Harvard Mathematics News
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Seminars at the Harvard Mathematics DepartmentWed, 11 Dec 2019 16:58:03 -0500NUMBER THEORY SEMINAR : Liang Xiao (BICMR) speaks on <b>Slopes of modular forms and ghost conjecture of Bergdall and Pollack</b> on Dec 11, 19, 3:00 pm in Science Center 507:
Abstract: In classical theory, slopes of modular forms are p-adic valuations of the eigenvalues of the Up-operator. On the Galois side, they correspond to the p-adic valuations of eigenvalues of the crystalline Frobenius on the Kisin's crystabelian deformations space. I will report on a joint work in progress in which we seems to have proved a version of the ghost conjecture of Bergdall and Pollack. This has many consequences in the classical theory, such as some cases of Gouvea-Mazur conjecture, and some hope towards understanding irreducible components of eigencurves. On the Galois side, our theorem can be used to prove certain integrality statement on slopes of crystalline Frobenius on Kisin's deformation space, as conjectured by Breuil-Buzzard-Emerton. This is a joint work with Ruochuan Liu, Nha Truong, and Bin Zhao.
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http://www.math.harvard.edu/1 Sun, 11 Dec 2019 15:00:00 -0400 HARVARD-MIT-MSR COMBINATORICS SEMINAR:Ross Berkowitz (Yale University ) speaks on <b>A local limit theorem for cliques</b> on Dec 11, 19, 4:15 pm - 5:15 pm in MIT 2-139:
Abstract: Let $X$ denote the number of cliques of size 7 in a uniformly random graph on $n$ vertices. What is the probability that $X$ is exactly equal to its mean? A central limit theorem estimating the behavior of $X$ on a large scale is a classic result, and in this talk we extend this to understanding the pointwise distribution of $X$. Additionally, we will examine some random variables where central limit theorems do not extend to nice local limit theorems.
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http://www.math.harvard.edu/2 Sun, 11 Dec 2019 16:15:00 -0400 CMSA QUANTUM MATTER AND QUANTUM FIELD THEORY SEMINAR:Yuya Tanizaki (NCSU) speaks on <b>Constraints on possible dynamics of QCD by symmetry and anomaly</b> on Dec 11, 19, 10:30 am - 12:00 pm in CMSA, 20 Garden St, G10:
Abstract: Low-energy dynamics of Quantum Chromodynamics (QCD) is an important subject for nuclear and hadron physics, but it is a strongly coupled system and difficult to solve. In this situation, symmetry and also anomaly give us an important guide to make a solid conclusion on possible behaviors of QCD. We find a new discrete anomaly in massless QCD, which says that the baryon number current is anomalously broken under the background gauge field for vector-like flavor symmetry and discrete axial symmetry. To match this anomaly in the chiral-symmetry broken phase, the existence of baryons as Skyrmions is mandatory, and Skyrmion current must show the same anomaly. This is satisfied in the ordinary scenario of chiral symmetry breaking, but not satisfied in an exotic scenario of chiral symmetry breaking proposed by Stern about two decades ago. Since the anomaly matching is applicable even with the sign problem, Stern phase is excluded even for the finite-density QCD at zero temperatures.
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http://www.math.harvard.edu/3 Sun, 11 Dec 2019 10:30:00 -0400 CMSA CONDENSED MATTER/MATH SEMINAR :Yuya Tanizaki (NCSU) speaks on <b>Modifying instanton sums in QCD</b> on Dec 12, 19, 11:50 am – 1 :00 pm in CMSA, 20 Garden St, G10:
Abstract: In the path integral formulation, we need to sum up all possible field configurations to define a QFT. If the configuration space is disconnected, we must specify how they should be summed by giving extra data to specify the QFT. In this talk, we try to restrict the possible number of instantons in SU(N) gauge theories, Yang-Mills theory and QCD, and we find out the vacuum structures of them. For consistency with locality, we have to introduce an extra topological degrees of freedom, and the theory acquires the 3-form symmetry. The existence of this 3-form symmetry leads to extra vacua, and moreover it turns out to give an interesting selection rule for the domain-wall excitation/vacuum decay.
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http://www.math.harvard.edu/4 Sun, 12 Dec 2019 11:50:00 -0400 HARVARD-MIT-MSR COMBINATORICS SEMINAR:Dapeng Weng (Michigan State University) speaks on <b>Cluster Duality of Grassmannian and a Cyclic Sieving Phenomenon of Plane Partitions</b> on Dec 13, 19, 3:30 pm - 4:30 pm in Science Center 530:
Abstract: Fix two positive integers $a$ and $b$. Scott showed that the homogeneous coordinate ring of the Grassmannian $Gr_{a, a+b}$ has the structure of a cluster algebra. This homogeneous coordinate ring can be decomposed into a direct sum of irreducible representations of $GL_{a+b}$ which correspond to non-negative integer multiples of the fundamental weight $w_a$. We introduce a periodic configuration space $Conf_{a+b,a}$ equipped with a natural potential function $W$ and prove that the tropicalization of $(Conf_{a+b,a},W)$ canonically parametrizes bases for the irreducible summands of the homogeneous coordinate ring of $Gr_{a,a+b}$, as predicted by the cluster duality conjecture of Fock and Goncharov. We identify the parametrizing set of each irreducible summand with a collection of plan partitions of size $a\times b$. As an application, we use this identification to show a cyclic sieving phenomenon of plane partitions under a certain sequence of toggling operations. This is joint work with Linhui Shen.
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http://www.math.harvard.edu/5 Sun, 13 Dec 2019 15:30:00 -0400