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The derived category of a Fano threefold Y of Picard rank 1 and index 2 admits a semiorthogonal decomposition. This defines a non-trivial subcategory Ku(Y) called the Kuznetsov component, which encodes most of the geometry of Y. I will present joint work with M. Altavilla and M. Petkovic, in which we describe certain moduli spaces of Bridgeland-stable objects in Ku(Y), via the stability conditions constructed by Bayer, Macrì, Lahoz and Stellari. Furthermore, in our work we study the behavior of the Abel-Jacobi map on these moduli space. As an application in the case of degree d = 2, we prove a strengthening of a categorical Torelli Theorem by Bernardara and Tabuada.

Zoom: https://harvard.zoom.us/j/96709211410?pwd=SHJyUUc4NzU5Y1d0N2FKVzIwcmEzdz09

Zoom: https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

How can we use the theory of dynamical systems in analyzing the capabilities of neural networks? Understanding the representational power of Deep Neural Networks (DNNs) and how their structural properties (e.g., depth, width, type of activation unit) affect the functions they can compute, has been an important yet challenging question in deep learning and approximation theory. In a seminal paper, Telgarsky reveals the limitations of shallow neural networks by exploiting the oscillatory behavior of a simple triangle function and states as a tantalizing open question to characterize those functions that cannot be well-approximated by small depths.
In this work, we point to a new connection between DNNs expressivity and dynamical systems, enabling us to get trade-offs for representing functions based on the presence of a generalized notion of fixed points, called periodic points that have played a major role in chaos theory (Li-Yorke chaos and Sharkovskii’s theorem). Our main results are general lower bounds for the width needed to represent periodic functions as a function of the depth, generalizing previous constructions relying on specific functions.

Based on two recent works:
with Ioannis Panageas, Sai Ganesh Nagarajan, Xiao Wang from ICLR’20 (spotlight):  https://arxiv.org/abs/1912.04378
with Ioannis Panageas, Sai Ganesh Nagarajan from ICML’20: https://arxiv.org/abs/2003.00777

Zoom: https://harvard.zoom.us/j/98231541450

The generic vanishing theorem of Green-Lazarsfeld says that for general elements in the Picard variety of a projective manifold, their cohomology groups vanish in all degrees. Moreover, the cohomological jumping locus, that is, the locus where generic vanishing fails, is a union of torsion translated abelian subvarieties. If one replaces the Picard variety by the character variety of rank 1 local systems, then one can study a similar phenomenon, which are works by Simpson and Budur-Wang topologically and Esnault-Kerz arithmetically.  In this talk, I will focus on the same phenomenon but from algebraic perspectives by using D-modules. More precisely, I will discuss zero loci of Bernstein-Sato ideals and explain why the zero loci can be treated as the algebraic analogue of topological jumping loci by using relative D-modules. Then I will prove a conjecture of Budur that zero loci of Bernstein-Sato ideals are related to the topological jumping loci in the sense of Riemann-Hilbert Correspondence. This is based on joint work with Nero Budur, Robin van der Veer and Peng Zhou.

Zoom: https://harvard.zoom.us/j/91794282895?pwd=VFZxRWdDQ0VNT0hsVTllR0JCQytoZz09

The underdoped Cuprate exhibits a rich variety of unusual properties that have been exposed after years of experimental investigations. They include a pseudo-gap near the anti-nodal points and “Fermi arcs” of gapless excitations, together with a variety of order such as charge order, nematicity and possibly loop currents and time reversal and inversion breaking. I shall argue that by making a single assumption of strong pair fluctuations at finite momentum (Pair density wave), a unified description of this phenomenology is possible. As an example, I will focus on a description of the ground state that emerges when superconductivity is suppressed by a magnetic field, which supports small electron pockets. [Dai, Senthil, Lee, Phys Rev B101, 064502 (2020)] There is some support for the pair density wave hypothesis from STM data that found charge order at double the usual wave-vector in the vicinity of vortices, as well as evidence for a fragile form of superconductivity persisting to fields much above Hc2. I shall suggest a more direct experimental probe of the proposed fluctuating pair density wave.

Zoom: https://harvard.zoom.us/j/977347126

From the viewpoint of spin glass theory, restricted Boltzmann machines represent a veritable challenge, as to the lack of convexity prevents us to use Guerra’s bounds. Therefore even the replica symmetric approximation for the free energy presents some challenges. I will present old and new results around the topic along with some open problems.

Zoom: https://harvard.zoom.us/j/98520388668?pwd=c1hVZk5oc3B6ZTVjUUlTN0J2dmdsQT09

Password: rmtpt2020

Zoom: https://harvard.zoom.us/j/96047767096?pwd=M2djQW5wck9pY25TYmZ1T1RSVk5MZz09

The local Langlands conjectures predict that (packets of) irreducible complex representations of p-adic reductive groups (such as GL_n(Q_p), GSp_2n(Q_p), etc.) should be parametrized by certain representations of the Weil-Deligne group.  A special role in this hypothetical correspondence is held by the supercuspidal representations, which generically are expected to correspond to irreducible objects on the Galois side, and which serve as building blocks for all irreducible representations.  Motivated by recent advances in the mod-p local Langlands program (i.e., with mod-p coefficients instead of complex coefficients), I will give an overview of what is known about supersingular representations of p-adic reductive groups, which are the “mod-p coefficients” analogs of supercuspidal representations.  This is joint work with Florian Herzig and Marie-France Vigneras.

Zoom: https://harvard.zoom.us/j/96767001802

Password: The order of the permutation group on 9 elements.

The purest forms of functional programming use monads to define computations that happen within contexts. For instance, the IO monad, which is a standard object in Haskell as well as languages inspired by Haskell, is used to handle processes that require interaction with the outside world. Monads are the dread of many fledgling programmers learning functional programming for the first time, but they are actually familiar constructions from category theory. This talk will discuss the definitions of monad in functional programming and category theory and describe how they are manifested in the context of a Haskell program that reads in and prints an integer.

Zoom: https://harvard.zoom.us/j/96759150216?pwd=Tk1kZlZ3ZGJOVWdTU3JjN2g4MjdrZz09

The twisted bilayer graphene (TBG) near the magic angle around 1 degree hosts topological flat moiré electron bands, and exhibits a rich tunable strongly interacting physics. Correlated insulators and Chern insulators have been observed at integer fillings nu=0,+-1,+-2,+-3 (number of electrons per moiré unit cell). I will first talk about the enhanced U(4) or U(4)xU(4) symmetries of the projected TBG Hamiltonian with Coulomb interaction in various combinations of the flat band limit and two chiral limits. The symmetries in the first chiral and/or flat limits allow us to identify exact or approximate ground/low-energy (Chern) insulator states at all the integer fillings nu under a weak assumption, and to exactly compute charge +-1, +-2 and neutral excitations. In the realistic case away from the first chiral and flat band limits, we find perturbatively that the ground state at integer fillings nu has Chern number +-mod(nu,2), which is intervalley coherent if nu=0,+-1,+-2, and is valley polarized if nu=+-3. We further show that at nu=+-1 and +-2, a first order phase transition to a Chern number 4-|nu| state occurs in an out-of-plane magnetic field. Our calculation of excitations also rules out the Cooper pairing at integer fillings nu from Coulomb interaction in the flat band limit, suggesting other superconductivity mechanisms. These analytical results at nonzero fillings are further verified by a full Hilbert space exact diagonalization (ED) calculation. Furthermore, our ED calculation for nu=-3 implies a phase transition to possible translationally breaking or metallic phases at large deviation from the first chiral limit.

Zoom: https://harvard.zoom.us/j/977347126