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PRODID:Mathematics Seminar Calendar
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UID:20210506T224952-27017@math.msu.edu
DTSTAMP:20210506T224952Z
SUMMARY:Quantum Graphs
DESCRIPTION:Speaker\: Priyanga Ganesan, Texas A&M University\r\nQuantum graphs are an operator space generalization of classical graphs. In this talk, I will motivate the idea of a quantum graph and its significance in quantum communication. We will look at the different notions of quantum graphs that arise in operator systems theory, non-commutative topology and quantum information theory. I will then introduce a non-local game with quantum inputs and classical outputs, that generalizes the non local graph coloring game. This is based on joint work with Michael Brannan and Samuel Harris.
LOCATION:Online (virtual meeting)
DTSTART:20210301T190000Z
DTEND:20210301T195000Z
URL:https://math.msu.edu/Seminars/TalkView.aspx?talk=27017
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UID:20210506T224952-27015@math.msu.edu
DTSTAMP:20210506T224952Z
SUMMARY:The landscape law for tight binding Hamiltonians
DESCRIPTION:Speaker\: Shiwen Zhang, U Minnesota\r\nThe localization landscape theory, introduced in 2012 by Filoche and Mayboroda, considers the so-called the landscape function u solving Hu=1 for an operator H. The landscape theory has remarkable power in studying the eigenvalue problems of H and has led to numerous ``landscape baked’’ results in mathematics, as well as in theoretical and experimental physics. In this talk, we will discuss some recent results of the landscape theory for tight-binding Hamiltonians H=-\Delta+V on Z^d. We introduce a box counting function, defined through the discrete landscape function of H. For any deterministic bounded potential, we give estimates for the integrated density of states from above and below by the landscape box counting function, which we call the landscape law. For the Anderson model, we get a refined lower bound for the IDS, throughout the spectrum. We will also discuss some numerical experiments in progress on the so-called practical landscape law for the continuous Anderson model. This talk is based on joint work with D. N. Arnold, M. Filoche, S. Mayboroda, and Wei Wang.
LOCATION:
DTSTART:20210309T200000Z
DTEND:20210309T210000Z
URL:https://math.msu.edu/Seminars/TalkView.aspx?talk=27015
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UID:20210506T224952-27014@math.msu.edu
DTSTAMP:20210506T224952Z
SUMMARY:Stationary random Schrödinger operators at small disorder
DESCRIPTION:Speaker\: Christopher Shirley, Paris-Saclay University\r\nIn this presentation, we will study Schrödinger operators with stationary potential and the existence of stationary Bloch waves for several types of stationarity and in particular in the random case. We will see that the Bloch waves of the unperturbed operator seem to vanish at weak disorder in the case of short-range correlations. This phenomenon suggests a resonance problem, difficult to study due to the lack of compactness.\r\n\r\nWe therefore investigate this problem using Mourre theory.
LOCATION:Online (virtual meeting)
DTSTART:20210315T180000Z
DTEND:20210315T190000Z
URL:https://math.msu.edu/Seminars/TalkView.aspx?talk=27014
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UID:20210506T224952-28039@math.msu.edu
DTSTAMP:20210506T224952Z
SUMMARY:Complex analysis applied to operator algebras
DESCRIPTION:Speaker\: Brent Nelson, Michigan State University\r\nGiven a positive definite matrix $D\in M_n(\mathbb{C})$ with $\text{Tr}(D)=1$, one can define a linear functional $\varphi\colon M_n(\mathbb{C})\to \mathbb{C}$ by $\varphi(x):=\text{Tr}(Dx)$ which we call a faithful state. This positive definite matrix also encodes a noncommutative dynamical system through $x\mapsto D^{it} x D^{-it}$ for $t\in \mathbb{R}$. From the perspective of operator algebras, it is useful to encode this dynamical system as... well, an algebra of operators. More precisely, as a $*$-algebra $\mathcal{M}$ containing $M_n(\mathbb{C})$ in a way that remembers the action of $\mathbb{R}$. In the general (infinite dimensional) setting, this is accomplished using crossed products and Tomita–Takesaki theory. In this talk, I will apply these methods to the more modest finite dimensional case, and show how a little bit of complex analysis allows one to find the analogue of $\text{Tr}$ on this larger $*$-algebra $\mathcal{M}$. (This talk will assume some familiarity with linear algebra and complex analysis, but nothing further.)
