Dependence of the density of states on the probability distribution for discrete random Schrödinger operators, II

Speaker: 

Christoph Marx

Institution: 

Oberlin College

Time: 

Friday, April 20, 2018 - 1:00pm

Location: 

rh 340N

We prove the Hölder-continuity of the density of states measure (DOSm) and the integrated density of states (IDS) for discrete random Schrödinger operators with finite-range potentials with respect to the probability measure. In particular, our result implies that the DOSm and the IDS for smooth approximations of the Bernoulli distribution converge to the corresponding quantities for the Bernoulli-Anderson model. Other applications of the technique are given to the dependency of the DOSm and IDS on the disorder, and the continuity of the Lyapunov exponent in the weak-disorder regime for dimension one. The talk is based on joint work with Peter Hislop (Univ. of Kentucky) 

Weaving Biological Snapshots into Stories through Mathematical Modeling

Speaker: 

Eberhard Voit

Institution: 

Georgia Tech

Time: 

Monday, October 1, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

Most biological experiments result in snapshots that illustrate select aspects of a phenomenon of interest. These snapshots may be very complicated, consisting of thousands of peaks in a mass spectrogram, the expression levels of whole genomes, or an impressive image visualizing the localization of different proteins. Yet, each result is separately frozen in time, even if experiments are performed in sequence and yield time series measurements. Extracting a storyline from these data requires a cognitive process in the form of conceptual or formal models. Among these, dynamic models have an unparalleled capability of weaving multiple, often heterogeneous biological results into chains of events with which a cell or organism responds to a change in its environment. This presentation illustrates with three vignettes the process of stitching static results into explanatory descriptions of cellular strategies.

                  The first vignette pertains to the enzyme NADPH oxidase 1 (Nox1) in vascular smooth muscle cells. This enzyme plays an important role in the control of reactive oxygen species and their involvement in vascular physiology and pathophysiology. In order to function properly, Nox1 needs to be available in an optimal state, where it is able to respond rapidly to upstream signals. Nox1 consists of four subunits, whose assembly has been studied extensively. By contrast, the disassembly process has largely been ignored. A detailed computational analysis shows that this disassembly process of NOX1 is crucial for a cell to regain its ready-to-respond state in an optimal fashion.

                  The second vignette addresses the coordination of first responders to heat stress in yeast. One of these fast responding units is the metabolic pathway system of sphingolipids. These lipids are not only integral components of membranes but also serve as important signaling molecules that can initiate longer-term genomic responses. Computational analysis of sphingolipid time series data reveals that essentially all enzymes in the pathway system respond to heat stress in a finely-tuned manner and as well-coordinated modules. Within two minutes, key sphingolipids emerge in much enlarged concentrations, subsequently triggering the expression of key response genes. After 30 minutes, the sphingolipid concentrations have returned to the profile they exhibit under optimal, cool conditions which, however, is governed by a greatly altered enzyme activity profile, due to the ongoing heat stress.  

                  The final vignette describes the regulation of glycolysis in the dairy bacterium Lactococcus lactis, which must solve a complicated control task. The bacterium faces erratic changes in its favorite substrate, glucose, including periods when glucose is absent altogether. The challenge is that glycolysis is an essentially linear pathway and that glucose uptake requires one of its intermediate metabolites, phosphoenolpyruvate (PEP), to get started. Without regulation, the glucose entering glycolysis would be sequentially catalyzed all the way to end products, thereby depleting the pool of PEP and preventing the uptake of newly available glucose. A dynamic model, based on time series data of representative glycolytic metabolites, explains the intricate regulatory processes that lead to the controlled termination of glycolysis, when glucose runs out, and to a metabolic state where the cell is immediately ready to respond to the availability of new substrate.  

Selmer groups, Tate-Shafarevich groups, and ranks of abelian varieties in quadratic twist families

Speaker: 

Zev Klagsbrun

Institution: 

CCR-La Jolla

Time: 

Thursday, May 31, 2018 - 3:00pm to 4:00pm

Location: 

RH 306

We determine the average size of the Φ-Selmer group in any quadratic twist family of abelian varieties having an isogeny Φ of degree 3 over any number field. This has several applications towards the rank statistics in such families of quadratic twists. For example, it yields the first known quadratic twist families of absolutely simple abelian varieties over Q, of dimension greater than one, for which the average rank is bounded; in fact, we obtain such twist families in arbitrarily large dimension. In the case that E/F is an elliptic curve admitting a 3-isogeny, we prove that the average rank of its quadratic twists is bounded; if F is totally real, we moreover show that a positive proportion of these twists have rank 0 and a positive proportion have 3-Selmer rank 1. We also obtain consequences for Tate-Shafarevich groups of quadratic twists of a given elliptic curve. This is joint work with Manjul Bhargava, Robert Lemke Oliver, and Ari Shnidman.

Selmer groups, Tate-Shafarevich groups, and ranks of abelian varieties in quadratic twist families

Speaker: 

Zev Klagsbrun

Institution: 

CCR-La Jolla

Time: 

Thursday, May 31, 2018 - 3:00pm to 4:00pm

Location: 

RH 306

We determine the average size of the Φ-Selmer group in any quadratic twist family of abelian varieties having an isogeny Φ of degree 3 over any number field. This has several applications towards the rank statistics in such families of quadratic twists. For example, it yields the first known quadratic twist families of absolutely simple abelian varieties over Q, of dimension greater than one, for which the average rank is bounded; in fact, we obtain such twist families in arbitrarily large dimension. In the case that E/F is an elliptic curve admitting a 3-isogeny, we prove that the average rank of its quadratic twists is bounded; if F is totally real, we moreover show that a positive proportion of these twists have rank 0 and a positive proportion have 3-Selmer rank 1. We also obtain consequences for Tate-Shafarevich groups of quadratic twists of a given elliptic curve. This is joint work with Manjul Bhargava, Robert Lemke Oliver, and Ari Shnidman.

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