam proposed an integer sequence with simple rules: the first two terms are 1 and 2 and each subsequent term is the smallest integer that can be uniquely written as the sum of two distinct earlier terms. This gives the sequence: 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28… The first trillion Ulam numbers have been computed, but much about the sequence remains a mystery. The most common gap size between consecutive Ulam numbers is two, but gap sizes as large as 1,373 have appeared far into the sequence. Finding an upper bound on the gap size and growth rate remains an open question. Recently, there has been renewed interest into the sequence and its generalizations (s-Additive sequences) after Stefan Steinerberger discovered a “hidden signal” using Fourier series with Ulam numbers as frequencies. He found a persisting signal in the noise which gives rise a very regular distribution function. In this talk, we will delve into the world of Ulam numbers, what has been proven so far (including some surprising results when we choose different initial values), and possible avenues for further study.
February 24
Speaker: Chad Berner
Title: Hilbert spaces of entire functions
The Paley-Wiener space is a space of entire functions that are of exponential type and are square integrable on the real axis. It turns out that these entire functions are fourier transforms and show us the isometry of L^2 through the fourier transform. Furthermore, Louis de Branges generalized these spaces into Hilbert spaces of entire functions that are based off of entire functions of a more general sense than just exponentials. These generalized spaces also come equipped with nice properties such as results in function interpolation.
March 3
Speaker: Aniket Banerjee
Title: Formulation of a mathematical model showing the dynamics of different biotypes of Soybean Aphids
The soybean aphid, Aphis glycines (Hemiptera: Aphididae), is an invasive pest that can cause severe yield loss to soybeans in the northcentral United States. A tactic to counter this pest is the use of aphid-resistant soybean varieties. However, the occurrence of virulent biotypes can alter plant physiology and impair the use of this management strategy. Soybean aphids can alter soybean physiology primarily by two mechanisms, feeding facilitation and the obviation of resistance, favoring subsequent colonization by additional conspecifics. We developed a non-local, differential equation population model, to explore the dynamics of these biological mechanisms on soybean plants co-infested with virulent and avirulent aphids. We then use demographic parameters from laboratory experiments to perform numerical simulations via the model. These simulations successfully mimic various aphid dynamics observed in the field. Our model showed an increase in colonization of virulent aphids increases the likelihood that aphid-resistance is suppressed, subsequently increasing the survival of avirulent aphids, producing an indirect, positive interaction between the biotypes. These results suggest the potential for a "within plant" refuge that could contribute to the sustainable use of aphid resistant soybeans.
March 10
Speaker: Thomas Griffin
Title: Icosystems – Modeling a Swarm
Swarms are collections of loosely coupled distinct agents each following "simple" rules. Swarms do not use central coordination and individual agents need not be aware of the swarm’s overall function/goal. As each agent performs its task, the swarm collective can display unusual and unexpected emergent behaviors. For those swarms defying analytic evaluation, simulation remains as the only method to reveal these emergent behavior. A number of swarm behaviors will be demonstrated with different simple rules to follow, and then combined to discuss some more interesting dynamics and real world applications.
March 24
Speaker: Brian Zilli
Title: Computable Blaschke Products
We present some results of Matheson and McNicholl concerning the computability of Blaschke products and discuss the potential for further results on the computability of the set of accumulation points of Blaschke products.
April 7
Speaker: Sarah McCarty
Title: Image of Dual Ridgelet Transform
We present results characterizing the image of the Dual Ridgelet Transform in the single variable case. The Dual Ridgelet Transform builds functions from a simple activation function and integration against compactly supported, signed, finite measures, which has applications in neural networks.
April 14
Speaker: Diego Rojas
Title: From Classical to Computable – Effectivizing Objects and Theorems in Analysis
Since the publication of Alan Turing's 1937 paper on the computable real numbers, mathematicians have looked for ways to adapt concepts found in mathematical analysis (and other fields) to a framework suitable for computation via a process called effectivization. Through effectivization, we are able to develop computable versions of theorems regarding objects such as real numbers, continuous real-valued functions, and measures. In this talk, I will be going over some examples of computable analogues to objects found in classical analysis, and I will explain how the effectivization process works in each instance. I will also give examples of computable versions of certain theorems from classical analysis.
