Events
Department of Mathematics and Statistics
Texas Tech University
We present a framework for simulating measure-preserving, ergodic dynamical systems with pure point spectrum by a finite-dimensional quantum system amenable to implementation on a quantum computer. The framework is based on a quantum feature map for representing classical states by density operators on a reproducing kernel Hilbert space, H, of functions on classical state space with Banach algebra structure. Simultaneously, a mapping is employed from classical observables into self-adjoint operators on H such that quantum mechanical expectation values are consistent with pointwise function evaluation. With this approach, quantum states and observables on H evolve under the action of a unitary group of Koopman operators in a consistent manner with classical dynamical evolution. Moreover, the state of the quantum system can be projected onto a finite-rank density operator on a 2^n-dimensional tensor product Hilbert space associated with n qubits, enabling efficient implementation in a quantum circuit. We illustrate our approach with quantum circuit simulations of low-dimensional dynamical systems, as well as actual experiments on the IBM Quantum System One.
This week's Analysis seminar may be attended at 4:00 PM CDT (UT-5) via this Zoom link.
Meeting ID: 976 4978 7908
Passcode: 973073
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Recent research has revealed the essential role that microbial metabolites play in host-microbiome interactions. Although statistical and machine-learning methods have been employed to explore microbiome-metabolome interactions in multiview microbiome studies, most of these approaches focus solely on the prediction of microbial metabolites, which lacks biological interpretation. Additionally, existing methods face limitations in either prediction or inference due to small sample sizes and highly correlated microbes and metabolites. To overcome these limitations, we present a transfer-learning method that evaluates microbiome-metabolome interactions. Our approach efficiently utilizes information from comparable metabolites obtained through external databases or data-driven methods, resulting in more precise predictions of microbial metabolites and identification of essential microbes involved in each microbial metabolite. Our numerical studies demonstrate that our method enables a deeper understanding of the mechanism of host-microbiome interactions and establishes a statistical basis for potential microbiome-based therapies for various human diseases.
Please virtually attend this week's Statistics seminar at 4:00 PM (CDT, UT-5) via this zoom link
Meeting ID: 982 5262 2072
Passcode: 235469
Thanks to a result of Baez and Hoffnung, the category of diffeological spaces is equivalent to the category of concrete sheaves on the site of cartesian spaces. By thinking of diffeological spaces as kinds of sheaves, we can therefore think of diffeological spaces as kinds of infinity sheaves. We do this by using a model category presentation of the infinity category of infinity sheaves on cartesian spaces, and cofibrantly replacing a diffeological space within it. By doing this, we obtain a new generalized cocycle construction for diffeological principal bundles, a new version of Čech cohomology for diffeological spaces that can be compared very directly with two other versions appearing in the literature, which is precisely infinity sheaf cohomology, and we show that the nerve of the category of diffeological principal G-bundles over a diffeological space X for a diffeological group G is weak equivalent to the nerve of the category of G-principal infinity bundles over X. arXiv:2202.11023.
Bring your own lunch and discuss (mostly) biology-related topics in math. Students, postdocs, and faculty welcome from any discipline.
This week's PDGMP seminar may be attended at 3:00 PM CDT (UT-5) via this Zoom link.
Meeting ID: 933 4646 9342
Staged tree models are discrete statistical models, generalizing
Bayesian networks, and are described by algebraic varieties, that is
the zero set of a collection of polynomials. A particularly nice class
of varieties for computational purposes are the toric varieties. If
the staged tree model variety is toric, it offers a Markov basis,
helpful for hypothesis testing, and facilitates the study of maximum
likelihood estimates. In 2021 G\"orgen, Maraj, and Nicklasson showed
that the variety for staged tree models with one color having at most
3 children at each vertex, and depth at most 3, are toric. We showed
that their conjecture fails in depth 4, using methods from geometry
and representation theory. In this talk I'm going to introduce staged
tree models, the associated variety to the model, define and explain
how to compute its symmetry Lie algebra, and will introduce an
algorithmic way to prove or disprove if certain ideals can be
generated by binomials. This is a joint work with Aida Maraj.
Abstract. Schrödinger operators with random potentials are very important models in quantum mechanics, in the study of transport properties of electrons in solids. In this talk, we study the approximation of eigenvalues via the landscape theory for some random Schrödinger operators as well as some related models. The localization landscape theory, introduced in 2012 by Filoche and Mayboroda, considers the landscape function u solving Hu=1 for an operator H. The landscape theory has remarkable power in studying the eigenvalue problems for a large class of operators and has led to numerous “landscape baked” results in mathematics, as well as in theoretical and experimental physics. We first give a brief review of the localization landscape theory. Then we focus on some recent progress of the landscape-eigenvalue approximation for operators on general graphs. We show that the maximum of the landscape function is comparable to the reciprocal of the ground state eigenvalue, for Anderson or random hopping models on certain graphs with growth and heat kernel conditions, as well as on some fractal-like graphs such as the Sierpinski gasket. There will be precise asymptotic behavior of the ground state energy for some 1D chain models, as well as numerical stimulations for excited states energies.
About the speaker. Dr. Shiwen Zhang joined the Department of Mathematics & Statistics at University of Massachusetts Lowell as an Assistant Professor in August 2022. Shiwen obtained his PhD from University of California, Irvine in 2016, under the supervision of Svetlana Jitomirskaya. From 2016 to 2022, he worked as a postdoc at Michigan State University, and then at University of Minnesota. Shiwen’s research field is analysis, including Mathematical Physics, dynamical systems, PDEs, and spectral theory of Schrödinger operators. In particular, Shiwen is interested in quantum localization in a disordered medium, dynamical and fractal properties of quasiperiodic operators. More recently, Shiwen works in collaboration with experts in numerical analysis and physicists, on problems of scientific computing of eigenvalues and eigenfunctions, arising from models in physics and semi-conductor devices.
When: 4:00 pm (Lubbock's local time is GMT -5)
Where: room MATH 011 (basement)
ZOOM details:
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* Meeting ID: 940 7062 3025
* Passcode: applied
In this talk, we shall define the Gaussian hypergeometric function and express the special values of 2_F_1, 3_F_2, and 4_F_3- Gaussian hypergeometric function in terms of traces of the Frobenius endomorphisms of certain families of elliptic curves. To obtain the value of 4_F_3- Gaussian hypergeometric function, we shall first discuss the finite field analogs of classical summation identity connecting F_3- classical Appell series and 4_F 3- classical hypergeometric series. As an application, we obtain the summation formula satisfied by the 4_F_3- Gaussian hypergeometric function.
Please attend this talk for a Department Post-Doc position at 10:30 AM CDT (UT-5) via this Zoom link.
Meeting ID: 941 6773 6910
Passcode: interview