khovanov homology is a knot invariant, which generalizes and more specifically categorifies the jones
polynomial of knots. however, in the construction of khovanov homology we seem to make several
choices in the frobenius algebra that we consider. in this talk, we will discuss a distinct alternative to
khovanov homology known as lee theory, in which we make different choices on our frobenius algebra.
these different choices break the q-grading, but allow us to create a filtration, and use this to define a
rasmussen's s-invariant. we will describe several properties and applications of this invariant.

the notion of an elliptic partial differential equation (pde) goes back at least to 1908, when it appeared in a paper j. hadamard. in this talk we present a recently discovered structural condition, called $p$-ellipticity, which generalizes classical ellipticity. it was co-discovered independently by carbonaro and dragicevic on one hand, and pipher and myself on the other, and plays a fundamental role in many seemingly unrelated aspects of the $l^p$ theory of elliptic complex-valued pde. so far, $p$-ellipticity has proven to be the key condition for:

(i) convexity of power functions (bellman functions)
(ii) dimension-free bilinear embeddings,
(iii) $l^p$-contractivity and boundedness of semigroups $(p_t^a)_{t>0}$
associated with elliptic operators,
(iv) holomorphic functional calculus,
(v) multilinear analysis,
(vi) regularity theory of elliptic pde with complex coefficients.

during the talk, i will describe my contribution to this development, in particular to (vi).

given an affine coxeter group w, the corresponding shi arrangement is a refinement of the corresponding coxeter hyperplane arrangements which was introduced by shi to study the kazhdan-lusztig cells for w. shi showed, in particular, that each region of a shi arrangement contains exactly one element in w of minimal length. garside shadows in w were introduced to study the word problem of the corresponding artin-tits (braid) group and turns out to produce automata to study the combinatorics of reduced words in w.

in this talk, we will discuss the following conjecture: the set of minimal length elements of the regions in a shi arrangement is a garside shadow. the talk will be illustrated by the example of the affine symmetric group.

we first talk about gage's area-preserving flow (gapf) for simple, smooth and star-shaped curves. it is shown that gapf drives a centrosymmetric initial curve into a circle as time tends to infinity. then we generalize gapf to deform one convex curve to another. this work gives a partial answer to yau's problem on realizing whitney-graustein theorem via curvature flows.

this talk will be a short review of the construction of the stochastic integral and some basic properties. the purpose is to remind the group of how the integral is defined, the importance of the ito isometry, and the ito formula.

the growth in scope and complexity of modern data sets presents the field of statistics and data science with numerous inferential and computational challenges, among them how to deal with various forms of heterogeneity. latent variable models provide a principled approach to modeling heterogeneous collections of data. however, due to the over-parameterization, it has been observed that parameter estimation and latent structures of these models have non-standard statistical and computational behaviors. in this talk, we provide new insights into these behaviors under mixture models, a building block of latent variable models.

from the statistical viewpoint, we propose a general framework for studying the convergence rates of parameter estimation in mixture models based on wasserstein distance. our study makes explicit the links between model singularities, parameter estimation convergence rates, and the algebraic geometry of the parameter space for mixtures of continuous distributions.

from the computational side, we study the non-asymptotic behavior of the em algorithm under the over-specified settings of mixture models in which the likelihood need not be strongly concave, or, equivalently, the fisher information matrix might be singular. focusing on the simple setting of a two-component mixture fit with equal mixture weights to a multivariate gaussian distribution, we demonstrate that em updates converge to a fixed point at euclidean distance $\mathcal{o}((d/n)^{1/4})$ from the true parameter after $\mathcal{o}((n/d)^{1/2})$ steps where $d$ is the dimension.

from the methodological standpoint, we develop computationally efficient optimization-based methods for the multilevel clustering problem based on wasserstein distance. experimental results with large-scale real-world datasets demonstrate the flexibility and scalability of our approaches. if time allows, we further discuss a novel post-processing procedure, named merge-truncate-merge algorithm, to determine the true number of components in a wide class of latent variable models.

low-rank matrix recovery from structured measurements has been a topic of intense study in the last decade and many important problems such as blind deconvolution and matrix completion have been formulated in this framework. an important benchmark method to solve these problems is to minimize the nuclear norm, a convex proxy for the rank. a common approach to establish recovery guarantees for this convex program relies on the construction of a so-called approximate dual certificate. however, this approach provides only limited insight in various respects. most prominently, the noise bounds exhibit seemingly suboptimal dimension factors. in this talk, we will discuss a more geometric viewpoint to analyze the blind deconvolution scenario. we find that for both these applications the dimension factors in the noise bounds are not an artifact of the proof, but the problems are intrinsically badly conditioned. we show, however, that bad conditioning only arises for very small noise levels: under mild assumptions that include many realistic noise levels we derive near-optimal error estimates for blind deconvolution under adversarial noise. at the end, we will briefly discuss how the results can be extended to the scenario of matrix completion.

in this talk, we will focus on how one can deduce some
geometric invariants of an abelian variety or a k3 surface by
studying their frobenius polynomials. in the case of an abelian variety,
we show how to obtain the decomposition of the endomorphism algebra, the
corresponding dimensions, and centers. similarly, by studying the
variation of the geometric picard rank, we obtain a sufficient criterion
for the existence of infinitely many rational curves on a k3 surface of
even geometric picard rank.

we will discuss stabilization of the betti numbers of the moduli space of
sheaves on surfaces, and in the special case of the projective plane we
will produce lower bounds on the second chern class of the sheaves such
that the betti numbers of their moduli space stabilizes.

can an integer $n$ be represented as a sum of two squares $n=x^2+y^2$? if so, how many different representations are there? we begin with the answers to these classical questions due to fermat and jacobi. we then illustrate hurwitz's class number formula for binary quadratic forms, and put all these classical formulas under the modern perspective of the siegel-weil formula. we explain how the latter perspective led gross-keating to discover a new type of identity between arithmetic intersection numbers on modular surfaces and derivatives of certain eisenstein series. after outlining the influential program of kudla and rapoport for generalization to higher dimensions, we report a recent proof (joint with w. zhang) of the kudla-rapoport conjecture and hint at the usage of the uncertainty principle in the proof.

dear math enthusiast, have you ever been afraid that mathematics
contains nothing of substance? are you afraid of realizing that, maybe,
all the proofs near and dear to you are actually easy corollaries of some
deeper, horrendous technicality with no interesting geometric,
number-theoretic, algebraic, combinatorial, or analytic interpretation?
well, you are in luck! we will be providing the simulation of such an
experience. in this week's food for thought talk, we will present
interesting problems, such as a solution to a version of fermat's last
theorem in a field with large enough characteristic, only to realize that
these problems boil down to dry logical tricks!