Seminars and Colloquia Schedule

The Bulk and the Extremes of Minimal Spanning Acycles and Persistence Diagrams of Random Complexes

Series
Stochastics Seminar
Time
Thursday, January 21, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/751242993/PASSWORD (To receive the password, please email Lutz Warnke)
Speaker
Sayan MukherjeeDuke University

Frieze showed that the expected weight of the minimum spanning tree (MST) of the uniformly weighted graph converges to ζ(3). Recently, this result was extended to a uniformly weighted simplicial complex, where the role of the MST is played by its higher-dimensional analogue -- the Minimum Spanning Acycle (MSA). In this work, we go beyond and look at the histogram of the weights in this random MSA -- both in the bulk and in the extremes. In particular, we focus on the `incomplete' setting, where one has access only to a fraction of the potential face weights. Our first result is that the empirical distribution of the MSA weights asymptotically converges to a measure based on the shadow -- the complement of graph components in higher dimensions. As far as we know, this result is the first to explore the connection between the MSA weights and the shadow. Our second result is that the extremal weights converge to an inhomogeneous Poisson point process. A interesting consequence of our two results is that we can also state the distribution of the death times in the persistence diagram corresponding to the above weighted complex, a result of interest in applied topology.

Based on joint work with Nicolas Fraiman and Gugan Thoppe, see https://arxiv.org/abs/2012.14122

Global solutions for the energy supercritical NLS

Series
CDSNS Colloquium
Time
Friday, January 22, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see add'l notes for link)
Speaker
Mouhamadou SyU Virginia

Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09

In this talk, we will discuss the global well-posedness issue of the defocusing nonlinear Schrödinger equation (NLS). It is known that for subcritical and critical nonlinearities, the equation is globally well-posed on Euclidean spaces and some bounded domains. The supercritical nonlinearities are by far less understood; few partial or conditional results were established. On the other hand, probabilistic approaches (Gibbs measures, fluctuation-dissipation ...) were developed during the last decades to deal with low regularity settings in the context of dispersive PDEs. However, these approaches fail to apply the supercritical nonlinearities.  The aim of this talk is to present a new probabilistic approach recently developed by the author in the context of the energy supercritical NLS. We will review some known results and briefly present earlier probabilistic methods, then discuss the new method and the almost sure global well-posedness consequences for the energy supercritical NLS. The results that will be presented are partly join with Xueying Yu.

 

Combinatorial aspects of RNA design

Series
Mathematical Biology Seminar
Time
Friday, January 22, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Yann PontyEcole Polytechnique France

BlueJeans Link: https://bluejeans.com/348270750

RiboNucleic Acids (RNAs) are ubiquitous, versatile, and overall fascinating, biomolecules which play central roles in modern molecular biology. They also represent a largely untapped potential for biotechnology and health, substantiated by recent disruptive developments (mRNA vaccines, RNA silencing therapies, guide-RNAs of CRISPR-Cas9 systems...). To address those challenges, one must effectively  perform RNA design, generally defined as the determination of an RNA sequence achieving a predefined biological function.

I will focus in this talk on algorithmic results and enumerative properties stemming from the inverse folding, the problem of designing a sequence of nucleotides that fold preferentially and uniquely (with respect to base-pair maximization) into a target secondary structure. Despite the NP-hardness of the problem (+ absence of a Fixed Parameter-Tractable algorithm) we showed that it can be solved in polynomial time for restricted families of structures. Such families are dense in the space of designable 2D structures, so that any structure that admits a solution for the inverse folding can be efficiently designed in an approximated sense.

We show that any 2D structure avoiding two forbidden motifs can be modified into a designable structure  by adding at most one extra base-pair per helix. Moreover, both the modification and the design of a sequence for the modified structure can be computed in linear time. Finally, if time allows, I will discuss combinatorial consequences of the existence of undesignable motifs. In particular, it implies an exponentially decreasing density of designable structures amongst secondary structures. Those results extend to virtually any design objectives and energy models.

This is joint work with Cédric Chauve, Jozef Hales, Jan Manuch, Ladislav Stacho (SFU, Canada), Alice Héliou, Mireille Régnier, and Hua-Ting Yao (Ecole Polytechnique, France).

Prime gaps, probabilistic models and the Hardy-Littlewood conjectures

Series
Combinatorics Seminar
Time
Friday, January 22, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/751242993/PASSWORD (To receive the password, please email Lutz Warnke)
Speaker
Kevin FordThe University of Illinois at Urbana-Champaign

Motivated by a new probabilistic interpretation of the Hardy-Littlewood k-tuples conjectures, we introduce a new probabilistic model of the primes and make a new conjecture about the largest gaps between the primes below x.  Our bound depends on a property of the interval sieve which is not well understood.  We also show that any sequence of integers which satisfies a sufficiently uniform version of the Hardy-Littlewood conjectures must have large gaps of a specific size.  This work is joint with Bill Banks and Terry Tao.