Title: The moduli space of complex Hadamard matrices

Abstract: During the past two weeks, Dr. Nicoara derived conditions that the first and second order "derivatives" of a sequence of commuting squares must satisfy if they exist. In the case of the commuting square obtained from the finite group Z_n, the sequence of commuting squares corresponds to a sequence of Hadamard matrices converging to the Fourier matrix F_n. We will present the main result from a paper by Dr. Nicoara and Dr. White that states the second order conditions follow automatically from the first order conditions in this case. We also state a conjecture of Bengtsson on the existence of matrices satisfying the third order conditions of the commuting square obtained from the finite group Z_n in the case where n is the product of three distinct primes.

]]>Title: Classification and symmetries of knotsAbstract: For the topologist, a (classical) knot is a smooth simple closed curve in 3-dimensional space. Two knots are equivalent if one can continuously deform one to the other: stretching and bending are allowed but cutting is prohibited. We can picture a knot as a length of rope with its ends glued together, and with a small amount of practice we can draw pictures of knots, seemingly making the subject approachable. However, knot theory is teeming with problems that are easy to ask and very hard to solve. Even the problem of deciding whether two given knots are equivalent can be tricky, and has caught people out over the years. A very useful approach is to transform a knot theoretical problem into a (hopefully) easier problem in algebra. Some of the simpler algebraic invariants will be presented.

]]>Affiliation: Wayne State U

Title: Construction of H^{2}(curl) conforming elements and their application

Abstract: In 1980 and 1986, Nedelec proposed H(curl)-conforming elements to solve electromagnetic equations that contains the “curl” operator. It is more or less as the H^{1}-conforming elements (or C^{0} elements) for elliptic equations that contains the “grad” operator. As is well known in the finite element method literature, in order to solve 4th-order elliptic equations such as the bi-harmonic equation, H^{2}-conforming elements (or C^{0}-elements) were developed. Recently, there have been some research in solving electromagnetic equations which involve four “curl” operators. Hence, construction of H(curl curl)-conforming elements becomes necessary. In this work, we construct H(curl curl)-conforming elements for rectangular and triangular meshes and apply them to solve quad-curl equations as well as related eigenvalue problems.

Title: Least-area Polyhedral Tiles of Spaces

Abstract: The cube is the least-area unit-volume polyhedron of six sides. What about other numbers of sides? We'll discuss what's known and make some guesses about other cases.

]]>TITLE: Isoperimetric Problems

ABSTRACT: The Ancient Greeks proved that the circle is the least-perimeter way to enclose given area in the plane. What about other spaces? Such isoperimetric questions are now playing as large a role as ever throughout mathematics and applications. We'll discuss some open questions and recent results, some by undergraduates.

]]>Title: "A New Theory of Fractional Differential Calculus and Fractional Sobolev Spaces"

Abstract: This talk presents a new theory of weak fractional differential calculus and fractional Sobolev spaces in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer order weak derivatives, it also helps to unify multiple existing fractional derivative definitions. Various calculus rules are established for the weak fractional derivatives. Based on the weak fractional derivative notion, new fractional order Sobolev spaces are introduced and many important properties, such as density theorem, extension theorem and trace theorem, of those Sobolev spaces are established. Moreover, their relationships with existing fractional Sobolev spaces are also established. The new theory lays down a solid theoretical foundation to systematically and rigorously develop a fractional calculus of variations theory and a fractional PDE theory in a subsequent talk.

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Abstract: In this presentation we shall explore two new families of fractional PDEs obtained as Euler-Lagrange equations of fractional calculus of variations problems. Several new fractional differential operators will be introduced, including the fractional $p$-Laplacian, Laplacian, and Neumann boundary operator. In each family of problems, we consider one-sided differentiation as well as differentiation in each direction; both in the weak sense. The first family of problems connects minimization problems with prescribed boundary conditions to associated fractional PDEs via the calculus of variations. The second family of problems establishes the connection between minimization problems with natural boundary conditions and fractional PDEs with Neumann boundary data. We prove the existence and uniqueness of weak solutions in the newly developed fractional Sobolev space(s) $\leftidx{^{\pm}}{W}{^{\alpha,p}}$. We also consider fractional PDEs for which there is no associated minimization problem. In addition to proving existence and uniqueness of solutions, we discuss the issue of choosing appropriate initial conditions and our interpretation of an initial value problem.