April 21
Speake: Yifan Hu
Title: Stability Analysis for Spacetime Discontinuous Galerkin Methods
Can we treat time as an additional dimension in solving time-dependent partial differential equations? What are the advantage and disadvantage of spacetime formulations for Discontinuous Galerkin methods? How do we know these numerical methods are stable? In this talk, we will try to answer these questions by inspecting Locally Implicit Discontinuous Galerkin (LIDG) methods and Regionally Implicit Discontinuous Galerkin (RIDG) methods, and share some common approaches to find the maximal timestep size that produces stable schemes.
April 28
Speaker: Evan Camrud
Title: The Carleman Matrix and Iterated Functions
The Carleman matrix is a homomorphism (ALMOST a representation) mapping function composition into infinite matrices (hence why it's ALMOST a representation). We will introduce such matrices, their generalizations, and how they relate to calculating iterates of functions.
November 5
Speaker: Chad Berner
Title: The Fragman-Lindelof Principle and a Representation for Some Analytic Functions in the Upper Half-Plane.
The maximum modulus theorem from complex analysis has a corollary called the Phragmen-Lindelof principle on simply connected sets due to analytic functions on these sets having a primitive, which is a result of Runge's theorem. This allows us to provide a representation of analytic functions on the upper half plane whose real part is non-negative with continuous extension to the closed half plane.
November 12
Speaker: Evan Camrud
Title: Chaos – An Introduction
This will be an introduction to chaos theory, broadly defined. We will cover definitions and examples of mixing, sensitivity to initial conditions, and strange attractors of nonlinear dynamical systems.
November 19
Speaker: Lee Przybylski
Title: Measuring Effects of Starting Pitching
Betting on baseball is challenging. One feature unique to the sport is that moneylines usually list probable starting pitchers. To take advantage of this, we develop a generalized linear mixed effects model using retrosheet data from several seasons. The model includes effects for teams, starting pitchers, and venue. Being able to assess a pitcher's performance independent of his team is also challenging. By estimating effects for each starting pitcher, fitting the model provides another way to measure a starting pitcher's effectiveness. In this talk we also provide some background on popular pitching metrics, such as ERA, FIP, and opponent WOBA, and compare these metrics to our estimated pitcher effects. As was done with FIP back in 2001, we also investigate how well these starting pitcher effects reflect an individual pitchers performance by looking at their correlation across seasons.
December 3
Speaker: Herionexy Mounier
Title: Optimal Line Packings with Heisenberg Symmetry
Zauner's Conjecture has been a subject of study in Frame Theory for several years. It states that for any complex Hilbert space of dimension d, there exists a d x d^2 equiangular tight frame (ETF). However, we proved a consequence of this conjecture by showing the existence of some particular ETFs. This was done by considering an orbit of some action of the Heisenberg group of a finite abelian group over the set of d x d matrices with complex entries. In this talk, I will introduce the concept of an ETF, together with the main result of this research.
December 10
Speaker: Brian Zilli
Title: A Brief Introduction to Concepts in Computable Analysis
We say that a real-valued function is ‘computable’ if there exists a computer program which, given a real number computed to a certain degree of precision, returns the function value to a desired degree of precision. We will define this concept rigorously, see some examples, and examine the interesting construction of a function which is computable and continuously differentiable but whose derivative is not computable.
January 28
Speaker: Mary Vaughan
Title: Fractional Powers of Nondivergence Form Elliptic Operators
In this talk, we will show how to define fractional powers of nondivergence form differential operators in a pointwise and weak sense. We will then discuss some tools required to prove desired regularity estimates, such as Harnack inequality.
February 4
Speaker: Animesh Biswas
Title: Decreasing Rearrangement and Measure Preserving Transformation.
In this talk, we will show how to define decreasing rearrangement of a measurable function. We will also define measure preserving transformation. Finally we will try to prove that in any finite measure space, a function can be written as a composition of its rearrangement and a measure preserving transformation.
February 11
Speaker: Manas Bhatnagar
Title: Critical Thresholds in One Dimensional Euler-Poisson Equations with Nonlocal Forces
We will look at the existing critical threshold results in systems with nonlocal forces. We will see what makes the system complicated by adding a nonlocal term in the momentum equation. Lastly, we will also look at some open problems in this area.