LOCATION:Online (virtual meeting)
DTSTART:20210318T210000Z
DTEND:20210318T215000Z
URL:https://math.msu.edu/Seminars/TalkView.aspx?talk=28039
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UID:20210506T224952-27013@math.msu.edu
DTSTAMP:20210506T224952Z
SUMMARY:Spectral gaps in graphene structures
DESCRIPTION:Speaker\: Rui Han , Louisiana State University\r\nWe will present a full spectral analysis for a model of graphene in magnetic fields with constant flux through every hexagonal comb. In particular, we provide a rigorous foundation for self-similarity by showing that for any irrational flux, the spectrum of graphene is a zero measure Cantor set. I will also discuss the spectral decomposition, Hausdorff dimension of the spectrum and existence of Dirac cones. This talk is based on joint works with S. Becker and S. Jitomirskaya.
LOCATION:Online (virtual meeting)
DTSTART:20210412T180000Z
DTEND:20210412T190000Z
URL:https://math.msu.edu/Seminars/TalkView.aspx?talk=27013
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UID:20210506T224952-29049@math.msu.edu
DTSTAMP:20210506T224952Z
SUMMARY:Groups, Group Actions, and von Neumann Algebras
DESCRIPTION:Speaker\: Rolando de Santiago, Purdue University\r\nGiven a group $G$ acting on measure space $(X,\mu)$ Murray and von Neumann’s group-measure space construction describes a von Neumann algebra $L^\infty(X,\mu)\rtimes G $ which encodes both the group, the space and the action. The special case where the space is a singleton and the action is trivial produces the group von Neumann algebra $L(G) $. \r\n\r\nIn this talk, we will aim to describe properties of $L^\infty(X,\mu)\rtimes G $ in terms of the group, the space and the action; compute $L^\infty(X,\mu)\rtimes G $ in special cases; and describe how the group-measure space varies or the group von Neumann algebra varies with $G$. All this serves to illustrate the fundamental problem in this area: von Neumann algebras tend to have poor memory of their generating data.\r\n\r\nThis talk assumes a working knowledge of group theory and linear algebra, and while knowledge of measure theory may be helpful, it is not required.
LOCATION:Online (virtual meeting)
DTSTART:20210415T210000Z
DTEND:20210415T215000Z
URL:https://math.msu.edu/Seminars/TalkView.aspx?talk=29049
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UID:20210506T224952-27012@math.msu.edu
DTSTAMP:20210506T224952Z
SUMMARY:Mathematical aspects of topological insulators.
DESCRIPTION:Speaker\: Alexis Drouot, University of Washington\r\nTopological insulators are phases of matter that act like extraordinarily stable waveguides along their boundary. They have a rich mathematical structure that involves PDEs, spectral theory, and topology. I will survey some results and discuss some (semi-classical) directions of research.
LOCATION:Online (virtual meeting)
DTSTART:20210419T180000Z
DTEND:20210419T190000Z
URL:https://math.msu.edu/Seminars/TalkView.aspx?talk=27012
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UID:20210506T224952-27016@math.msu.edu
DTSTAMP:20210506T224952Z
SUMMARY:Tight-binding limits in strong magnetic fields and the integer quantum Hall effect
DESCRIPTION:Speaker\: Jacob Shapiro, Princeton University\r\nTight-binding models of non-interacting electrons in solids are commonly used when studying the integer quantum Hall effect or more generally topological insulators. However, up until recently there was no proof that the topological properties of these discrete Schrodinger operators agree with those of the continuum models of which they are the tight-binding limit. Before tending to this question, we first tackle the basic issue of the double-well eigenvalue splitting in strong perpendicular constant magnetic fields in 2D. Once this is understood, we set up norm-resolvent convergence of a scaled continuum Schrodinger operator on L^2(R^2) (a magnetic Laplacian plus a lattice potential) to its tight-binding limit and finally show why the Chern numbers of these two models, discrete and continuum respectively, must agree. This talk is based on joint collaborations with C. L. Fefferman and M. I. Weinstein.
LOCATION:Online (virtual meeting)
DTSTART:20210426T180000Z
DTEND:20210426T190000Z
URL:https://math.msu.edu/Seminars/TalkView.aspx?talk=27016
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