President's Day
Monday, Feb 23, 2026
Homin Lee (Korea Institute of Advanced Studies)
Title: Singularity conjecture in higher rank
Abstract: In this talk, we discuss about random walk on a subgroup of simple Lie groups and hitting measures on their boundary. For instance, Furstenberg showed that, using discretization of Brownian motion, there is a driving measure on SL(2,Z) so that the hitting measure (Furstenberg measure) on the boundary circle is absolutely continuous with respect to Lebesgue measure. On the other hand, it is conjectured that if the driving measure is finitely supported then the hitting measure (Furstenberg measure) is singular.
We will discuss background and motivations on singularity conjecture and related results. And then, we will discuss about our singularity result on higher rank case. This is the joint work with Wouter van Limbeek and Giulio Tiozzo.
Tuesday, Mar 3, 2026 in TMCB 203 at 3 pm (Note: different day of the week and room)
Josefien Kuijper (University of Toronto)
Title: All K-theory is squares K-theory: constructing a derived Euler characteristic
Abstract: Combinatorial (or “cut-and-paste”) K-theory is a modern approach to the study of classical scissors congruence groups, inspired by algebraic K-theory of Waldhausen categories, and can be applied to other geometric settings as well, such as the categories of varieties and semi-algebraic sets. We present the K-theory of squares category as a framework that unifies Waldhausen K-theory as well as many instances of combinatorial K-theory. As an application, we lift the Euler characteristic for definable sets in an o-minimal structure to a map of K-theory spectra.
Monday, Mar 9, 2026
Nur Saglam (University of Georgia)
Title: Construction of Exotic 4-Manifolds Using Finite Order Cyclic Group Actions
Abstract: In this talk, we will discuss the construction of exotic 4-manifolds using Lefschetz fibrations over S^2, which are obtained by finite order cyclic group actions on Sg. We will first apply various cyclic group actions on Sg for g>0, and then extend it diagonally to the product manifolds SgxSg. These will give singular manifolds with cyclic quotient singularities. Then, by resolving the singularities, we will obtain families of Lefschetz fibrations over S^2. Following the resolution process, we will determine the configurations of the singular fibers and the monodromy of the total space. In some cases, deformations of the Lefschetz fibrations give rise to nice applications using the rational blow-down operation, which provides exotic examples. This is a joint work with A. Akhmedov and M. Bhupal.
Monday, Mar 16, 2026
Jason Day (University of Houston)
Title: Measures of maximal entropy for shift spaces via dimension theory
Abstract: To study the probabilistic properties of a dynamical system, we must pair the system with an invariant probability measure. Among these measures, measures of maximal entropy or MMEs are of particular importance as they can gather the most information about the dynamics of the system. I will discuss a new method to construct MMEs for shift spaces that uses dimension theoretic techniques analogous to Hausdorff dimension. This construction can also be applied to construct MMEs for Sinai billiards. By coding the Sinai billiard, we can use this construction to show that any Sinai billiard has a unique MME. This is joint work with Vaughn Climenhaga.
Monday, Mar 23, 2026
Liam Ashton (Purdue University)
Title: Decomposing the Goldman-Turaev Lie bialgebra along a simple separating curve
Abstract: The vector space spanned by free homotopy classes of curves in an oriented surface carries a Lie algebra structure with bracket defined by resolving transverse intersections between two curves. Incorporating self-intersections yields a compatible cobracket obtained by splitting a curve at its double points. Together, these operations form the Goldman-Turaev Lie bialgebra of a surface, a construction that has influenced diverse areas of geometry, topology, and mathematical physics, yet its structure remains only partially understood.
In this talk, I will describe a way of decomposing the Goldman-Turaev Lie bialgebra along any simple separating curve on the surface. This splitting produces two new algebraic structures built from intersecting paths with fixed endpoints on the separating curve. Each of these structures features the surprising appearance of a "double bracket," a 2 to 2 operation satisfying identities introduced by Van der Bergh in the context of non-commutative geometry, and a corresponding "double cobracket" which has been previously undefined. I will also explain how to recover the original Lie bialgebra from the resulting two pieces of structure.
Tuesday, Mar 31, 2026 in TMCB 203 at 3 pm (Note: different day of the week and room)
Wooyeon Kim (Korea Institute of Advanced Studies)
Title: Distribution of the values of ternary quadratic forms
Abstract: The Oppenheim conjecture, proved by Margulis in 1986, states that for a non-degenerate indefinite irrational quadratic form Q in n= 3 variables, the image set Q(Z^n) of integral vectors is a dense subset of the real line. Determining the distribution of values of an indefinite quadratic form at integral points asymptotically is referred to as quantitative Oppenheim conjecture. This problem was resolved by Eskin, Margulis, and Mozes for quadratic forms in n= 4 variables.
In this talk, we discuss the case of ternary quadratic forms (n=3). The approach uses ideas from homogeneous dynamics, where the problem is translated into an equidistribution problem for certain unipotent flows on the space of 3-dimensional lattices. The main ingredient is a strong quantitative non-divergence estimate, which controls how long these unipotent flows can spend deep in the cusp of the space of lattices.
Monday, Apr 6, 2026
Abraham Harris (Brigham Young University)
Title: TBD
Abstract: TBD
Monday, Apr 13, 2026
Joseph James (Brigham Young University)
Title: TBD
Abstract: TBD
Fall 2025
Seminars will be on Wednesdays at 1:00 pm - 1:50
pm in 225 TMCB. If you would like to participate,
please email Dr. Kent.
*-speakers to be confirmed.
