In a recent paper of Pierce, Turnage-Butterbaugh, and Wood, the authors proved an effective Chebotarev density theorem for families of number fields. Notably D_4-quartic fields have not been treated in their paper. In this talk, I will explain the importance of the case for D_4-quartic fields and will present my proof of a Chebotarev density theorem for certain families of D_4-quartic fields. The key tools are a lower bound for the number of fields in the families and a zero-free region for almost all fields in the families.

The world is in dire need of more formalisms with which to do differential geometry. Just kidding! More seriously, over the years I have encountered some complicated formulae in differential geometry and have, successfully I believe, used canonical objects sitting on frame bundles to simplify them and their proofs. I will give at least one example of such a formula, namely Simon's calculation of the Laplacian of the second fundamental form of a minimal submanifold and show how the formalism makes it and its proof simple.

Another Job Audition.

Biological physical networks, especially those involved in resource delivery and distribution, often exhibit a hierarchical structure. Quantifying this structure is crucial to obtaining a better understanding of the processes underlying the network formation, and such a quantification has long been obtained using the Horton-Strahler ordering scheme. The scheme assigns an integer order to each edge in the network based on the topology of branching such that the order increases from distal parts of the network (e.g., mountain streams or capillaries) to the ``root'' of the network (e.g., the river outlet or the aorta). However, Horton-Strahler ordering cannot be applied to networks with loops because they they create a contradiction in the edge ordering in terms of which edge precedes another in the hierarchy. In this talk I will present a generalization of the Horton-Strahler order to weighted planar reticular networks, where weights are assumed to correlate with the importance of network edges, e.g., weights estimated from edge widths may correlate with flow capacity. The new method assigns hierarchical levels not only to edges of the network, but also to its loops, and classifies the edges into reticular edges, which are responsible for loop formation, and tree edges. I will show that the sensitivity of the hierarchical levels to weight perturbations can be analyzed in a rigorous way. I will also discuss applications of this generalized Horton-Strahler ordering to the study of leaf venation and other biological networks.

In representation theory, you learn about compact Lie groups. In algebraic geometry, you learn about complex algebraic varieties. In this talk, we will see how these two notions come together in the world of Abelian varieties. These are important objects in algebraic geometry, which carry some pretty remarkable and unexpected properties. We will try to cover as many of these as possible. The talk will focus on complex Abelian varieties, so some familiarity with complex analysis would be helpful. Those currently in the Riemann surfaces course may find some interesting parallels.

Consider the following three questions: How many uniformly random permutations are needed to invariably generate the symmetric group? How many iterations of a Monte-Carlo algorithm are needed to decide whether a polynomial in Z[x] splits? How many sumsets formed from independent Poisson multistep are needed to have a chance of an empty intersection? And their answer: Four. What is so special about four? Not much. These all are special cases of the ubiquitous Ewens sampling formula. Come and find out why.

The nephron is the functional unit of the kidney. The flow rate in each nephron is regulated, in part, by tubuloglomerular feedback, a negative feedback loop. In some parameter regimes, this feedback system can exhibit oscillations that approximate limit-cycle oscillations. However, nephron flow in spontaneously hypertensive rats (SHR) can exhibit highly irregular oscillations similar to deterministic chaos. We used a mathematical model of tubuloglomerular feedback (TGF) to investigate potential sources of the irregular oscillations and the associated complex power spectra in SHR. A bifurcation analysis of the TGF model equation was performed by finding roots of the characteristic equation, and numerical simulations of model solutions were conducted to assist in the interpretation of the analysis. Four potential sources of spectral complexity in SHR were identified: (1) bifurcations that produce qualitative changes in solution type, leading to multiple spectrum peaks and their respective harmonic peaks; (2) continuous lability in delay parameters, leading to broadening of peaks and their harmonics; (3) episodic lability in delay parameters, leading to multiple peaks and their harmonics; and (4) coupling of small numbers of nephrons, leading to broadening of peaks, multiple peaks, and their harmonics. We conclude that the complex power spectra in SHR may be explained by the inherent complexity of TGF dynamics, which may include solution bifurcations, variation in TGF parameters, and coupling between small numbers of neighboring nephrons.