February 18
Speaker: Nicole Buczkowski (University of Nebraska, Lincoln)
Title: Local and Nonlocal Energy Minimization
Today we will be going over some basics of the nonlocal framework and nonlocal operators. We will also go into the classical Euler-Lagrange equations as well as their nonlocal counterparts. We will further explore the growth conditions necessary for a minimizer of an energy functional to yield a weak solution to its corresponding Euler-Lagrange equation.
March 3
Speaker: Evan Camrud
Title: Some Fixed-Point Theorems and Their Applications
Fixed point theorems are some of the most general and widely-used theorems in analysis. They are consistently applied in differential equations, but have also found a foothold in discrete dynamical systems and functional analysis, namely in the invariant subspace problem. This talk will introduce some of the most common fixed point theorems, and give examples of them in applied mathematics.
March 10
Speaker: Hayley Olson (University of Nebraska, Lincoln)
Title: Poincaré Inequalities in a Nonlocal Vector Calculus
The development of a nonlocal vector calculus rose out of the desire to describe standard systems of differential equations to discontinuous functions: this is pertinent in applications such as dynamic fracturing and image processing. This opened up a need to show well-posedness of systems with nonlocal calculus operators. In this talk, we introduce two nonlocal variations of the Poincaré inequality from Mikil Foss' Nonlocal Poincaré Inequalities for Integral Operators with Integrable Nonhomogeneous Kernels and discuss their use in showing well-posedness of systems with nonlocal operators.
September 3
Speaker: Nathan Harding.
Title: The Kaczmarz Algorithm with Nonuniform Relaxation Parameters
We modify the proof of Natterer for the relaxed Kaczmarz algorithm with nonuniform relaxation parameters. We show (1) that the Kaczmarz method converges to the solution of minimal norm if the system of equations is consistent and (2) that the Kaczmarz method converges to a parameter-dependent, weighted least squares solution if the system of equations is inconsistent. Our method further extends results for altered Kaczmarz algorithms. We also address randomized parameter selection. We end with some suggestive numerical experiments based on our method for further work in optimal parameter selection.
September 10
Speaker: Animesh Biswas
Title: Regularity Theory for Nonlocal Space-Time Master Equations
We analyze recent novel regularity theory for fractional power of parabolic operators in divergence form. These equations are fundamental in continuous time random walk models and appear as generalized Master equation. These equations are non-local in nature and were studied by Luis Caffarelli and Luis Silvestre. We developed a parabolic method of semigroups that allows us to prove a local extension problem. As a consequence we obtain interior and boundary Harnack inequalities and sharp interior and global parabolic Schauder estimates. For the latter, we also prove a characterization of the correct intermediate parabolic Hölder spaces in the spirit of Sergio Campanato. This is a joint work with Marta de Leon-Contreras (Universidad Autonoma de Madrid, Spain) and Pablo Raul Stinga (Iowa State University)
September 17
Speaker: Pranamesh Chakraborty
Title: Applications of Trend Filtering and Maximum Likelihood in Traffic Incident Detection
By 2020, the volume of traffic data is expected to rise to 11 petabytes, while a billion traffic cameras will be installed worldwide. This presentation will focus on using this wealth of information available from large-scale traffic data and closed-circuit television cameras towards developing an efficient automatic incident detection (AID) framework. First, a graph-based trend filtering method of AID framework will be introduced that can leverage large-scale traffic data along with the topology of the road network to learn the historical pattern. The second part of this presentation will focus on using maximum-likelihood estimation based semi-supervised learning technique to detect incidents from vehicle trajectories extracted from cameras. The results demonstrate that this framework can achieve superior performance in detecting such anomalies using these data sources and provide rapid incident responses.
September 24
Speaker : Wumaier Maimaitiyiming
Title: Fisher Information Regularization Schemes for Wasserstein Gradient Flows
The variational scheme for computing Wasserstein gradient flows builds upon the Jordan–Kinderlehrer–Otto framework with the Benamou- Brenier’s dynamic formulation of the quadratic Wasserstein metric and adds a regularization by the Fisher information. This regularization can be derived in terms of energy splitting and is closely related to the Schrödinger bridge problem. It improves the convexity of the variational problem and automatically preserves the non-negativity of the solution. As a result, it allows us to apply sequential quadratic programming to solve the sub-optimization problem. We further save the computational cost by showing that no additional time interpolation is needed in the underlying dynamic formulation of the Wasserstein-2 metric, and therefore, the dimension of the problem is vastly reduced. This scheme can be allied to porous media equation, nonlinear Fokker-Planck equation, aggregation diffusion equation, and Derrida-Lebowitz-Speer-Spohn equation.
