The Rayleigh-Ritz method approximates the eigenvalues of a large
Hermitian matrix $B$ with Ritz values, which are the eigenvalues of $B$'s
restriction to a smaller trial subspace $S$. We can view the $k$-th
Ritz value as a real-valued function ("Ritz energy landscape") on the
manifold of all possible s-dimensional trial subspaces, the
Grassmannian $Gr_s(C^n)$ (or $Gr_s(R^n)$ for the real symmetric case).
Motivated by questions from quantum chemistry and spectral
optimization, this talk explores the topology of the Ritz energy
landscape. A Morse function is called "perfect" if it describes the
topology of its domain in the most efficient way possible, meaning the
number of its critical points of each type exactly matches the
corresponding Betti number of the space. We demonstrate that for a
matrix $B$ with distinct eigenvalues, the Ritz landscape is indeed
perfect. While the function itself is not everywhere smooth and its
critical points are not isolated --- and not even Morse-Bott --- its
critical structure is nevertheless well-defined and ultimately
reflects the topology of the Grassmannian in a minimal, perfect way.
To be more precise, we show that the filtration of the Grassmannian by
the sublevel sets of the $k$-th Ritz value is homologically perfect.
The proof proceeds by introducing a suitable perturbation which
ensures that points of non-smoothness are not critical (by a theorem
of Zelenko and the presenter) and that the remaining smooth critical
points are isolated.
Based on a joint work with Mark Goresky (IAS).
An important special case of Schubert calculus for the Grassmannian
concerns flags osculating the rational normal curve, which is equivalent
to the Bethe Ansatz in the Gaudin model for $\mathfrak{gl}_d$. The most natural
family of these problems are indexed by partitions $\lambda$. Liao and
Rybnikov recently studied a subgroup of the monodromy group for the
Bethe Ansatz equivalent to the action of the cactus group on standard
Young tableau of shape $\lambda$. When $\lambda$ is a hook or is symmetric,
they showed that it was not 2-transitive, but was otherwise giant
(contains the alternating group).
I will describe this background and then give some geometric arguments
which refine their work on hooks and symmetric partitions, and then
present some computational evidence that Harris's principle holds in
that the monodromy is a large as possible. This is based on joint work with Leonid Rybnikov (UdeM).
n this talk, I will present a detailed study of the curvature and symplectic asphericity properties of symmetric products of curves. I show that these spaces can be used to answer nuanced questions arising in the study of closed Riemannian manifolds with positive scalar curvature. For example, symmetric products of curves sharply distinguish between two distinct notions of macroscopic dimension introduced by Gromov and Dranishnikov. As a natural generalization of this circle of ideas, I will also address the Gromov–Lawson and Gromov conjectures in the Kaehler projective setting and draw new connections between the theories of the minimal model, positivity in algebraic geometry, and macroscopic dimensions. This is joint work with Alexander Dranishnikov and Ekansh Jauhari.
The study of the closest point(s) on a statistical model from a given distribution in the probability simplex with respect to a fixed Wasserstein metric yields a polyhedral norm-distance optimization problem.
There are two components to the complexity of computing the Wasserstein distance between a data point and a model.
One is the combinatorial complexity, governed by the combinatorics of the Lipschitz polytope of the finite metric to be used.
Another is the algebraic complexity, which is governed by the polar degrees of the Zariski closure of the model.
In this talk, I will discuss the formulas for the polar degrees of rational normal scrolls and graphical models whose underlying graphs are star trees.
If time permits, I will discuss our efforts to compute polar degrees for graphical models with four binary random variables, including the path on four vertices and the four-cycle, as well as for small, no-three-way-interaction models.
We show that the homogeneous ideals of secant varieties of smooth projective curves and surfaces in sufficiently ample embeddings are determinantally presented. The same result holds for the first secant varieties of arbitrary smooth projective varieties in sufficiently ample embeddings. This completely settles a conjecture of Eisenbud-Koh-Stillman for curves and partially resolves a conjecture of Sidman-Smith in higher dimensions. To establish the results, we employ the geometry of Hilbert schemes of points. Based on our method, we also prove that the homogeneous ideals of arbitrary projective schemes in sufficiently ample embeddings are generated by quadrics of rank three, confirming a conjecture of Han-Lee-Moon-Park. This is joint work with Daniele Agostini.
In 1994 and 2006 survey articles, Rosenberg and Stolz stated a series of conjectures by "persisting" non-positive scalar curvature metrics from a closed manifold $ X $ to the product space between $ X $ and a real line, the 2-plane, or a circle, respectively.
There are two classical methods to study this type of problem: one is the method of index-theoretic obstructions, and the other is through geometric measure theory initiated by minimal hypersurfaces, then developed by e.g. Gromov's $ \mu $-bubble method.
