Abstract: This is a joint work with Jonathan Rosenberg.Let $(M,L)$ be a non-spin spin$^c$ manifold. A metric $g$ on $M$ and a connection $A$ on $L$ determine the associated spin$^c$ Dirac operator $D_L$ on $M$. The Lichnerowicz-Schr\"odinger formula takes the form $$ D_L^2 =\nabla^*\nabla+\frac{1}{4}R^{tw}_{(g,A)}, $$ where $R^{tw}_{(g,A)}=R_g+ 2ic(\Omega)$ is the zeroth order operator (which is called "twisted scalar curvature"). Here $R_g$ is the scalar curvature of $g$ and $2ic(\Omega)$ comes from the curvature $2$-form $\Omega$ of the connection $A$.
There is an affirmative existence result: a closed non-spin simply-connected spin$^c$-manifold $(M,L)$ of dimension $n\geq 5$ admits a pair $(g,A)$ such that $R^{tw}_{(g,A)}$ is greater than $0$ if and only if the index $\alpha^c(M,L):=\textrm{ind} D_L$ vanishes in the complex K-theory $KU_n$.
It turns out there is also a scalar-valued "generalized scalar curvature" $R^{gen}_{(g,A)}$ with the property that the positivity condition of the operator $R^{tw}_{(g,A)}$ is equivalent to the positivity of the scalar function $R^{gen}_{(g,A)}$. I will present several interesting results concerning the curvature $R^{gen}_{(g,A)}$, in particular, on the topology of the relevant space of metrics/connections when this curvature is positive.
Abstract: I will discuss a construction of maps from the homology of Bianchi spaces for an imaginary quadratic field F to second K-groups of ray class fields of F. These maps are “Eisenstein” in the sense that they factor through the quotient by the action of an Eisenstein ideal way from the level. These long-expected maps are direct analogues of known explicit maps in the setting of modular curves and cyclotomic fields. We use a refinement of a method Venkatesh and I developed for constructing Eisenstein cocycles, which I’ll explain. This is joint work with E. Lecouturier, S. Shih, and J. Wang.
Abstract: The field of computability theory studies the complexity of uncomputable problems. In this study, a special role is played by the Halting Problem—i.e. the problem of determining whether a given program stops after a finite number of steps or runs forever. Not only is it the first problem proved to be uncomputable, it also seems to be the simplest "natural" uncomputable problem. Martin's Conjecture is a long-standing open question in computability theory which partially explains why the Halting Problem plays such a special role. A key idea behind Martin's Conjecture is to view the Halting Problem not just as an individual problem, but as an operator on problems, which takes any problem to a strictly harder one. Martin's Conjecture consists of a classification of such operators, which says, in part, that the Halting Problem is the minimal non-trivial operator. I will discuss the background and motivation for Martin's Conjecture, as well as recent progress by Benjamin Siskind and myself which essentially completes a proof of the conjecture for a special class of operators called "order-preserving."
Abstract: Sudden, persistent changes in ecosystem state or configuration, known in ecology as regime shifts, are difficult to predict and a cause of great concern. A large, stable prey population may suddenly collapse to an alternative low-density state in response to a stochastic perturbation, for example, or stochasticity may trigger outbreaks in pest populations that were previously stably suppressed. To explain phenomena like these, ecologists have drawn heavily on deterministic theory that emphasizes the nonlinearities that give rise to bifurcation-induced tipping points, while marginalizing the complex role of stochasticity in driving transitions between states. In this talk, I will discuss how different types of tipping points arise, and how we can use potential functions (including their extensions, such as the quasi-potential) to derive stronger stability concepts that allow us to move beyond classical deterministic theory. Given the pervasive influence of large perturbations in nature, this view promises to yield improved insights into the factors that stabilize or destabilize ecological systems.
Abstract: Algebraic topology is the study of shapes through invariants called cohomology theories. Algebraic geometry explores the geometry of solution sets of polynomial equations over commutative rings. Over the past 50 years, significant progress has been made in bridging these fields, resulting in a rich theory of cohomology theories in algebraic geometry. To bring unity to this landscape, Grothendieck introduced the theory of motives, which are essentially geometric pieces of cohomologies of smooth compact varieties. Unfortunately, not all cohomology theories (e.g. algebraic de Rham cohomology) can be fruitfully understood from this perspective, as they behave in a seemingly pathological fashion on open varieties. In this talk, I explain a new method of systematically overcoming this, by modifying a cohomology theory in such a way that its values on smooth projective varieties remain unchanged.