This is a paper I've been reading recently, this paper starts with a general class of PDEs and reformulate the problem as a Wasserstein gradient flow, then introduces its variational formulation, finally solves the discrete minimization problem with the help of fisher information regularization.
October 1
Speaker: Mary Vaughan
Title: Viscosity Solutions of Elliptic Equations (Part 1)
Following Luis Silvestre's notes on Viscosity Solutions of Elliptic Equations, we will provide an introduction to fully nonlinear PDEs and viscosity solutions.
October 8
Speaker: Lee Przybylski
Title: ATD Algorithms for Traffic Incident Detection and Ship Tracking
ATD stands for "Algorithms for Threat Detection." The objective of the program is to develop new algorithms for analyzing spatiotemporal data sets. As part of this program, and in collaboration with Steven Harding, Paranamesh Chakraborty, and Eric Weber, we have developed various algorithms for detecting traffic incidents and tracking the paths of ships. This talk will discuss the details of developing and implementing those algorithms.
October 15
Speaker: Evan Camrud
Title: Not a Proof of the Riemann Hypothesis
Solving one of the Millennium Prize Problems is hard, but that doesn't mean trying to can't be fun. This talk will be a brief survey of the Riemann zeta function, Mellin transform, and other special function theory, and will culminate in my latest attempt at tackling the Riemann hypothesis. (Spoiler: it didn't work.)
October 22
Speaker : Caleb Camrud
Title: A Brief Introduction to Continuous Logic
While classical logic works well in formulating theories about discrete mathematical structures, its application to continuum-valued analytic structures is not very intuitive. Thus, in an attempt to more easily apply logical language and model theoretic results to analytic structures, continuous logic was invented. This introductory-level talk will not assume any background in logic, and will introduce the language of theories and deductions in classical logic before describing the corresponding notions in continuous logic.
October 29
Speaker: Makrand Khanwale
Title: Provable Energy Stability for Second Order Time Integration Schemes for Cahn-Hilliard Navier Stokes Equations
I will talk about conservation laws (especially momentum transport) for single and multiphase flows. Starting with a brief introduction about various forms of momentum equations, I will present the generalisation to a specific two-phase formulation we use to simulate two-phase flows. We use a diffuse interface approach, which utilizes a thermodynamically consistent set of coupled Cahn Hilliard Navier-Stokes equations. I will talk about the strategy for proving energy stability of time integration schemes for such complicated operators with the example of a second order scheme in the semi-discrete form. I will also talk about the uniqueness of the advection-diffusion operator we use to track the interface in the Browder-Minty context.
November 5
Speaker: Mary Vaughan
Title: Viscosity Solutions of Elliptic Equations (Part 2)
Continuing our discussion on Luis Silvestre's notes on Viscosity Solutions of Elliptic Equations, we will introduce some tools and techniques which are useful when working with viscosity solutions and then prove a comparison principle, an essential tool for uniqueness of solutions.
November 12
Speaker: Manas Bhatnagar
Title: On the Energy Estimates Used to Prove Local Existence in Traffic Flow Models and Hyperbolic Balance Laws
We will take a look at the local existence and uniqueness results for a traffic flow model. To do critical threshold analysis (which is what I do), we need a little more than just existence. We need to know what goes wrong when the solution ceases to exist. One method is to derive energy estimates for the concerned physical quantities. We will prove these estimates. In the process, we will need some special Sobolev/interpolation inequalities which I will prove (time permitting).
December 3
Speaker: Nyle Sutton
Title: Squaring the Circle – A Brief Discussion of Matrix Compression and the Story So Far
Given two bounded linear operators A and B, which act on Hilbert spaces H and K, the relation between the numerical range inclusion relation W(B) ⊆ W(A) and the condition that B can be dilated to an operator of the form A⊗I is complex and nuanced. In this talk, I will discuss some known results in the area of Matrix Compressions and the applications of positive and completely positive maps to this question.