TBA

Oh rhythm of my heart is beating like a drum...... with the words "I love you" rolling off my tongue....... No never will I roam for I know my place is home..... where the ocean meets the sky...... I'll be sailing.....

A body of literature has developed concerning “cloaking by anomalous localized resonance”. Most analytical work in this area has relied on separation of variables, and has therefore been restricted to radial geometries. In this talk, we will discuss a new approach based on a pair of dual variational principles, and apply it to some non-radial examples. In our examples, as in the radial setting, the spatial location of the source plays a crucial role in determining whether or not resonance occurs. The talk assumes minimal background knowledge.

I will discuss the rich mathematical structures which arise when one asks how to define acceleration in the absence of a preferred coordinate system. I will introduce the Yang-Mills equations, which specialize to give electromagnetism and much of the physics of the standard model. I'll discuss aspects of the geometry, topology, and analysis of the Yang-Mills equations and how too much symmetry can actually make an analysis problem more difficult.

A polytope is a bounded subset of R^d which is the intersection of finitely many half-spaces. Given a polytope P, we can consider integer dilations of P, and ask how many integer points are contained in each dilation, as a function of the dilation factor. Under the right conditions, this counting function is a polynomial with some very interesting and unexpected properties. To demonstrate the usefulness of these results, we will give alternative proofs to some well known results from far outside the realm of geometry.

To a Legendrian knot, one can associate an $\mathcal{A}_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two ends. We study the functor and establish a long exact sequence relating the corresponding Legendrian cohomology categories of the two ends. As applications, we prove that the functor between augmentation categories is injective on objects, and find new obstructions to the existence of exact Lagrangian cobordisms. The main technique is a recent work of Chantraine, Dimitroglou Rizell, Ghiggini and Golovko on Cthulhu homology.

A toric variety is a special class of algebraic variety, which can be expressed by a picture that makes some of it's properties much easier to analyze. These varieties are an area of ongoing research in algebraic geometry, due in part to their applications in many different fields (such as mathematical physics). In this talk I will discuss how we can use a fan to construct a toric variety as a quotient. This talk will be accessible to all graduate students and will focus on examples, to illustrate the usefulness of this method.

Beyond Krull dimension, rings and modules have various "dimensions": depth, height, projective/injective dimensions, flat dimension, global dimension, weak dimension, among others. These notions are defined homologically in terms of lengths of resolutions, and Ext and Tor functors provide a way to measure them. I'll talk about how they are related with each other. I'll start from geometic interpretation of Krull dimension and height, followed by regular sequences and depth. Then I'll introduce Cohen-Macaulay modules and Gorenstein modules as modules having particularly nice homological properties. Just as in the case of completion where analysis is introduced to algebra to prove some highly nontrivial results, homological techniques have proved to be very powerful in modern commutative algebra, producing such surprising results as homological characterization of regular rings (Serre, Auslander, Buchsbaum). I'll briefly introduce the notion of canonical modules and the question of finiteness of injective resolution. Finally, I'll talk about how these notions can be globalized to scheme and sheaves, which makes geometry "kind of equivalent" to algebra.

The instanton equations appear in gauge theory and generalize both the Maxwell equations and the harmonic equation. Their study has been and continues to be a very fertile ground for interactions between physicists and mathematicians. The object of this talk is a description of instanton solutions on S^1xR^3 due to Hurtubise and myself using the Nahm transform, a non-linear transformation that takes a system of PDE and produces a system of ODE or even a system of algebraic equations. This description allows us to answer existence questions for calorons.

The theory of theta functions, which are defined by certain Fourier series, was developed by great mathematicians like Jacobi and Riemann. Among the numerous applications of this theory are certain results in projective geometry pertaining to complex tori. In this talk, we will focus on the 1-dimensional case and briefly discuss the higher-dimensional story towards the end.

