January 31st
Speaker: Billy Duckworth
Title of talk: Density of Partial Geometric Sums
Abstract: We 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 21st
Speaker: Chad Berner
Title of talk: Frame-like Fourier expansions for finite Borel measures
Abstract: 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 28th
Speaker: Billy Duckworth
Title of talk: Infinite Dimensional Banach Spaces have at least Continuum Hamel Dimension
Abstract: 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 14th
Speaker: Chad Berner
Title of talk: Operator orbit frames and frame-like expansions
Abstract: 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 28th
Speaker: Khiem Nguyen
Title of talk: The function guessing game
Abstract: 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 4th
Speaker: Micah Coats
Title of talk: The Nikodym Metric Space: treating sets as points in a metric space.
Abstract: If (X, M, µ) is a finite measure space, then a metric on sets can be defined by p(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 11th
Speaker: Billy Duckworth
Title of talk: Choquet's Theorem
Abstract: 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 18th
Speaker: Billy Duckworth
Title of talk: Orbit Almost Periodicity
Abstract: 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.
April 25th
Speaker: Chad Berner
Title of talk: The beautiful Kaczmarz Algorithm
Abstract: The Kaczmarz Algorithm is a process to reconstruct vectors using some inner product data in a simple way using projections. In this talk, we provide examples of collections of vectors with this reconstructive property as well as necessary conditions for this property. Then we will try to understand this algorithm in the setting of Hilbert C*-modules, which generalize Hilbert spaces. Additionally, we also provide examples of collections of vectors with this property as well as necessary conditions to make this algorithm feasible.
May 2nd
Speaker: Micah Coats
Title of talk: Equivalence of compactness and sequential compactness in metric spaces
Abstract: A fundamental result from topology is that the definition of compactness and sequential compactness are equivalent in metric spaces. A proof for this result is provided in today’s talk.
September 13th
Speaker: Billy Duckworth
Title of talk: A Little Bit about Banach Limits
Abstract: Banach limits are a way of extending the notion of taking limits from convergent sequences to all bounded sequences. We will show that one can in fact "take limits" of every bounded sequence; from here we will see that the space of absolutely summable sequences is not the dual of the space of bounded sequences. The concept of "almost convergence" will also be introduced.
September 20th
Speaker: Chad Berner
Title of talk: Frames and applications to Fourier series
Abstract: In the Hilbert space setting, an orthonormal basis is a rigid structure that allows for stable reconstruction of all vectors. However, it is often the case in practice that this structure is too rigid to deal with noise and error. Additionally, constructing an orthonormal basis in general can be computationally challenging. Frames on the other hand, are more forgiving in signal recovery because of their redundancy. We will study frames as a relaxion of an orthonormal basis and try to understand their most important approximation properties. Furthermore, we apply this theory to Fourier approximation and even consider Fourier approximations that are more relaxed than frame approximations.
October 4th
Speaker: Yifan Yu
Title of talk: Adaptive Mesh Refinement Algorithms and Applications to Vlasov Simulations
Abstract: 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 25th
Speaker: Micah Coats
Title of talk: Subharmonic in the distributional sense implies mean value subharmonic
Abstract: 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 1st
Speaker: Khiem Nguyen
Title of talk: Complete Ordered Fields are unique up to isomorphism
Abstract: 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 8th
Speaker: Billy Duckworth
Title of talk: First Introduction to Filters in Analysis
Abstract: 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 22nd
Speaker: Alex Heitzman
Title of talk: On One-Sided Nonlocal Antiderivatives
Abstract: 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 12th
Speaker: Billy Duckworth
Title of talk: Computability for Analysis
Abstract: 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 26th
Speaker: Joe Miller
Title of talk: Multiplication Operators on Weighted Banach Spaces Part I
Abstract: 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 2nd
Speaker: Joe Miller
Title: Multiplication Operators on Weighted Banach Spaces Part II
Abstract: 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 9th
February 16th
Speaker: Chad Berner
Title: Backward shift invariant subspaces of the Hardy space
Abstract: 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.
February 23rd
March 1st
Speaker: Billy Duckworth
Title: The Weird and Wonderful World of Ultrametric Analysis
Abstract: 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 8th
Speaker: Micah Coats
Title: Using Riesz-Schauder Theory to solve an Applied Mathematics 2 qual problem
Abstract:
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 22nd
Speaker: Mitch Haeuser
Title: A Kaczmarz Algorithm for Solving a Distributed System of Equations on a Lattice
Abstract: 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 29th
Speaker: Sydney Miyasaki
Title: Introduction to Analytic Combinatorics
Abstract: 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 5th
Speaker: Chase Foxen
Title: The Calculus of Variations
Abstract: 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 12th
Speaker: Mitch Haeuser
Title: Regularity theory for equations with nonlocal boundary conditions
Abstract: 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.
April 19th