In this talk, we introduce a different, conformal geometry and PDE approach to partially answer a series of Rosenberg-Stolz conjectures, and Gromov-Lawson type scalar and mean curvature comparison results for all dimensions from "transposing the positivity" point of view. We also show its application in complex geometry. Some of my work involves collaboration with S. Rosenberg.
We argue that protected data of $4d$ $N=2$ SCFTs admits a purely algebro-geometric characterization. We conjecture that both the Macdonald index (and hence the Schur index) and the Higgs branch are encoded by a bifiltered affine scheme determined by OPE nilpotency and decoupling relations. Focusing on Argyres–Douglas theories, where the Higgs branch is a point, we show that this geometric construction suffices to reconstruct the full Macdonald index. This is based on recent works with Craig Lawrie and Jaewon Song.
A 3-dimensional subspace $f$ of real polynomials defines a map $f\colon {\mathbb P}^1 \to {\mathbb P}^2$ whose image is a rational plane curve.
It is maximally inflected when all of its flexes are real, equivalently, when its Wronski determinant has only real roots.
We associate two {\it a priori} distinct signs ($\pm 1$) to $f$: the Welschinger invariant of the rational curve and the degree of
the Wronski map at $f$. Extensive computation suggests that these signs coincide. While studying the conjecture,
we were led to a deeper conjecture: We define a mixed Wronskian, a function ${\mathbb P}^1 \to {\mathbb P}^1$. The
inverse image of the positive reals encodes the real geometry of $f$ and conjecturally is an object called a web.
We conjecture that known bijections between webs and standard Young tableaux and between tableaux with maximally inflected curves
recovers the curve.
This talk will explain this picture with compelling evidence and beautiful
pictures. It is joint work with Brazelton, Karp, Le,
Levinson, McKean, Peltola, and Speyer.
Let $G$ be a connected simply-connected simple algebraic group over $\mathbb{C}$ and let $T$ be a maximal torus, $B\supset T$ a Borel subgroup and $K$ a maximal compact subgroup. Then, the product in the (algebraic) based loop group $\Omega(K)$ gives rise to a comultiplication in the topological $T$-equivariant $K$-ring $K_T^{\rm{top}}(\Omega(K))$. Recall that $\Omega(K)$ is identified with the affine Grassmannian $\mathcal{X}$ (of $G$) and hence we get a comultiplication in
$ K_T^{\rm{top}}(\mathcal{X})$. Dualizing, one gets the Pontryagin product in the $T$-equivariant $K$-homology $K^T_0(\mathcal{X})$, which in-turn gets identified with the convolution product (due to S. Kato).
Now, $ K_T^{\rm{top}}(\mathcal{X})$ has a basis $\{\xi^w\}$ over the representation ring $R(T)$
given by the ideal sheaves corresponding to the finite codimension Schubert varieties $X^w$ in $\mathcal{X}$. We make a positivity conjecture on the comultiplication structure constants in the above basis. Using some results of Kato, this conjecture gives rise to an equivalent conjecture on the positivity of the multiplicative structure constants in $T$-equivariant quantum $K$-theory $QK_T(G/B)$ in the Schubert basis Let $G$ be a connected simply-connected simple algebraic group over $\mathbb{C}$ and let $T$ be a maximal torus, $B\supset T$ a Borel subgroup and $K$ a maximal compact subgroup. Then, the product in the (algebraic) based loop group $\Omega(K)$ gives rise to a comultiplication in the topological $T$-equivariant $K$-ring $K_T^{\rm{top}}(\Omega(K))$. Recall that $\Omega(K)$ is identified with the affine Grassmannian $\mathcal{X}$ (of $G$) and hence we get a comultiplication in
$ K_T^{\rm{top}}(\mathcal{X})$. Dualizing, one gets the Pontryagin product in the $T$-equivariant $K$-homology $K^T_0(\mathcal{X})$, which in-turn gets identified with the convolution product (due to S. Kato).
Now, $ K_T^{\rm{top}}(\mathcal{X})$ has a basis $\{\xi^w\}$ over the representation ring $R(T)$
given by the ideal sheaves corresponding to the finite codimension Schubert varieties $X^w$ in $\mathcal{X}$. We make a positivity conjecture on the comultiplication structure constants in the above basis. Using some results of Kato, this conjecture gives rise to an equivalent conjecture on the positivity of the multiplicative structure constants in $T$-equivariant quantum $K$-theory $QK_T(G/B)$ in the Schubert basis
The border rank of tensors is a widely studied topic with practical applications to theoretical computer science and algebraic statistics.
Lower bounds on the border rank of the matrix multiplication tensor were obtained using techniques from representation theory and
algebraic geometry.
In this talk, we will prove non-trivial border rank lower bounds for a class of $GL(V)$-invariant tensors
using Young flattenings constructed by Wu.
We will see how this comes down to proving results on ranks of certain maps between Schur functors,
the proofs of which surprisingly use deep results in representation theory and commutative algebra
