Wonwoo Kang (University of Illinois - Urbana Champaign) will speak about Skein relations for punctured surfaces

Abstract: Since the introduction of cluster algebras by Fomin and Zelevinsky in 2002, these algebras have garnered significant attention, particularly for their surface-type variants. These algebras enable the construction of various combinatorial structures, such as snake graphs, \(T\)-paths, and posets, which serve as tools for proving key properties like positivity and the existence of bases. In this talk, we will present a cluster expansion formula that employs poset representatives for arcs on triangulated surfaces. Leveraging these posets and the expansion formula, we will establish skein relations that resolve intersections or incompatibilities between arcs. As a result, we will demonstrate that bangles and bracelets form spanning sets and exhibit linear independence, thereby proving the existence of the bangle and bracelet bases in punctured surfaces with boundaries and closed surfaces of genus 0. This is a joint work with Esther Banaian and Elizabeth Kelley.

This presentation will describe opportunities at the National Security Agency that are available to undergraduate and graduate students in mathematics (who are US citizens). The speaker will discuss summer internships, full-time positions, and fellowships that students can apply to.

Note: Free refreshments. The talk starts at 5:40.

Proof is central to doing, communicating, and recording mathematics, which is why it is emphasized across various K–12 and advanced undergraduate education frameworks. However, little is known about students’ experiences with proof in early undergraduate courses. This study examined college calculus instructors’ conceptions of the nature of proof. Specifically, the study explores how instructors attend to, balance, and prioritize its dual aspects: the public rigor necessary for formal validation of statement and the private insights that foster an intuitive understanding of concepts that explain why statements are true. The study’s findings contribute to understanding of proof practice in the early undergraduate setting. Implications for educational practice and research are discussed.

In 1825 Gauss developed a measure of curvature for surfaces and then went on to establish a beautiful result on geodesic triangles and curvature. Later he developed a more technical approach to surface curvature, which became the standard. In this talk we will follow Gauss’s earlier approach, to understand the insight behind the result that he described as “among the most elegant” in the theory of curved surfaces.

Note: Free refreshments. The talk starts at 5:40.

In this talk we will present the (sketch of a) proof for an important technique in sub-Riemannian geometry known as the Extended Hadamard Technique. It represents one of the strongest general techniques to obtain the cut time and cut locus of geodesics in a sub-Riemannian manifold. We will apply it to the 2D example of the Grushin plane. Time permitting, we will also discuss the limitations of this technique and one or two other tactics that have been employed in the theory of sub-Riemannian cut times.

Hyun Kyu Kim (KIAS) will speak about Canonical bases for moduli spaces of local systems on a surface

Abstract: For a punctured surface S and a split reductive algebraic group G such as \(SL_n\) or \(PGL_n\), Fock and Goncharov (and Shen) consider two types of moduli spaces parametrizing G-local systems on S together with certain data at punctures. They show that these spaces have special coordinate charts, hence are birational to cluster varieties. Fock and Goncharov’s duality conjectures predict the existence of a canonical basis of the algebra of regular functions on one of these spaces, enumerated by the tropical integer points of the other space. I will give an introductory overview of this topic, briefly explain recent developments involving quantum topology and mirror symmetry of log Calabi-Yau varieties, and present some open problems if time allows.

Note: Free refreshments. The talk starts at 5:40.

In this talk, I will share practical advice for developing effective teaching practices, engaging diverse student populations, and balancing instructional responsibilities with research and service. Drawing on my experiences as a TA, instructor, and mentor, I will discuss ideas for course design, alternative assessments, and active learning strategies that foster student success. Specifically, we will explore opportunities available to graduate students that can enhance your prospects in the teaching job market.

Whether you are just starting as a TA or envisioning a long-term academic career, this talk will provide insights to help you grow as an educator while navigating the challenges of teaching in higher education.

The competition is for mathletes from local middle and junior high schools.  Members of the Department of Mathematics will help with the competition to be held in McHugh Hall 102.

This is the rescheduled event. Note a new location.

Tyler Genao (Ohio State University) will speak about Uniform polynomial bounds on torsion from rational geometric isogeny classes
Abstract: In 1996, Merel showed that for any elliptic curve \(E\) defined over a number field \(F\) of degree \(d\in\mathbb^+\), the size of the torsion group of \(E\) over \(F\) is bounded by a constant \(B:=B(d)\) which depends only on \(d\), and that conjecturally is in fact a polynomial in \(d\).
In this talk, I will discuss recent joint work with Abbey Bourdon which shows that \(B\) is polynomial in \(d\) for torsion from the family \(\mathcal_\) of elliptic curves which are geometrically isogenous to at least one rational elliptic curve. For torsion from the subfamily \(\mathcal_\) of elliptic curves with rational \(j\)-invariant, our results strengthen prior work of Clark and Pollack.

In the 19th century, Cantor discovered that infinities come in various sizes. In particular, he showed that the size (or cardinality) of the set of real numbers is strictly greater than the size of the set of natural numbers, even though both sets are infinite.

This discovery raised the question of whether there could be a size of infinity that is strictly between the size of the natural numbers and the size of the real numbers. The continuum hypothesis, written as CH, is the statement that there is no such infinite size. In this talk, we will explore these ideas in more detail and consider how mathematicians have tried to figure out whether CH is true or not.

Note: Free refreshments. The talk starts at 5:40.

Join us in the Logic Colloquium for a talk by Tyler Knowlton (Linguistics & Cognitive Science, U Delaware).

Details t.b.a.

https://logic.uconn.edu/calendar/