Homological perturbation theory for cyclic L-infinity algebras
Homological perturbation theory is a way of transporting algebraic structures from a complex to a homotopy retract. For example, a homotopy retract of a dg algebra has a canonical A-infinity structure (under suitable convergence hypotheses), and a homotopy retract of a dg Lie algebra has a canonical L-infinity structure. There is also a canonical A-infinity/L-infinity morphism from the original dg (Lie) algebra to its homotopy retract.
The problem I address in this talk, motivated by questions in mathematical physics, is to extend this story to cyclic L-infinity algebras: according to Kontsevich, these are essentially the same thing as symplectic formal derived stacks, and the variant of homological perturbation theory that I will explain has a differential geometric flavour, since it is based on integrating a vector field on this derived stack.
Time permitting, I will mention how to carry out similar arguments in the associative, and commutative, settings. In fact, all of these constructions extend to algebras over a cyclic Koszul operad.
I will not assume prior familiarity with L-infinity algebras, or homological perturbation theory.
A categorical approach to real Lie groups
The representation theory of real reductive Lie groups, such as SL(2,R), has played a fundamental role in mathematics for decades, with beautiful connections to number theory (through automorphic forms and the Langlands program), mathematical physics (through conservation laws and quantum states) and harmonic analysis (through nonabelian Fourier transform). Because of its varied applications, it has been studied through many different lenses – there is a rich set of geometric, algebraic, and analytic tools to describe admissible and unitary representations. In this talk, I will discuss a categorical perspective on this classical story, which emphases the combinatorial role of the Weyl group and its subgroups. Specifically, I will introduce an algebraic family of categories constructed from only Weyl group data, which encode information about characters of admissible representations of a real reductive group. These categories arise as module categories over the monoidal category of Soergel bimodules.
More than two thirds of the zeros of the Riemann zeta function are simple and on the critical line
The talk will be aimed at a general mathematical audience
The Riemann Hypothesis asserts that all non-trivial zeros of the Riemann zeta function lie on the critical line Re(s)=1/2. In the absence of a proof of the Riemann hypothesis, a fundamental problem has been to determine what proportion of the zeros can be shown to lie on the critical line.
Three weeks ago, an internal research version of Claude developed by Anthropic announced a breakthrough result, later verified by Anthropic mathematicians Levent Alpöge and Ralph Furman, showing unconditionally that more than (67.25%) of the zeros are simple and lie on the critical line. Their argument is technically intricate and its main mechanism is not transparent. I will present a new, shorter and conceptually simpler proof of this result.
[Number theory talk] On the average of class numbers of real quadratic fields
The average behaviour of class numbers of real quadratic fields is still poorly understood. Let $S(X)$ denote the sum of the class numbers of real quadratic fields with discriminant at most $X$. The class number formula gives the elementary bound $S(X)\ll X^{3/2}$, while a classical result of Siegel gives the best known upper bound $S(X)\ll X^{3/2}/\log X$. On the other hand, Hooley conjectured that $S(X)$ should be of order $X(\log X)2$. There is therefore a substantial gap between what is known and what is expected. In this talk, I will discuss this problem and some recent progress on it.
Langlands duality for 3-manifolds and Donaldson–Thomas theory
Algebraic Geometry talk
Verlinde formula for chiral homology
In two-dimensional conformal field theory, local holomorphic observables can be modeled by vertex operator algebras. Their global counterparts, the spaces of conformal blocks, form a twisted D-module over the moduli space of smooth curves. In certain cases, conformal blocks are known to extend to stable curves in such a way that they satisfy a gluing formula relating the conformal blocks of a nodal curve to those of its normalization. When these D-modules are flat connections, this reduces the computation of their rank to the genus-zero case, leading to the famous Verlinde formula. I will describe a geometric implementation of these ideas in the language of Beilinson and Drinfeld’s chiral and factorization algebras. In particular, this framework yields a gluing formula for the derived enhancement of conformal blocks given by chiral homology.
