Current contact: Brandi Henry and Rebekah Palmer

The seminar takes place on Fridays at 1:00 pm in Room 617 on the sixth floor of Wachman Hall. Pizza and refreshments are available beforehand in the lounge next door.

• Friday February 1, 2019 at 14:30, Wachman Hall 617
Cannon's conjecture

Thomas Ng, Temple University

Abstract: The class of all groups (even finitely generated ones) are known to have wildly inefficient, in fact often unsolvable, algorithmic properties. It is thus helpful to specialize to well-behaved subclasses. In the early 1900s Dehn studied groups as geometric objects and showed that finitely generated groups arising as the fundamental group of manifolds in dimensions less that 4 are often very algorithmically efficient. Given the existence of these well-behaved groups, one is naturally led to ask how unique they are or under what conditions are they uniquely determined. Following Gromov's notion of boundary for a hyperbolic group introduced in 1987, Cannon conjectured that for hyperbolic groups the condition that its boundary is a 2-sphere uniquely determines 3-manifold fundamental groups. In this talk we will briefly review the notions of Cayley graph, hyperbolicity, and boundary of a group. We will then survey some of the recent results that inspired and attempt to solve the Cannon conjecture.

• Friday February 8, 2019 at 14:30, Wachman Hall 617
Resolution of singularities

Rebekah Palmer, Temple University

Abstract: To study structures that are not quite smooth, it is convenient to transform them so that we can apply the many tools developed for smooth structures. Given a variety with singularities, we ask if there is a consistent method of transformation into a non-singular variety. In this talk, we will introduce ourselves to some varieties with singularities, demonstrate how to uniquely resolve algebraic curves, and further discuss approaches to resolving higher-dimensional varieties.

• Friday February 15, 2019 at 14:30, Wachman Hall 617
Bead Movement in Biofilms: Effects of Curli Amyloids and Analysis of Density Dependence

Brandi Henry, Temple University

• Friday February 22, 2019 at 14:30, Wachman Hall 617
Classifying Actions of $T_n \otimes T_n$ on Path Algebras of Quivers

Delaney Aydel, Temple University

Abstract: Let $T_n$ denote the $n$th Taft algebra. We fully classify inner-faithful actions of $T_n \otimes T_n$ on four-vertex Schurian quivers as extensions of the actions of $\mathbb{Z}_n \times \mathbb{Z}_n$. One example will be presented in full, with the remaining results briefly given.

• Friday March 29, 2019 at 14:30, Wachman Hall 617
New off-ramp coupling conditions on the road

Najmej Salehi, Temple University

• Friday April 19, 2019 at 14:30, Wachman Hall 617
The Class Number One Problem

Dong Bin Choi, Temple University

Abstract: In 1952, Kurt Heegner proved (up to minor gaps) that the imaginary quadratic fields $K$ with class number $h(K) = 1$ are $\mathbb{Q}(\sqrt{d}$ with $d = -1, -2, -3, -7, -11, -19, -43, -67, -163$. To understand the problem in context, we begin from integer quadratic forms $Q(x,y) = ax^2 + bxy + cy^2$ and the question of what numbers $n$ occur as solutions to $ax^2 + bxy + cy^2 = n$ for a given $a, b, c$. This depends on the equivalence class of the quadratic form, which turns out to be closely related to ideal classes of the quadratic integer ring with discriminant $b^2 - 4ac$. We touch on some number-theoretic notions used in Heegner's proof, such as the genus of a quadratic form and the $j$-invariant of lattices. We sketch Heegner's proof (following Kezuka 2012), and conclude with solutions to some generalizations of the problem.

• Friday September 6, 2019 at 13:00, Wachman Hall 617
Welcome Back

• Friday September 13, 2019 at 13:00, Wachman Hall 617
Explicit power laws in analytic continuation problems via reproducing kernel Hilbert spaces.

Narek Hovsepyan, Temple University

Abstract: The need for analytic continuation arises frequently in the context of inverse problems. We consider several such problems and show that they exhibit a power law precision deterioration as one moves away from the source of data. We introduce a general Hilbert space-based approach for determining these exponents. The method identifies the "worst case" function as a solution of a linear integral equation of Fredholm type. In special geometries, such as the circular annulus, an ellipse or an upper half-plane the solution of the integral equation and the corresponding exponent can be found explicitly.

