Format results
Special cycles on moduli spaces of unitary shtukas, and higher derivatives of L-functions-IV (Online)
Benjamin HowardICTS:31884Quantum Gravity and Effective Topology
Renate Loll Radboud Universiteit Nijmegen
2-dimensional topological field theories via the genus filtration
Jan SteinebrunnerQuantum Chemistry in the Universe’s Coldest Test Tube
Alan Jamison Institute for Quantum Computing (IQC)
PIRSA:25050015Special cycles on moduli spaces of unitary shtukas, and higher derivatives of L-functions-III (Online)
Benjamin HowardICTS:31883Learning and testing quantum states of fermionic systems
Antonio Mele Freie Universität Berlin
Degenerate automorphic forms and Euler systems-V (Online)
Marco Sangiovanni VincentelliICTS:31899Euler Systems have proven to be versatile tools for understanding Selmer groups and their connections to special values of L-functions. However, despite the key role they have played in making progress toward foundational conjectures in number theory like the Birch–Swinnerton-Dyer and Bloch– Kato Conjectures, only a handful of provably non-trivial Euler systems have been constructed to date. A significant obstacle to constructing Euler Systems lies in producing candidate Galois cohomology classes. This lecture series presents a method to overcome this obstacle that does not rely on rare (known) motivic classes. We will focus on building ´etale cohomology classes originating from automorphic data: Eisenstein series and Theta series. This framework not only retrieves most classical Euler systems but can also be applied to construct an Euler system for the adjoint of an elliptic modular form.
References:
• C. Skinner, L-values and nonsplit extensions: a simple case, https://msp.org/ent/2024/3-1/p03.xhtml
• H. Darmon etal, p-adic L-functions and Euler systems: a tale in two trilogies.An Introduction to the GGP conjectures - IV
Dipendra PrasadICTS:31874Spill-over from the last lecture, and finish with some low dimensional examples, including the fundamental work of Waldspurger; illustrative examples from finite fields.
Special cycles on moduli spaces of unitary shtukas, and higher derivatives of L-functions-IV (Online)
Benjamin HowardICTS:31884The arithmetic Siegel-Weil formula, conjectured by Kudla-Rapoport and proved by Li-Zhang, expresses the degrees of certain 0-cycles on integral models of unitary Shimura varieties in terms of the nondegenerate Fourier coefficients of the central derivative of an Eisenstein series.
Feng-Yun-Zhang proved a higher derivative version of this arithmetic Siegel-Weil formula in the function field setting, now expressing degrees of 0-cycles on moduli spaces of unitary shtukas to the nondegenerate Fourier coefficients of higher central derivatives of an Eisenstein series.
The goal of my lecture series is (1) to explain all of this background, (2) extend the results of Feng-Yun-Zhang to include some degenerate coefficients, and (3) deduce from this extension an arithmetic application: the nonvanishing of higher central derivatives of certain Langlands L-functions implies the nonvanishing of classes in the Chow groups of moduli spaces of shtukas.
