Seminars and Colloquia by Series

Monte Carlo methods for the Hermitian eigenvaue problem

Series
Applied and Computational Mathematics Seminar
Time
Monday, January 25, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE https://bluejeans.com/884917410
Speaker
Robert WebberCourant Institute

In quantum mechanics and the analysis of Markov processes, Monte Carlo methods are needed to identify low-lying eigenfunctions of dynamical generators. The standard Monte Carlo approaches for identifying eigenfunctions, however, can be inaccurate or slow to converge. What limits the efficiency of the currently available spectral estimation methods and what is needed to build more efficient methods for the future? Through numerical analysis and computational examples, we begin to answer these questions. We present the first-ever convergence proof and error bounds for the variational approach to conformational dynamics (VAC), the dominant method for estimating eigenfunctions used in biochemistry. Additionally, we analyze and optimize variational Monte Carlo (VMC), which combines Monte Carlo with neural networks to accurately identify low-lying eigenstates of quantum systems.

The asymptotic dimension of big mapping class groups

Series
Geometry Topology Seminar
Time
Monday, January 25, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Yvon VerberneGeorgia Institute of Technology

Please Note: Dan Margalit is inviting you to a scheduled Zoom meeting. https://zoom.us/j/94410378648?pwd=TVV6UDd0SnU3SnAveHA1NWxYcmlTdz09 Meeting ID: 944 1037 8648 Passcode: gojackets

In 2010, Bestvina-Bromberg-Fujiwara proved that the mapping class group of a finite type surface has finite asymptotic dimension. In contrast, we will show the mapping class group of an infinite-type surface has infinite asymptotic dimension if it contains an essential shift. This work is joint with Curtis Grant and Kasra Rafi.

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.

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

Please Note: 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).

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

Please Note: 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.

 

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

Large deviations of the greedy independent set algorithm on sparse random graphs

Series
Combinatorics Seminar
Time
Friday, January 15, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/751242993/PASSWORD (To receive the password, please email Lutz Warnke)
Speaker
Brett KolesnikUniversity of California, Berkeley

We study the greedy independent set algorithm on sparse Erdős-Rényi random graphs G(n,c/n). This range of p is of interest due to the threshold at c=e, beyond which it appears that greedy algorithms are affected by a sudden change in the independent set landscape. A large deviation principle was recently established by Bermolen et al. (2020), however, the proof and rate function are somewhat involved. Upper bounds for the rate function were obtained earlier by Pittel (1982). By discrete calculus, we identify the optimal trajectory realizing a given large deviation and obtain the rate function in a simple closed form. In particular, we show that Pittel's bounds are sharp. The proof is brief and elementary. We think the methods presented here will be useful in analyzing the tail behavior of other random growth and exploration processes.

Based on https://arxiv.org/abs/2011.04613

Identifying Dehn Functions of Bestvina--Brady Groups From Their Defining Graphs

Series
Geometry Topology Seminar
Time
Monday, January 11, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Yu-Chan ChangEmory University

Please Note: https://zoom.us/j/8833025617?pwd=R1FvQWp1MVlRSTVBdFZNejE3ZURmUT09 Meeting ID: 883 302 5617

Bestvina--Brady groups are subgroups of right-angled Artin groups, and their Dehn functions are bounded above by quartic functions. There are examples of Bestvina--Brady groups whose Dehn functions are linear, quadratic, cubic, and quartic. In this talk, I will give a class of Bestvina--Brady groups that have polynomial Dehn functions, and we can identify the Dehn functions by the defining graphs of those Bestvina--Brady groups. 

Flip processes on finite graphs and dynamical systems they induce on graphons

Series
Combinatorics Seminar
Time
Friday, December 11, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/751242993/PASSWORD (To receive the password, please email Lutz Warnke)
Speaker
Jan HladkyCzech Academy of Sciences

We introduce a class of random graph processes, which we call flip processes. Each such process is given by a rule which is just a function $\mathcal{R}:\mathcal{H}_k\rightarrow \mathcal{H}_k$ from all labelled $k$-vertex graphs into itself ($k$ is fixed). Now, the process starts with a given $n$-vertex graph $G_0$. In each step, the graph $G_i$ is obtained by sampling $k$ random vertices $v_1,\ldots,v_k$ of $G_{i-1}$ and replacing the induced graph $G_{i-1}[v_1,\ldots,v_k]$ by $\mathcal{R}(G_{i-1}[v_1,\ldots,v_k])$. This class contains several previously studied processes including the Erdos-Renyi random graph process and the random triangle removal.

Given a flip processes with a rule $\mathcal{R}$ we construct time-indexed trajectories $\Phi:\mathcal{W}\times [0,\infty)\rightarrow\mathcal{W}$ in the space of graphons. We prove that with high probability, starting with a large finite graph $G_0$ which is close to a graphon $W_0$, the flip process will follow the trajectory $(\Phi(W_0,t))_{t=0}^\infty$ (with appropriate rescaling of the time).

These graphon trajectories are then studied from the perspective of dynamical systems. We prove that two trajectories cannot form a confluence, give an example of a process with an oscilatory trajectory, and study stability and instability of fixed points.

Joint work with Frederik Garbe, Matas Sileikis and Fiona Skerman.

Asymptotic dimension of minor-closed families and beyond

Series
Graph Theory Seminar
Time
Tuesday, December 8, 2020 - 15:45 for 1 hour (actually 50 minutes)
Location
https://us04web.zoom.us/j/77238664391. For password, please email Anton Bernshteyn (bahtoh ~at~ gatech.edu)
Speaker
Chun-Hung LiuTexas A&M University

The asymptotic dimension of metric spaces is an important notion in geometric group theory. The metric spaces considered in this talk are the ones whose underlying spaces are the vertex-sets of (edge-)weighted graphs and whose metrics are the distance function in weighted graphs. A standard compactness argument shows that it suffices to consider the asymptotic dimension of classes of finite weighted graphs. We prove that the asymptotic dimension of any minor-closed family of weighted graphs, any class of weighted graphs of bounded tree-width, and any class of graphs of bounded layered tree-width are at most 2, 1,and 2, respectively. The first result solves a question of Fujiwara and Papasoglu; the second and third results solve a number of questions of Bonamy, Bousquet, Esperet, Groenland, Pirot and Scott. These bounds for asymptotic dimension are optimal and generalize and improve some results in the literature, including results for Riemannian surfaces and Cayley graphs of groups with a forbidden minor.

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