]]>Speaker: Jimmy Scott, University of Tennessee

Abstract: TBD

]]>SPEAKER: Guy David, Ball State University

Abstract: TBD

]]>Speaker: Anastasios Stefanou, Ohio State University

Abstract: TBD

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Abstract: In this talk I will present on recent results regarding a variant of Gamow's liquid drop model, initially developed to predict the shape of atomic nuclei, with an anisotropic surface energy. This universal model can be considered as a perturbation of the anisotropic isoperimetric problem by nonlocal long-range interactions. Under suitable regularity and ellipticity assumptions on the surface tension, Wulff shapes are minimizers in this problem if and only if the surface energy is isotropic. Moreover for smooth anisotropies, in the small nonlocality regime, minimizers converge to the Wulff shape in $C^1$-norm with a quantitative rate of convergence. It is also possible to obtain a quantitative expansion of the energy of any minimizer around the energy of the Wulff shape which yields a geometric stability result for minimizers. For certain crystalline surface tensions one can determine the global minimizer explicitly and obtain its exact energy expansion in terms of the nonlocality parameter. Results will be drawn from projects joint with Rustum Choksi, Robin Neumayer, and Oleksandr Misiats.

]]>SPEAKER: Chad Topaz, Williams College

ABSTRACT: From nanoparticle assembly to synchronized neurons to locust swarms, collective behaviors abound anywhere in nature that objects or agents interact. Investigators modeling collective behavior face a variety of challenges involving data from simulation and/or experiment. These challenges include exploring large, complex data sets to understand and characterize the system, inferring the model parameters that most accurately reflect a given data set, and assessing the goodness-of-fit between experimental data sets and proposed models. Topological data analysis provides a lens through which these challenges may be addressed. This talk consists of three parts. First, I introduce the core ideas of topological data analysis for newcomers to the field. Second, I highlight how these topological techniques can be applied to models arising from the study of groups displaying collective motion, such as bird flocks, fish schools, and insect swarms. The key approach is to characterize a system's dynamics via the time-evolution of topological invariants called Betti numbers, accounting for persistence of topological features across multiple scales. Finally, moving towards a theory of reduced topological descriptions of complex behavior, I present open questions on the topology of random data, complementing research in random geometric graph theory.

]]>Affiliation: ORNL

Title: TBA

Abstract: TBA

]]>SPEAKER: Dallas Albritton, UMN

ABSTRACT: TBD

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Title: TBC

]]>SPEAKER: Brian Katz, Augustana College

ABSTRACT: TBD

]]>SPEAKER: Markus Pflaum, University of Colorado Boulder

ABSTRACT: TBD

]]>SPEAKER: Mike Perlmutter, Michigan State University

ABSTRACT: The scattering transform is a mathematical framework for understanding convolutional neural networks (CNNs), which was originally introduced by Stephane Mallat for functions defined on Rn. Similarly to more traditional CNNs, the scattering transform features an alternating cascade of convolutions and nonlinearities. However, it differs by using pre-designed, wavelet filters rather than filters that are learned from training data. This leads to a network that provably has desirable invariance and stability properties.

In addition to these theoretical properties, the scattering transform can achieve near state of the art numerical results in settings such as quantum chemistry where the wavelets can be designed in correspondence with the underlying physics. Moreover, since it does not learn its filters from data, it is well-suited to limited data environments. I will provide an overview of Mallat’s original construction and also discuss recent variations of the scat- tering transform which are custom designed for certain tasks such as handling data with a non-Euclidean geometric structure.

Title: TBA

]]>SPEAKER: Shelby Scott, University of Tennessee

ABSTRACT: TBD

]]>SPEAKER: TBD

ABSTRACT: TBD

]]>Title: Stuck in Traffic

Abstract: We discuss how mathematics can play a role in analyzing large-scale traffic patterns. In particular, we discuss, time permitting, up to three problems; application of matrix factorization to traffic data, the topology of congestion, and understanding the interplay of traffic and safety. A traffic dataset from New York city will be used to illustrate all problems. The problems will be accessible to a broad mathematical audience.

TIME: 3:35 PM - 4:35 PM

ROOM: Ayres 405

HOST: Vasileios Maroulas

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