Prime divisors of Hecke eigenvalues of Ikeda lifts
In this talk, we will discuss bounds on the largest prime factor of Fourier coefficients of normalized Hecke eigenforms. Further, we will study the positivity of Hecke eigenvalues of Ikeda lifts and bounds on the large prime divisors of these Hecke eigenvalues. These are joint works with Yuri Bilu, and Sanoli Gun.
Derived operadic centers in deformation quantization
It is well known that Hochschild cohomology and, in particular, the algebraic structure it carries, plays a central role in studying the infinitesimal noncommutative deformations of geometric spaces. We construct a canonical solution to Deligne’s conjecture for Hochschild cochains on a scheme, even for the singular case, by exhibiting the Hochschild complex as an ∞-operadic center. We show that this equips the Hochschild complex with a universal E2-algebra structure that precisely agrees with the classical Gerstenhaber bracket and cup product on cohomology in the affine and smooth cases. Our motivation stems from the mysterious appearance of the square root of the Todd genus in Kontsevich’s formality theorem in deformation quantization, as well as the conjectural relationships between these objects and the motivic Galois group.
Group actions on curves and applications
In this talk we will focus on one arithmetic setting: the action of a finite group G on a curve X. The idea is to pass to its Jacobian variety and let the representation theory of G do the work. This splits the Jacobian, up to isogeny, into smaller factors. We will explain this method and give some examples.
As an application, I will turn to Tate--Shafarevich groups. These play an important role in the study of rational points on abelian varieties, but they are notoriously difficult to compute, or even to prove finite. For an elliptic curve, the Tate--Shafarevich group has square order when finite. This is not always true for abelian varieties of higher dimension, a fact that was long overlooked and often misstated in the literature. We will show that every square-free positive integer appears as the square-free part of the order of the Tate--Shafarevich group of an abelian variety over the rationals.
The Carleson project: a collaborative formalization
Mathematical formalization consists of digitizing mathematical definitions and results using a `proof assistant', a computer program capable of checking whether a proposition can be deduced from a set of inference rules and a collection of basic axioms. In recent years, the community of mathematicians working on formalization has grown rapidly and has reached milestones that demonstrate the ability to formalize results at the frontier of knowledge. Proof assistants have applications to mathematics research, teaching, and communication.
In this talk I will describe a collaborative project whose goal was to formalize a result from modern harmonic analysis. The talk will focus on the organization of the collaboration and the lessons we learned during the formalization process.
A well-known result in Fourier analysis establishes that the partial Fourier sums of a smooth periodic function $f$ converge uniformly to $f$, but the situation is a lot more subtle in more general settings (for instance, when $f$ is a continuous function). However, in 1966 Carleson proved that they do converge at almost all points for $L^2$ periodic functions on the real line. Carleson’s proof is famously hard to read, and there are no known easy proofs of this theorem.
We formalized in Lean a generalization of Carleson’s theorem in the setting of doubling metric measure spaces (proven by the Bonn harmonic analysis group in 2023), and Carleson's original result as a corollary.
Chromatic Homotopy Theory: Why do we keep talking about it?
For half a century, chromatic homotopy theory has provided us one framework for organizing computations and the search of large-scale phenomena. We can reasonably ask why. As one answer, I'll give a highly anecdotal and perhaps idiosyncratic survey of the field, past and present and future, with the explicit aim of highlighting the essential contributions of Neil Strickland in making it all go.
Galois and separable extensions of Tambara functors
I will explain how to transfer the notions of separability and Galois extensions from ordinary to equivariant algebra. The definition of separability of Tambara functors is a straightforward generalization. I'll show that the Burnside Tambara functor is separably closed. The notion of Galois extensions of Tambara functors is closely modelled on the one for commutative rings and it turns out that Galois extensions are separable. I'll discuss some examples and show how to interpret Nullstellensatzian objects in the category of Tambara functors in terms of Galois extensions.
From cubical structures to the string orientation
Witten constructed a modular form-valued genus for any manifold with a $U<6>$-structure. Motivated by this, Ando, Hopkins and Strickland characterized all ring maps from $MU<6>$ to even-periodic spectra in terms of cubical structures and characterized the Witten genus in these terms.
In this talk, I will explore how one can combine their insights with equivariant topological modular forms to try to construct an equivariant string orientation, further refining the Witten genus.