Abstract: The study of matrix algebras is of fundamental importance in non-commutative analysis, especially in disciplines such as operator algebras. In recent decades, there has been a significant effort to uncover the analytic and probabilistic behavior of large N-limits of matrices. Voiculescu initiated his free probability theory with a crucial insight on the limiting joint spectral distribution of pairs of independent random Gaussian ensembles. This limiting distribution actually is concretely seen inside the von Neumann algebra associated to the free group on two generators. Since this result, there have been several deep contributions in this line of research, including theories of entropy, strong convergence, etc, with powerful applications to the study of operator algebras. In this talk I will describe my recent contributions to this area studying the so-called ultraproduct of matrix algebras, with applications to the internal structure of the free group von Neumann algebras, continuous model theory, and new considerations related to the famous Connes embedding problem which has been recently resolved using quantum complexity theory.
Abstract: I will discuss analytic aspects of the representation theory of graded affine Hecke algebras, which originally arose in the context of certain completely integrable models. This has applications to the automorphic spectrum of reductive groups, and to the unitarisability of certain representations of real reductive groups.
Abstract: One of the main goals of statistical physics is to observe how spins displayed along a lattice Z^d interact together and fluctuate. When the spins belong to a discrete set (for example the celebrated Ising model where spins \sigma_x belong to {-1,+1}), the nature of the phase transitions which arise as one varies the temperature is now rather well understood. When the spins belong instead to a continuous space (for example the unit circle S^1 for the so-called XY model, the unit sphere S^2 for the classical Heisenberg model etc.), the nature of the phase transitions differs drastically from the discrete symmetry setting. The case where the (continuous) symmetry is non-Abelian is currently more mysterious than when the symmetry is Abelian. In the later case, phase transitions are caused by a change of behaviour of certain monodromies in the system called "vortices". They are called topological phase transitions for this reason. In this talk, after an introduction to the mathematics of spin systems with a continuous symmetry, I will present some recent results on these spins systems. One proof will happen to rely on an intriguing Bayesian statistics problem (!). The talk will not require any background in statistical physics/probability and will be based on joint works with Juhan Aru, Paul Dario, Avelio Sepúlveda and Tom Spencer.
Abstract: Quantum eigenvalue(phase) estimation is one of the most important quantum primitives. While numerous quantum algorithms have been proposed to tackle this problem, they often demand substantial quantum resources, making them impractical for early fault-tolerant quantum computers. The talk will begin with a quantum oracle that transforms the quantum eigenvalue estimation problem into a classical signal processing problem. I will then introduce a simple classical subroutine for solving this problem, which surprisingly achieves state-of-the-art complexity results. Additionally, I will review the performance of traditional classical algorithms for this problem and share new results gained from our study. No prior knowledge on quantum computing is needed in this talk.
Abstract: Addressing contemporary problems of collective action—from pandemic management to climate change—requires that we understand the dynamic interplay between information and behavior. In this talk, I will discuss two models of cooperative behavior coupled with dynamics of information spread. In the first model, we will consider cooperation driven by the spread of social reputations. Using methods from evolutionary game theory and dynamical systems, we develop a mathematical model of cooperation that integrates a mechanistic description of how reputations spread through peer-to-peer gossip. We show that sufficiently long periods of gossip can stabilize cooperation by facilitating consensus about reputations. In the second model, we will examine the dynamics of prosociality under political polarization. We develop a stochastic model of game-theoretic opinion dynamics in a multi-dimensional space of political interests. We show that while increasing the diversity of interests can improve both cooperation and social cohesion, strong partisan bias reduces the effective dimensionality of the opinion space via self-sorting along party lines, yielding greater in-group cooperation at the cost of increasing polarization. Taken together, these studies contribute to our understanding of when and how communication and opinion contagion facilitate cooperation.
Abstract: I will discuss the statement and proof of an equivariant main conjecture in Iwasawa theory, building upon recent work of Dasgupta-Kakde on the Galois module structure of the Selmer groups defined by Burns-Kurihara-Sano. I will make connections with my earlier joint results with Greither in geometric Iwasawa theory and give applications to the Galois module structure of the even Quillen K-groups of rings of algebraic integers. This is based on recent joint work with Rusiru Gambheera.
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