The Colloquium is held on Mondays at 4pm in 208 University Hall.
Spring Quarter, 2023
- April 10, Anne Schilling (UC Davis)
New ideas about Markov chains
Abstract: We will discuss some new ideas from semigroup theory to analyze the stationary distribution and mixing time of finite Markov chains. An example for a Markov chain is card shuffling and a natural question is how often do you have to shuffle the deck before it is mixed or random. It turns out that semigroup theory can help answer these questions.
- April 24, David Nadler (UC Berkeley)
Commuting elements in Lie groups
Abstract: I will review some basics of Lie groups in particular classical results about functions on Lie groups invariant under conjugation. Then I will discuss recent work with Penghui Li and Zhiwei Yun developing analogous results for commuting pairs of group elements.
- May 1, Christine Berkesch (U Minnesota)
Geometry of toric syzygies
Abstract: Free resolutions, or syzygies, with a graded structure are algebraic objects that encode many geometric properties. This correspondence lies at the heart of classical projective algebraic geometry. By analogy, multigraded resolutions should also provide powerful geometric tools. I will discuss some foundational results from the classical story and give an overview of recent work to extend these tools to the multigraded setting of toric geometry.
- May 15, Sebastian Schreiber (UC Davis)
Coexistence and extinction in an autocorrelated world
: All species experience temporal fluctuations in environmental conditions e.g. temperature or mortality risk. These fluctuations often are autocorrelated e.g. warmer days tending to be followed by warmer days. How these autocorrelations influence ecological communities remains, largely, an open problem.
To tackle this problem, I will introduce stochastic difference equations on a compact state space. An invariant subset of this state space corresponds to extinction of one or more species. Coexistence corresponds to the existence of stationary distributions supported by the non-extinction set, while extinction corresponds to almost sure convergence to the extinction set. I will describe sufficient conditions for both of these outcomes using Lyapunov exponents that correspond to the average per-growth rates of rare species.
Using these methods, I will illustrate how autocorrelated fluctuations (i) can promote population persistence in patchy landscapes, (ii) can mediate coexistence, or (iii) result in unpredictable losses of species in ecological communities.
Fall Quarter, 2022
- October 13, Svetlana Jitomirskaya (UC Irvine)
Fractal properties of the Hofstadter butterfly, eigenvalues, and topological phase transitions
Abstract: We will give a brief introduction to the spectral theory of ergodic operators, assuming only introductory graduate courses as background. Then we will discuss several remarkable spectral phenomena present in the class of quasiperiodic operators and illustrate using the almost Mathieu (aka Harper’s) operator – a model behind the Hofstadter’s butterfly and Thouless theory of the Quantum Hall Effect. We will discuss the fascinating history of this model, that is now heavily studied in physics and linked to several Nobel and IMU prizes, and then will describe several recent results that resolve some long-standing conjectures.
- October 31, Ben Webster (U Waterloo and Perimeter Institute)
Canceled due to COVID
- November 14, Siu-Cheong Lau (Boston U)
Mirror correspondences between quiver representations and sheaves
Abstract: Quiver representations emerge from Lie theory and mathematical physics. In this talk, I will first explain the basic setting of quiver representations, with motivations coming from modeling branes and brains. Second, I will recall some well-known correspondences between quiver algebras and geometry of sheaves. Finally, I will talk about our project on constructing these algebro-geometric correspondences from mirror symmetry.
- November 21, Sebastian Schreiber (UC Davis)
- November 28, Mock AMS talks
Alonso Delfin ,Greg Knapp, Marissa Masden
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