Calendar

For many classes of graphs that arise naturally in discrete geometry (for example intersection graphs of segments or disks in the plane), the edges of these graphs can be defined algebraically using the signs of a finite list of fixed polynomials. We investigate the number of n-vertex graphs in such an algebraically defined class of graphs. Warren’s theorem (a variant of a theorem of Milnor and Thom) implies upper bounds for the number of n-vertex graphs in such graph classes, but all the previously known lower bounds were obtained from ad hoc constructions for very specific classes. We prove a general theorem giving a lower bound for this number (under some reasonable assumptions on the fixed list of polynomials), and this lower bound essentially matches the upper bound from Warren’s theorem.

Zoom link: https://harvard.zoom.us/j/99715031954?pwd=eVRvbERvUWtOWU9Vc3M2bjN3VndBQT09

Password: 1251442

Progress in satisfiability (SAT) solving has made it possible to
determine the correctness of complex systems and answer long-standing
open questions in mathematics. The SAT solving approach is completely
automatic and can produce clever though potentially gigantic proofs.
We can have confidence in the correctness of the answers because
highly trustworthy systems can validate the underlying proofs
regardless of their size.

We demonstrate the effectiveness of the SAT approach by presenting
some recent successes, including the solution of the Boolean
Pythagorean Triples problem, computing the fifth Schur number, and
resolving the remaining case of Keller’s conjecture. Moreover, we
constructed and validated a proof for each of these results. The
second part of the talk focuses on notorious math challenges for which
automated reasoning may well be suitable. In particular, we discuss
our progress on applying SAT solving techniques to the chromatic
number of the plane (Hadwiger-Nelson problem), optimal schemes for
matrix multiplication, and the Collatz conjecture.

https://harvard.zoom.us/j/99651364593?pwd=Q1R0RTMrZ2NZQjg1U1ZOaUYzSE02QT09

The étale cohomology of varieties over Q enjoys a Galois action. In the case of Hilbert modular varieties, Nekovář-Scholl observed that this Galois action on the level of cohomology extends to a much larger profinite group: the plectic group. They conjectured that this extension holds even on the level of complexes, as well as for more general Shimura varieties.

We present a proof of the analogue of this conjecture for local Shimura varieties. This includes (the generic fibers of) Lubin–Tate spaces, Drinfeld upper half spaces, and more generally Rapoport–Zink spaces. The proof crucially uses Scholze’s theory of diamonds.

will speak on:

“Some remarks on contact Calabi-Yau 7-manifolds”

Wednesday, October 13, 2021

4:00 – 5:00 PM  *Hong Kong time*

4:00 – 5:00 AM *Eastern Time*

Abstract: In geometry and physics it has proved useful to relate G2 and Calabi-Yau geometry via circle bundles. Contact Calabi-Yau 7-manifolds are, in the simplest cases, such circle bundles over Calabi-Yau 3-orbifolds. These 7-manifolds provide testing grounds for the study of geometric flows which seek to find torsion-free G2-structures (and thus Ricci flat metrics with exceptional holonomy). They also give useful backgrounds to examine the heterotic G2 system (also known as the G2-Hull-Strominger system), which is a coupled set of PDEs arising from physics that involves the G2-structure and gauge theory on the 7-manifold. I will report on recent progress on both of these directions in the study of contact Calabi-Yau 7-manifolds, which is joint work with H. Sá Earp and J. Saavedra.

Link: https://cuhk.zoom.us/j/91936025861

Meeting ID: 919 3602 5861
Passcode: 20211013

In multi-agent systems the complex interaction of fixed incentives can lead agents to outcomes that are poor (inefficient) not only for the group but also for each individual agent. Price of anarchy is a technical game theoretic definition introduced to quantify the inefficiency arising in these scenarios– it compares the welfare that can be achieved through perfect coordination against that achieved by self-interested agents at a Nash equilibrium. We derive a differentiable upper bound on a price of anarchy that agents can cheaply estimate during learning. Equipped with this estimator agents can adjust their incentives in a way that improves the efficiency incurred at a Nash equilibrium. Agents adjust their incentives by learning to mix their reward (equiv. negative loss) with that of other agents by following the gradient of our derived upper bound. We refer to this approach as D3C. In the case where agent incentives are differentiable D3C resembles the celebrated Win-Stay Lose-Shift strategy from behavioral game theory thereby establishing a connection between the global goal of maximum welfare and an established agent-centric learning rule. In the non-differentiable setting as is common in multiagent reinforcement learning we show the upper bound can be reduced via evolutionary strategies until a compromise is reached in a distributed fashion. We demonstrate that D3C improves outcomes for each agent and the group as a whole on several social dilemmas including a traffic network exhibiting Braess’s paradox a prisoner’s dilemma and several reinforcement learning domains.

Zoom ID: 950 2372 5230 (Password: cmsa)

Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models. In this talk, I will illustrate how this challenge can be overcome by elucidating the probabilistic connections between models of different levels of detail. These connections explain how stochastic partial differential equations (SPDE) arise naturally from particle models.

I will also present some novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE not only interpolate between particle models and PDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of the underlying population dynamics.

Zoom link: https://harvard.zoom.us/j/96657833341

Password: cmsa

Inspired by a result of Esnault and Srinivas on automorphisms of surfaces and recent advances in complex dynamics, Truong raised a question on the comparison of two dynamical degrees, which are defined using pullback actions of dynamical correspondences on numerical cycle class groups and cohomology groups, respectively. An affirmative answer to his question would surprisingly imply Weil’s Riemann hypothesis. In this talk, I first discuss a special case of Abelian varieties. Then I will introduce the so-called dynamical correspondence and its application to certain surfaces. This is based on joint work with Tuyen Truong.

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