Current Schedule

2/6/26

Just my Pot Luck!

2/20/26

Channing Bentz

Title: Derived Categories 

Abstract: When working in a general category, sometimes we would like to say that things "are the same" up to a weak equivalence. How do we do this? Localization! In this talk I will talk about localization of a category on a weak equivalence and then look at it's application to the category of chain complexes over an abelian category. This is what is known as the derived category and it has a lot of fun properties! 

2/27/26

Matt Burnham

Title: Spectral theory of K_t-decomposable graphs

Abstract: The q-Laplacian of a graph G is the matrix L^{(q)} = A + qD where A is the adjacency matrix and D the diagonal degree matrix. A graph is called K_t-decomposable if the edges of G can be partitioned into edge-disjoint copies of K_t. The 1-Laplacian A + D is a well-studied matrix called the signless Laplacian. In this talk, we give an overview of well-known results about the spectrum of the signless Laplacian, and then show how these results extend to the 1/(t-1)-Laplacian of K_t-decomposable graphs.

3/6/26

Micah Coats

Title: The Perimetric Inequality, some Hyperbolic Geometry, and Symmetrization of functions on Hyperbolic Space

Abstract: Steiner Symmetrization on a function f:R^n->R is a rearrangement of f such that measures of level sets are unchanged and where the rearrangement f* is radially symmetric and decreasing. It has been used as a tool for some optimization problems and classically was important for proving the isoperimetric inequality. In this talk we discuss a technique used to prove many fascinating properties of symmetrization that also has a hyperbolic analogue. A short discussion of hyperbolic geometry and its transformations is included to explain the technique.

3/13/26

Joanna Held

Title: Micro-macro decomposition scheme for the kinetic Vlasov system.  

Abstract: Kinetic theory is a classical description of the thermodynamic behavior of gases. In this model, the movement and collisions of particles govern the large-scale behavior of a system. However, kinetic equations like the Vlasov-Poisson which describes the behavior of an electrostatic plasma are computationally challenging as they exist in a high-dimensional phase space, these problems are multiscale in nature, and dynamics change based on which collisional regime the system is in. There are a variety of ways to handle these challenges, but one is to apply a so-called micro-macro framework to the Vlasov system in order to combine fluid representations while still resolving the kinetic or particle effects. In this talk, I will discuss this framework and show examples applying it to the 1D1V Vlasov-Poisson equation. 

3/20/26

Spring Break!

3/27/26

Dajuan Kinney

Title: Where are the balls? A Look Into The Mathematics of Juggling 

Abstract: Interest in the mathematics of juggling has grown significantly over the past three decades. This presentation provides a brief overview of several discrete mathematical methods used to describe and analyze juggling patterns. Representations such as \textbf{siteswaps}, \textbf{juggling cards}, and \textbf{state graphs} will be introduced as tools for encoding and studying patterns. The concept of \textbf{multiplex juggling patterns} will be defined, along with methods for counting them. Connections between juggling structures and classical combinatorial objects, including \textbf{Stirling numbers of the second kind} and \textbf{Bell numbers}, will also be explored. If time permits, audience members will have the opportunity to demonstrate their own juggling skills—and the presenter will be juggling as well. 

4/3/26

Joe Miller

Title: Certifying the block restricted isometry property for totally symmetric subspaces 

Abstract: In compressed sensing, the goal is to solve an underdetermined system Ax = b with the assumption that x is sparse, i.e. has many zeros. There are multiple efficient solvers which only work if A satisfies the restricted isometry property (RIP). It’s well-known that random matrices satisfy RIP with high probability, but it is an NP-hard problem to check if a specific matrix actually does. This led to the problem of constructing matrices which can be verified to satisfy RIP. This open problem has seen little progress in the last 15 years, stuck behind the “Gershgorin bottleneck”: a simple upper bound for our RIP constants. In this talk, we’ll see how representation theory of the symmetric group allows us to construct block matrices which satisfy block RIP and beat the Gershgorin bottleneck. 

4/10/26

Billy Duckworth

Title: Generic Properties of "Nice" Functions

Abstract: Much of our undergraduate exposure to functions gives us the illusion that all functions are "nice". When we reach our first analysis class we learn that general functions can be quite pathological, but our collections of "nice" functions are a safe sandbox to carry out most of our work. In this talk we will investigate just how scary generic "nice" functions can really be.

4/17/26

Laura Gamboa Guzman

Title: Automata and temporal logic: dealing with intervals and finite time.

Abstract: The most widely used formalism for logics that reason about time is Linear Temporal Logic (LTL), which allows reasoning about events over an infinite discrete timeline. The LTL language can be seen as an extension of the classic propositional logic, with connectives that let us express statements like "p is true until q becomes true" or "it will always be the case that p will become true sometime in the future". However, in many other frameworks, this approach is not the most suitable, as it becomes impractical when working with finite timelines and specifying fixed time intervals. These issues motivated the development of Mission-Time Temporal Logic (MLTL), whose language is similar to LTL's, but where each temporal operator is followed by a (closed) interval, allowing us to express statements such as "every request will be granted within 30 seconds" in a much simpler way. Although the MLTL logic can be seen as a fragment LTL, the approaches that have been developed to solve problems such as the satisfiability problem for LTL, which typically involves encoding a given LTL formula into a Büchi Automata do not work very well for MLTL, as the complexity of the known algorithms ends up being exponential in the length of the intervals and not just the number of temporal operators in the formula. In this talk, I will present ongoing work on encoding MLTL formulas using different types of automata to avoid an exponential increase in state space.

4/24/26

Kylie Schnoor

Title: Frames with Rational Eigensteps

Abstract: Frames can be seen as generalizations of bases that have many uses in both data science and machine learning applications. In this work, we explore a correspondence between frames with rational eigensteps and semistandard Young tableaux (SSYT), via the relation assigning a Gelfand-Tsetlin pattern to a frame via the frame's eigensteps. First, we begin with an introduction to frame theory as well as introducing the relevant combinatorial background. We will identify how certain key structures in SSYTs correlate with particular frame properties. Additionally, this correspondence leads to a novel way to construct the eigensteps of a frame coming solely from tableaux. This results in a combinatorial reinterpretation of known algorithms in frame theory. Leveraging combinatorial literature, we further discuss how a frame's Naimark complement may be represented combinatorially leading to a further discussion on the generalization of such a complement for non-tight frames. Further research points to an analysis of equiangular tight frames and their corresponding tableaux, as well as using more combinatorial operations to further analyze frames.

5/1/26

Micah Coats

Title: Minkowski space: an application of Pseudo-Riemannian Manifolds to Physics

Abstract: Special Relativity and its larger cousin, General relativity, is in large part the study of vector fields on a Pseudo-Riemannian manifold called the Minkowski space. Calculus of variations allows us to give meanings to the terms ‘electric and magnetic fields’ while the geometry of Minkowski space tells us how fields transform under coordinate transformations. Some beautiful consequences of these coordinate transformations are the various rules about how time and distance warp with speed, electric field transformations, and Einstein’s mass-energy law.

5/8/26

Kean Fallon

Title: Equiangular Tight Frames in Symplectic Spaces

Abstract: In the classical settings of real Euclidean and complex Hermitian vector spaces, equiangular tight frames (ETFs) are collections of optimally arranged lines that are not only mathematically beautiful but see a diverse array of applications. In this talk, we leave the comfort of these settings for the land of symplectic geometry. We will introduce symplectic ETFs and discuss research both old and new. In particular, we will compare and contrast the existence and construction of ETFs in real and complex symplectic space with those in the corresponding classical spaces.