Wednesday, Sept 10, 2025
Davi Obata (Brigham Young University)
Title:
Some rigidity results in random dynamics
Abstract:&Suppose that you have two dynamical systems and, at each step, you have some fixed probability of choosing one or the other. In this talk, I will survey some recent results on trying to understand statistical properties of such systems in the presence of chaotic behavior. In particular,I will focus on results that allow us to classify all of the stationary measures of such systems.
Wednesday, Sept 17, 2025
No seminar this week.
Wednesday, Sept 24, 2025
Curtis Kent (Brigham
Young University)
Title: Endpoint compactifications of infinite type surfaces
Abstract: We will discuss relationships between the fundamental group of the endpoint compactificaion of an infinite type surface and the mapping class group of a surface.
Wednesday, Oct 1, 2025
Samantha Sandberg-Clark (Ohio State Univesity)
Title: Generalized Non-Autonomous Parabolic Bifurcation
Abstract: Lavaurs showed that autonomous perturbations of g(z)=z+z^2 converge to, approximately, the identity under iteration. Vivas considered a related problem: what non-autonomous perturbations on the Möbius transformation h(z)=z/(1-z) give convergence to the identity under iteration. We build on this work by considering general Möbius transformations and the conditions needed to guarantee convergence.
Wednesday, Oct 8, 2025
Eric Swenson (Brigham Young University)
Title: Tits Rigidity for the join of three Cantor sets
Abstract:
Wednesday, Oct 15, 2025
Stephen McKean (Brigham Young University)
Title: Explicit bases for rational spin bordism
Abstract: Anderson, Brown, and Peterson calculated all spin and spin^c bordism groups in the 1960s, but their methods do not provide explicit manifolds representing these bordism classes. I will talk about joint work with Jonathan Buchanan, Arun Debray, and Cameron Krulewski, in which we give explicit manifolds representing the rationalization of these bordism classes. A fair amount of the talk will be spent defining bordism groups, spin structures, and so on.
Wednesday, Oct 22, 2025
Spencer Durham* (...)
Title: KAM and cohomological rigidity for high-rank affine parabolic actions
Abstract: Consider two commuting affine maps on the torus $a=Ax+\alpha$ and $b=Bx+\beta$. If we smoothly perturb this action, will it remain equivalent to the initial action up to a change of coordinates? In the case that A and B are partially hyperbolic (Damjanovic-Katok) or both identity (Moser) the answer is known, but in the case that A and B are parabolic (upper triangular with 1s on the diagonal), only limited results were previously available. I will discuss how KAM theory can be used to reduce this question to a question about the a well studied object in dynamical systems: the cohomological equation. To conclude, I will mention some results on the cohomological equation in the parabolic case that allow for the resolution of this problem. This is joint work with Bassam Fayad.
Wednesday, Oct 29, 2025
Jospeh James (Brigham Young University)
Title: Order of the Shape Kernel for Peano Continua
Abstract: It is conjectured that the shape kernel of a Peano continuum must either be trivial or uncountable. We show some additional hypotheses under which the conjecture is true and discuss the relationship of the Peanified Outer Wave Space to the class of spaces not covered by those hypotheses. The talk will begin with a quick overview of the shape kernel and some related results about the fundamental groups of Peano continua.
Wednesday, Nov 5, 2025
Yigal Kamel (University of Illinois Urbana-Champaign)
Title: Spin bordism, Reality, and C*-algebras
Abstract:
Bordism theories play a central role in both stable homotopy theory and application to topology and differential geometry. For instance, the various spin bordism spectra have been essential in the (partial) classification of manifolds that admit a positive scalar curvature metric by Lichnerowicz, Hitchin, Gromov—Lawson, Rosenberg, and Stolz; and the C_2-equivariant Real bordism spectrum plays a central role in the solution to the recently resolved Kervaire invariant problem concerning smooth structures on spheres by Hill—Hopkins—Ravenel. Spin and Real bordism share a common feature, which is that they are approximated by topological K-theory spectra via highly structured genera, a.k.a. orientations. In this talk, I will first introduce a new C_2-equivariant bordism theory called Real spin bordism, that simultaneously refines and unifies these orientations. Second, I will introduce a family of bordism theories for manifolds with "spin structure twisted by a C*-algebra, A,” which (weakly) orient the K-theory of A, and point towards potential applications to positive scalar curvature. This talk contains (pair-wise) joint work with Zach Halladay, Hassan Abdallah, and Fredrick Mooers.
Wednesday, Nov 12, 2025
Stephen Humphries (Brigham Young University)
Title: Some aspects of braid groups
Abstract: I would like to talk about a way to represent free groups as groups generated by transvections (inspired by a paper of Magnus) and use this to study representations of braid groups with an eye to constructing some polynomials and operators that one can use to find the intersection number of two simple closed curves on a planar surface. If there is time I will mention a connection to generalizations of the twin primes conjecture.
Wednesday, Nov 26, 2025
Thanksgiving break
Wednesday, Dec 3, 2025
Tyler Evans (Brigham Young University)
Title: Centralizers of Anosov Flows
Abstract: Given two $C^1$ flows, $X$ and $Y$, we say that $X$ and $Y$ commute if $X_t \circ Y_s = Y_s \circ X_t$ for all $s, t \in \mathbb{\R}$. The centralizer of the flow $X$, denoted $Z(X)$, is the collection of flows that commute with $Y$. We say that $Z(X)$ is trivial if for all $Y \in Z(X)$, there is some constant $c$ such that $Y = c X$. It is conjectured that for a typical flow, the centralizer is trivial (inspired by a conjecture of Smale for discrete dynamical systems). In this talk, we will show that Anosov flows on compact smooth manifolds have trivial centralizers.
Wednesday, Dec 10, 2025
Adam Call* (Brigham Young University)
Title: TBD
Abstract: TBD