All of the new results are joint work with Tony Feng and Mikayel Mkrtchyan.Quantum Gravity and Effective Topology
Renate Loll Radboud Universiteit Nijmegen
My presentation will introduce a new methodology to characterize properties of quantum spacetime in a strongly quantum-fluctuating regime, using tools from topological data analysis. Starting from a microscopic quantum geometry, generated nonperturbatively in terms of dynamical triangulations (DT), we compute the homology of a sequence of coarse-grained versions of the geometry as a function of the coarse-graining scale. This gives rise to a characteristic "topological finger print" of the quantum geometry. I discuss the results for Lorentzian and Euclidean 2D quantum gravity, defined via lattice quantum gravity based on causal and Euclidean DT. For the latter, our numerical analysis reproduces the well-known string susceptibility exponent governing the scaling behaviour of the partition function.[Joint work with Jesse van der Duin, Marc Schiffer and Agustin Silva, to appear.]2-dimensional topological field theories via the genus filtration
Jan SteinebrunnerBy a folk theorem (non-extended) 2-dimensional TFTs valued in the category of vector spaces are equivalent to commutative Frobenius algebras. Upgrading the bordism category to an (infinity, 1)-category whose 2-morphism are diffeomorphisms, one can study 2D TFTs valued in higher categories, leading for example to (derived) modular functors and cohomological field theories.I will explain how to describe such more general (non-extended) 2D TFTs as algebras over the modular infinity-operad of surfaces. In genus 0 this yields an E_2^{SO}-Frobenius algebra and I will outline an obstruction theory for inductively extending such algebras to higher genus. Specialising to invertible TFTs, this amounts to a genus filtration of the classifying space of the bordism category and hence the Madsen--Tillmann spectrum MTSO_2. The aforementioned obstruction theory identifies the associated graded in terms of curve complexes and thereby yields a spectral sequence starting with the unstable and converging to the stable cohomology of mapping class groups.An Introduction to the GGP conjectures - III
Dipendra PrasadICTS:31873A bit of representation theory of groups over local fields, parabolic induction, cuspidal representations. Review of the Local Langlands correspondence, L-functions and epsilon factors. L-packets, the Jacquet-Langlands correspondence, The GGP conjectures: both local and global conjectures.
Lectures on the Anticyclotomic main conjecture-II
Haruzo HidaICTS:31867We first prove, for a prime p>3 unramified in a CM quadratic extension of a totally real field F, h(M/F)L(\chi)|H(\psi)|h(M/F)F(\chi) (h(M/F)=h(M)/h(F)) in \Lambda for the congruence power serie H(\psi) of \psi lifting a fixed anti-cyclotomic character \chi and anticyclotomic Katz p-adic L-function L(\chi) of branch character \chi, built on the lectures by Tilouine proving this over \Lambda[1/p]. Here \Lambda is the many variable Iwasawa algebra of M. In the second lecture, we give a sketch of the proof of the reverse divisibility: H(\psi)|h(M/F)L(\chi) resulting in the main conjecture, as H(\psi)=h(M/F)F(\chi) for the anticyclotomic Iwasawa power series F(\chi) by the “R=T”-theorem.
Quantum Chemistry in the Universe’s Coldest Test Tube
Alan Jamison Institute for Quantum Computing (IQC)
PIRSA:25050015Abstract
How do chemical reactions change when they’re run at temperatures a billion times colder than a Canadian winter? What can we learn when we have perfect quantum control of the reactants? Before answering these questions, we’ll discuss the fascinating techniques of laser cooling that allow us to cool atoms and molecules to within a few billionths of a degree above absolute zero. We’ll then look at how molecules prepared at such temperatures allow us to control chemical reactions at the quantum level, beginning to open a new understanding of chemistry and new possibilities for technologies of the future.
About the Speaker
Dr. Alan Jamison is an Assistant Professor at the University of Waterloo, jointly appointed to the Department of Physics and Astronomy and the Institute for Quantum Computing (IQC). He leads the Jamison Lab, which investigates ultracold atoms and molecules to explore quantum many-body physics, quantum chemistry, and quantum information science. Dr. Jamison earned his B.S. in Mathematics from the University of Central Florida in 2007, followed by an M.S. and Ph.D. in Physics from the University of Washington in 2008 and 2014, respectively.
After completing his Ph.D., he joined the group of Nobel Laureate Wolfgang Ketterle at the Massachusetts Institute of Technology (MIT) as a postdoctoral researcher. At the University of Waterloo, Dr. Jamison's research centers on using ultracold atoms and molecules to investigate complex quantum systems. His lab aims to achieve precise control over chemical reactions at ultracold temperatures, providing insights into quantum chemistry and enabling advancements in quantum computing and simulation.
