The Christoffel-Darboux kernel is the reproducing kernel associated to the Hilbert space containing all polynomials up to a given degree. It can be naturally written in terms of any complete set of orthonormal polynomials. In classical analysis the Christoffel-Darboux kernel is useful for studying properties of the underlying measure with respect to which the Hilbert space of polynomials is defined. In this talk, we present the version of the Christoffel-Darboux kernel for $L^2$ spaces of tracial states on noncommutative polynomials. We view this kernel as a noncommutative function, and identify its values as maxima of certain sets of  non-negative matrices/operators.

In numerous cases, the classical version of the Christoffel-Darboux kernel can be used (after renormalization) to recover the measure to which it is associated as a weak derivative. This is done with the aid of the theory of plurisubharmonic functions. We use this same theory in order to introduce several noncommutative versions of the Siciak extremal function. We use the Siciak functions to prove that, in several cases of interest, the (properly normalized) limit of the evaluations of the Christoffel-Darboux kernel on matrix sets exists as a well-defined, quasi-everywhere finite plurisubharmonic function. Time permitting, we conclude with some conjectures regarding these objects. This is based on joint work with Victor Magron (LAAS) and Victor Vinnikov (Ben Gurion
University).

This talk describes work in progress computing the $H\underline{\mathbb{F}}_2$ homology of the $C_2$-equivariant Eilenberg-Maclane spaces associated to the constant Mackey functor $\underline{\mathbb{F}}_2$. We extend a Hopf ring argument of Ravenel-Wilson computing the mod p homology of non-equivariant Eilenberg-Maclane spaces to the $RO(C_2)$-graded setting. An important tool that arises in this equivariant context is the twisted bar spectral sequence which is quite complicated, lacking an explicit $E^2$ page and having arbitrarily long equivariant degree shifting differentials. We avoid working directly with these differentials and instead use a computational lemma of Behrens-Wilson along with norm and restriction maps to complete the computation.

Finding canonical metrics on compact Kähler varieties has been an intense topic of research for decades. A famous result of Yau says that every compact Kähler manifold with non-positive first Chern class admits a Kähler-Einstein metric (when the Chern class is negative this was also independently proved by Aubin). In this talk, I’ll present some recent joint works with Hamid Abban, Yuchen Liu and Chenyang Xu on the existence of Kähler-Einstein metrics when the first Chern class is positive and the variety is possibly singular (such varieties are called Fano varieties). I’ll focus on two particular aspects: the solution of the YauTian-Donaldson conjecture, which predicts that the existence of Kähler-Einstein metrics on Fano varieties is equivalent to an algebro-geometric stability condition called K-polystability, and a systematic approach (using birational geometry) to decide whether Kähler-Einstein metrics exist on explicit Fano varieties.

Ever find yourself lost in the woods? In this talk, we will speak for the trees with an overview of decision tree models and ensemble methods, including (but not limited to) random forests and XGBoost. We'll also discuss considerations of building such models and some applications. This talk does not require any background knowledge in machine learning.

We study exactness condition for Lasserre’s relaxation method for polynomial optimization problem with n variables under equality constraints defined by n polynomials. Under the assumption that the quotient ring has dimension equal to the product of the degrees of the n equality defining polynomials, we obtain an explicit bound on the order of Lasserre’s relaxation which guarantees exactness. When the common zero locus are real and all of multiplicity one, we describe the exact region as the convex hull of the moment map image of a vector subspace. For the relaxation of order equal to the explicit bound minus one, the convex hull coincides with the moment map image, and is diffeomorphic to its amoeba. Based on the theory of amoeba, we obtain an explicit description of the exact region, from which we further derive error estimations for relaxation of this specific order.

We overview the recently developed level set approach to the existence theory of minimal submanifolds and present some joint work with A. Pigati and D. Stern. The underlying idea is to construct minimal hypersurfaces as limits of nodal sets of critical points of functionals. After starting with a general overview of the codimension one theory, we will move to the higher codimension setting, and introduce the self-dual Yang-Mills-Higgs functionals. These are a natural family of energies associated to sections and metric connections of Hermitian line bundles, whose critical points have long been studied in gauge theory. We will explain to what extent the variational theory of these energies is related to the one of the (n - 2)-area functional and how one can interpret the former as a relaxation/regularization of the latter. We will mention some elements of the proof, with special emphasis on the role played by the gradient flow.

We present the probabilistic numeric solver BayesCG, for solving linear systems with real symmetric positive definite coefficient matrices. BayesCG is an uncertainty aware extension of the conjugate gradient (CG) method that performs solution-based inference with Gaussian distributions to capture the uncertainty in the solution due to early termination. Under a structure exploiting 'Krylov' prior, BayesCG produces the same iterates as CG. The Krylov posterior covariances have low rank, and are maintained in factored form to preserve symmetry and positive semi-definiteness. This allows efficient generation of accurate samples to probe uncertainty in subsequent computations.

Recent advances in the theory of smooth mod $p$ representations of a $p$-adic reductive group $G$ involve more and more derived methods.  It becomes increasingly clear that the proper framework to study smooth mod $p$ representations is the derived category $D(G)$.

I will talk about smooth mod $p$ representations and highlight their shortcomings compared to, say, smooth complex representations of $G$.  After explaining how the situation improves in the derived category, I will spend the remaining time on the left adjoint of the derived parabolic induction functor.
 

Let ${\mathbb{H}^n}$ be the hyperbolic 𝑛-space and Γ be a geometrically finite discrete subgroup in Isom$_+$(${\mathbb{H}^n}$) with parabolic elements. We investigate whether the geodesic flow (resp. the frame flow) over the unit tangent bundle T$^1$ (Γ \ ${\mathbb{H}^n}$) (resp. the frame bundle F(Γ \ ${\mathbb{H}^n}$)) mixes exponentially. This result has many applications, including spectral theory, prime geodesic theorems, orbit counting, equidistribution, etc.

I will start with a survey of the past results, methods, and related problems on this topic. Along the way, I will present the joint work with Jialun Li, Pratyush Sarkar.

For complex smooth threefolds, there are enumerative theories of curves defined using sheaves, such as Donaldson-Thomas (DT) theory using ideal sheaves and Pandharipande-Thomas (PT) theory using stable pairs. These theories are conjecturally related among themselves and conjecturally related to other enumerative theories of curves, such as Gromov-Witten theory. The conjectural relation between DT and PT theories is known only for Calabi-Yau threefolds by work of Bridgeland, Toda, where one can use the powerful machinery of motivic Hall algebras due to Joyce and his collaborators. Bryan-Steinberg (BS) defined enumerative invariants for Calabi-Yau threefolds $Y$ with certain contraction maps $Y\rightarrow X$. I plan to explain how to extend their definition beyond the Calabi-Yau case and what is the conjectural relation to the other enumerative theories. This conjectural relation is known in the Calabi-Yau case by work of Bryan-Steinberg using the motivic Hall algebra. In contrast to the DT/ PT correspondence, we manage to establish the BS/ PT correspondence in some non-Calabi-Yau situations.