Equivariant Twisted R-algebras via Thom Spectra
For a $C_2$-commutative ring spectrum $R$, a twisted $R$-algebra is an $R$-module with a multiplication whose order is switched by the $C_2$-action. In this talk, we shall construct various quotients of $R$ as twisted $R$-algebras, when $R$ is an even real commutative ring spectrum. These are constructed as Thom spectra of maps out of suitable $C_2$ -actions on $S^1$ and $U(n)$. One such example is given by $KR/2$ which is endowed with a twisted $KR$-algebra structure. In the context of twisted $R$-algebras, one may consider the real topological Hochschild homology, and for Thom spectra, one has a nice formula again as a Thom spectrum. We use this to obtain computations for the real topological Hochschild homology of $KR/2$ as a twisted $KR$-algebra. The computation also involves a splitting of the units spectrum, which is an analogue of the classical non-equivariant splitting.
Structured Real Orientations
In joint work with Ryan Quinn we developed a technique to produce structured equivariant orientations by lifting their underlying structured non-equivariant orientations. This allows us to lift many maps of interest, such as the Hirzebruch level-n genera or the Hahn-Shi orientations. In fact, our result is robust enough to lift structured equivalences resulting in a structured Real Snaith theorem, a structured Real Bökstedt periodicity, and finally the first multiplicative version of the Real Brown-Peterson spectrum.
Bökstedt periodicity and the even filtration
We will review Bökstedt periodicity by observing that the Eilenberg-Mac Lane $F_2$ is an $E_3$-MU-Thom spectrum and studying the even filtration. Along the way we will introduce the group schemes of cocycles $C_k(\hat{G}_a, G_m)$ studied by Ando-Hopkins-Strickland in Elliptic spectra, the Witten genus and the theorem of the cube.
Parametrized cohomology of the classifying space for $\mathbb{Z}/2$-line bundles
The Thom isomorphism fails for equivariant vector bundles in $RO(G)$-graded cohomology, even for $G=\mathbb{Z}/2$. Costenoble--Waner developed a parametrized equivariant cohomology theory incorporating Thom isomorphisms for all vector bundles. This cohomology theory extends the usual representation grading $RO(G)$ to $RO(\Pi_G B)$, representations of the equivariant fundamental groupoid. We compute the parametrized cohomology of the classifying space for real $\mathbb{Z}/2$-line bundles, $B_{C_2}O(1)$. This is joint work with Agnès Beaudry, Chloe Lewis, Sabrina Pauli, and Elizabeth Tatum.
Geometry of singularities and homotopy theory
I will outline a recent proof of the Donovan–Wemyss Conjecture, which strikingly links the birational geometry of threefold singularities with derived equivalences of finite-dimensional algebras. Central to this approach is a homological reinterpretation: contraction algebras arising from crepant resolutions of cDV singularities can be realized as derived endomorphism algebras of 2Z-cluster tilting objects in the singularity category. Leveraging the derived Auslander–Iyama correspondence, we deduce that derived equivalences of such algebras reflect isomorphisms of the underlying singularities. A key player in this story is the restricted universal Massey product, which acts as a fine invariant distinguishing derived structures. This is joint work with Gustavo Jasso and Bernhard Keller. This talk aims to showcase how geometry, representation theory, and homotopy theory intertwine in proving a deep and elegant conjecture.
Rational G-spectra with Weyl-finite isotropy
For a compact Lie group G, we may study G-equivariant cohomology theories, and it is conjectured that there is an abelian category A(G) so that there is an equivalence between rational G-spectra = dg A(G).
The category A(G) is (in a sense to be made precise) a category of sheaves over the space X(G) of conjugacy classes of subgroups of G. In fact X(G) is a finite disjoint union of blocks V(G,H) (where H is a subgroup with finite Weyl group) so A(G)=A(V(G,H1)) x …… x A(V(G,Hn)).
The talk will describe some progress understanding the structure of blocks in general and explain the algebraic model A(V(G,H)) for Weyl-finite blocks V(G,H). More generally it will describe the category of rational G-spectra with any specified Weyl-finite geometric isotropy.
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