I will introduce Serre spectral sequences, then compute some examples. The talk will be in most part following Allen Hatcher's notes on spectral sequences.
An initial result of Bourgain and Chang has lead to a number of striking advances in the understanding of polynomial extensions of Roth's Theorem.
The most striking of these is the result of Peluse and Prendiville which show that sets in [1 ,..., N] with density greater than (\log N)^{-c} contain polynomial progressions of length k (where c=c(k)). There is as of yet no corresponding result for corners, the two dimensional setting for Roth's Theorem, where one would seek progressions of the form(x,y), (x+t^2, y), (x,y+t^3) in [1 ,..., N]^2, for example.
Recently, the corners version of the result of Bourgain and Chang has been established, showing an effective bound for a three term polynomial Roth theorem in the finite field setting. We will survey this area. Joint work with Rui Han and Fan Yang.
The link for the seminar is the following
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
The Prague dimension of graphs was introduced by Nešetřil, Pultr and Rödl in the 1970s. Proving a conjecture of Füredi and Kantor, we show that the Prague dimension of the binomial random graph is typically of order $n/\log n$ for constant edge-probabilities. The main new proof ingredient is a Pippenger–Spencer type edge-coloring result for random hypergraphs with large uniformities, i.e., edges of size $O(\log n)$. Based on joint work with Kalen Patton and Lutz Warnke.
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.
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.
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.
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).
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.
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
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
