Events
Department of Mathematics and Statistics
Texas Tech University
Let \(R\) be a commutative noetherian ring. In the derived category of
\(R\), the level of a bounded \(R\)-complex \(M\) with respect to a
collection of objects \(C\) (often referred to as the \(C\)-level of
\(M\)) is the fewest number of mapping cones involving objects in
\(C\) needed to obtain \(M\). When \(C\) is a nice collection of
objects in \(D(R)\) (such as the projective modules), the \(C\)-level
of a complex can give a wealth of information about that complex and
the ring itself. For example, if \(R\) is local, then \(R\) is regular
if and only if the projective level of all bounded complexes is
finite. Recently, Christensen, Kekkou, Lyle, and Soto Levins have
found optimal upper bounds for the Gorenstein projective level of
bounded complexes with finitely generated homology. In my talk, I'll
show how to improve their result to find optimal upper bounds for the
projective, injective, flat, Gorenstein projective, Gorenstein
injective, and Gorenstein flat levels of all bounded R-complexes. As
an application of my results, I'll prove a version of the Bass Formula
for injective levels and for Gorenstein injective levels.
Abstract. Electrical Impedance Tomography (EIT) is a PDE-based inverse problem that aims to reconstruct electrical conductivity from boundary measurements of electrical currents and voltages. Mathematically, EIT is modeled by an elliptic equation with a variable conductivity function, and the goal is to infer this function from the Dirichlet-to-Neumann (DtN) map. One fundamental barrier in solving EIT is data scarcity, and in this talk we address this bottleneck by completing full data from partially observed DtN measurements.
Specifically, we train a conditional diffusion model to learn the distribution of DtN data and to infer full measurement vectors given partial observations. Our approach supports flexible source–receiver configurations and can be used as a plug-in preprocessing step with off-the-shelf EIT solvers. Under mild assumptions on the polygonal conductivity class, we derive nonasymptotic end-to-end bounds on the distributional discrepancy between the completed and ground-truth DtN measurements. In numerical experiments, we couple the proposed diffusion-based completion procedure with a deep learning–based inverse solver and compare its performance against the same solver using full measurement data. The results show that diffusion completion enables reconstructions that closely match the full-data baseline while using only 1% of the measurements. In contrast, standard baselines such as matrix completion require 30% of the measurements to achieve similar reconstruction quality.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room Math 011 (Math Basement)
ZOOM details:
- Choice #1: use this link
Direct Link that embeds meeting and ID and passcode.
- Choice #2: join meeting using this link
Join Meeting, then you will have to input the ID and Passcode by hand:
* Meeting ID: 949 9288 2213
* Passcode: Applied
Ixodes scapularis, the blacklegged tick, is the main North American vector for the bacteria Borrelia burgdorferi, which causes Lyme disease, the most prevalent vector-borne disease on the continent. Tick demographics are influenced by many factors which vary geographically, including its community of hosts and its questing behavior. Both of these factors differ significantly from the northeastern US to the southeastern US: southern ecosystems contain greater biodiversity, including higher reptile abundance, while northern communities are dominated by small mammals. Questing behavior also differs regionally, with southern populations more likely to seek hosts below the leaf litter, while northern ticks quest above it. This talk uses a stage-structured nonlinear system of difference equations that is the first to incorporate questing behavior and ratio-dependent host-attachment success. Excessive effective tick reproduction levels destabilize tick population density while questing more above the leaf litter stabilizes it; the alternative is a 2-cycle in which one cohort of ticks is lost but the other becomes more than twice as large as either cohort would normally be. The impact of this 1-cohort vs. 2-cohort outcome in population dynamics carries over into B. burgdorferi transmission dynamics and Lyme disease risk as well.
The Biomath seminar may be attended virtually Friday at 11:00 AM CST (UTC-6) via this Zoom link.
Meeting ID: 938 8653 3169
Passcode: 883472
abstract 2 PM CST (UTC-6)
Zoom link available from Dr. Brent Lindquist upon request.
TTU Math Circle Spring Flyer 6:30-7:30 PM Thursdays in the basement of Math, room 010