This is a joint work with Yury Grabovsky

• Friday September 27, 2019 at 13:00, Wachman Hall 617
Groups Acting on Trees

Rylee Lyman, Tufts University

Abstract: We introduce the Bass–Serre theory of groups acting on trees. Two common constructions on groups are the amalgamated free product and HNN extension. We begin by reviewing these constructions. Bass–Serre theory generalizes these constructions within the common framework of an action of the group on a tree. When the group is finitely generated, examining the quotient yields a presentation for the group, as well as other algebraic information. We describe the notion of the fundamental group of a graph of groups, a useful tool in the study of finitely generated groups, and, time permitting, famous theorems of Stallings and Dunwoody.

• Friday October 4, 2019 at 13:00, Wachman Hall 617
Modeling an Action Potential and Neuronal Behavior

Abstract: In this talk we explore the Hodgkin-Huxley conductance-based model which is used to describe the initiation and propagation of an action potential. We will also explore the different levels of modeling neuronal behavior and interactions: how is an action potential initiated within a neuron, how does an action potential induce communication between neurons, how do we model the interactions of neuronal networks? We will also present the current research that is being done to model intracellular behavior of a neuron.

• Friday October 11, 2019 at 13:00, Wachman Hall 617
Extension of groups

Khanh Le, Temple University

Abstract: In this talk, we will introduce the concept of an extension of groups. Roughly speaking, given two groups $G$ and $K$ what are the different ways to write down a short exact sequence $1 \to K \to E \to G \to 1$? To understand this question, we will naturally introduce the idea of group cohomology. After that, we explore different examples of extensions of groups.

• Friday October 18, 2019 at 13:00, Wachman Hall 617
Some invariant metrics under biholomorphic maps

Elie Abdo, Temple University

Abstract: Several theorems that hold in the theory of one complex variable cannot be generalized to the theory of several complex variables, one of them is the Riemann Mapping Theorem. In fact, invariant metrics are important tools that can be used to show that the open bi-disc and the open unit-ball in $\mathbb{C}^2$ are not biholomorphic. In this talk, we introduce some invariant metrics, the Kobayashi, Caratheodory and Sibony metrics, list some of their properties, and do some interesting examples.

• Friday October 25, 2019 at 13:00, Wachman Hall 617
Weyl's Law: From Isometries to Eigenvalues

Lindsay Dever, Bryn Mawr College

Abstract: Complex-valued, square-integrable functions on compact, hyperbolic 3-manifolds decompose uniquely into special functions called automorphic forms. These special functions are eigenfunctions of the Laplacian operator which are invariant under a discrete group of isometries. In general, it is a hard problem to determine the eigenvalues explicitly, but we do know how many eigenvalues to expect asymptotically. The asymptotic formula for the number of eigenvalues is known as Weyl's law. The proof of Weyl's law involves the Selberg trace formula, which connects these eigenvalues to geometric information about the manifold. In this expository talk, I will introduce hyperbolic 3-manifolds, define automorphic forms, and give a proof of Weyl's law for compact, hyperbolic 3-manifolds.

• Friday November 1, 2019 at 13:00, Wachman Hall 617
Optimal Finite Volume Limiter Functions

Abhijit Biswas, Temple University

Abstract: Hyperbolic conservation law has discontinuous solutions, even if the initial condition is smooth. We want to have schemes that are both accurate in smooth regions and non-oscillatory near discontinuities or sharp transition. Typical finite volume linear schemes can not fulfill those two desired properties simultaneously, even for linear advection problems. One way to get those two desired properties is to use limiter functions with a high order accurate linear scheme, for example, the Lax-Wendroff method. Many limiter functions have been introduced in the literature, and the general approach is to design a limiter function and demonstrate that it performs well on some test problems. Here we wish to investigate the inverse problem instead: given a portfolio of representative test cases, and a cost functional, determine the optimal limiter function.

• Friday November 8, 2019 at 13:00, Wachman Hall, 617
The adjoint action of group algebras

Ramy Yammine, Temple University

Abstract: In this talk we discuss the adjoint action of a group algebra k[G] on itself. We focus primarily on the ¨finite part¨ of this action, a group-subalgebra of k[G] that is better behaved. We will then discuss some interesting problems and questions about k[G] that reduce to its finite part.