Special cycles on moduli spaces of unitary shtukas, and higher derivatives of L-functions-III (Online)
Benjamin HowardICTS:31883The arithmetic Siegel-Weil formula, conjectured by Kudla-Rapoport and proved by Li-Zhang, expresses the degrees of certain 0-cycles on integral models of unitary Shimura varieties in terms of the nondegenerate Fourier coefficients of the central derivative of an Eisenstein series.
Feng-Yun-Zhang proved a higher derivative version of this arithmetic Siegel-Weil formula in the function field setting, now expressing degrees of 0-cycles on moduli spaces of unitary shtukas to the nondegenerate Fourier coefficients of higher central derivatives of an Eisenstein series.
The goal of my lecture series is (1) to explain all of this background, (2) extend the results of Feng-Yun-Zhang to include some degenerate coefficients, and (3) deduce from this extension an arithmetic application: the nonvanishing of higher central derivatives of certain Langlands L-functions implies the nonvanishing of classes in the Chow groups of moduli spaces of shtukas.
All of the new results are joint work with Tony Feng and Mikayel Mkrtchyan.Degenerate automorphic forms and Euler systems-IV (Online)
Marco Sangiovanni VincentelliICTS:31898Euler Systems have proven to be versatile tools for understanding Selmer groups and their connections to special values of L-functions. However, despite the key role they have played in making progress toward foundational conjectures in number theory like the Birch–Swinnerton-Dyer and Bloch– Kato Conjectures, only a handful of provably non-trivial Euler systems have been constructed to date. A significant obstacle to constructing Euler Systems lies in producing candidate Galois cohomology classes. This lecture series presents a method to overcome this obstacle that does not rely on rare (known) motivic classes. We will focus on building ´etale cohomology classes originating from automorphic data: Eisenstein series and Theta series. This framework not only retrieves most classical Euler systems but can also be applied to construct an Euler system for the adjoint of an elliptic modular form.
References:
• C. Skinner, L-values and nonsplit extensions: a simple case, https://msp.org/ent/2024/3-1/p03.xhtml
• H. Darmon etal, p-adic L-functions and Euler systems: a tale in two trilogies.Supporting neurodivergence and understanding neurodivergent ways of being (brought to PI by the Mental Health Working Group)
Helen HargreavesHelen Hargreaves, MSW, RSW will present a workshop for PI Residents, providing information on the basics of Emotion Theory, how to assess ones own needs and communicate them. This presentation will particularly focus on Autistic and other neurodivergent ways of experiencing emotions and stress and how to better support neurodivergent team members in the workplace. Helen Hargreaves is a Neurodivergent Therapist with over 15 years experience workings with Neurodivergent clients. She is the Director of Rainbow Brain, a social work group practice that focuses on providing queer, trans and neurodivergent affirming therapy. Please note that this will be a 1.5 hour session with presentation and experiential components.
Learning and testing quantum states of fermionic systems
Antonio Mele Freie Universität Berlin
Abstract: The experimental realization of increasingly complex quantum states in quantum devices underscores the pressing need for new methods of state learning and verification. Among the various classes of quantum states, fermionic systems hold particular significance due to their crucial roles in physics. Despite their importance, research on learning quantum states of fermionic systems remains surprisingly limited. In our work, we aim to present a comprehensive rigorous study on learning and testing states of fermionic systems. We begin by analyzing arguably the simplest important class of fermionic states—free-fermionic states—and subsequently extend our analysis to more complex fermionic states. We meticulously delineate scenarios in which efficient algorithms are feasible, providing experimentally practical algorithms for these cases, while also identifying situations where any algorithm for solving these problems must be inherently inefficient. At the same time, we present novel fundamental results of independent interest on fermionic systems, with additional applications beyond learning and characterizing quantum devices, such as many-body physics, resource theory of non-Gaussianity, and circuit compilation strategies. (Talk based on https://arxiv.org/pdf/2409.17953 , https://arxiv.org/pdf/2402.18665)