(Bio)chemical reaction networks are used to model processes that occur in cell biology. A fundamental problem is to characterize chemical reaction networks which admit multiple steady states. The existing literature has focused on identifying necessary conditions for multiple steady states. A natural (but significantly more difficult) problem is to determine sufficient conditions. In joint work with Anne Shiu, we suggest an approach for tackling this problem by defining certain minimal multistationary networks called 'atoms of multistationarity'. These 'atoms' are analogous to prime numbers in the theory of integers, in the sense that every multistationary network is either an atom or contains an atom. The talk will contain many examples from biology and will assume nothing more than calculus.

In this talk we will explore a mathematical data analysis tool called persistent homology and look specifically into how we can turn topological information into useful data for statistical techniques. The problem is one of translation: persistent homology outputs a module, but statistics is formulated for objects in metric, vector, Banach, and Hilbert spaces. We'll see some of the ways this issue can be dealt with in a special case (single-parameter persistence) and discuss which of those techniques are viable for a more general case (multiparameter persistence).

In this talk I will present a ODE model Dr. Schaeffer and I have developed in collaboration with Dr. Magwene of Duke's Department of Biology. The cAMP-PKA pathway is a key signal transduction pathway through which Yeast makes developmental decisions in response to environmental cues. A novel feature of our model is that for a wide range of parameters approach to steady state includes decaying oscillations. I aim to make this talk accessible to everyone and will give an overview of all relevant biology.

Many systems employ sensors to interpret the environment. The target-tracking task is to gather sensor data from the environment and then to partition these data into tracks that are produced by the same target. The goal of sensor fusion is to gather data from a heterogeneous collection of sensors (e.g, audio and video) and fuse them together in a way that enriches the performance of the sensor network at some task of interest. This talk summarizes two recent efforts that incorporate mildly sophisticated mathematics into the general sensor arena, and also comments on the joys and pitfalls of trying to apply math for customers who care much more about the results than the math. First, a key problem in tracking is to 'connect the dots:' more precisely, to take a piece of sensor data at a given time and associate it with a previously-existing track (or to declare that this is a new object). We use topological data analysis (TDA) to form data-association likelihood scores, and integrate these scores into a well-respected algorithm called Multiple Hypothesis Tracking. Tests on simulated data show that the TDA adds significant value over baseline, especially in the context of noisy sensor data. Second, we propose a very general and entirely unsupervised sensor fusion pipeline that uses recent techniques from diffusion geometry and wavelet theory to compress and then fuse time series of arbitrary dimension arising from disparate sensor modalities. The goal of the pipeline is to differentiate classes of time-ordered behavior sequences, and we demonstrate its performance on a well-studied digit sequence database. This talk represents joint work with many people. including Chris Tralie, Nathan Borggren, Sang Chin, Jesse Clarke, Jonathan deSena, John Harer, Jay Hineman, Elizabeth Munch, Andrew Newman, Alex Pieloch, David Porter, David Rouse, Nate Strawn, Adam Watkins, Michael Williams, Lihan Yao, and Peter Zulch.

Models for dynamical systems that include short-time or abrupt forcing can be written as impulsive differential equations. Applications include mechanical systems with impacts and models for electro-chemical spiking signals in neurons. We consider a model for spiking in neurons given by a nonlinear ordinary differential equation that includes a Dirac delta function. Ambiguities in how to interpret such equations can be resolved via perturbation methods and asymptotic analysis of delta sequences.

A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on~$\Gamma_0(N)$ to a simpler function of $k$ and~$N$, showing that the two are equal whenever $N$ is squarefree. We prove the converse of this theorem (with one small exception), thus providing a characterization of squarefree integers. We also establish a similar characterization of prime numbers in terms of the number of Hecke newforms of weight $k$ on~$\Gamma_0(N)$. It follows that a hypothetical fast algorithm for computing the number of such automorphic representations for even a single weight $k$ would yield a fast test for whether $N$ is squarefree. We also show how to obtain bounds on the possible square divisors of a number $N$ that has been found to not be squarefree via this test, and we show how to probabilistically obtain the complete factorization of the squarefull part of $N$ from the number of such automorphic representations for two different weights. If in addition we have the number of such Hecke newforms for even a single weight $k$, then we show how to probabilistically factor $N$ entirely. All of these computations could be performed quickly in practice, given the number(s) of automorphic representations and modular forms as input. (Joint work with Greg Martin.)