how do we make mathematicians? how do we become them ourselves? and why is it rare that we teach that way? in this talk, we will discuss the pedagogy of inquiry-based learning (ibl) and how it can make for a more holistic, equitable, and successful classroom. bring questions and an open mind - there may be a quiz!

we will study complete ricci-flat manifolds with euclidean volume growth. in the case when a tangent cone at infinity of the manifold has smooth cross section, the green function for the laplace equation can be used to define a functional which measures how fast the manifold converges to the tangent cone. using the Łojasiewicz inequality of colding-minicozzi for this functional, we describe how two arbitrarily far apart scales in the manifold can be identified in a natural way. i will also discuss a matrix harnack inequality for the green function when there is an additional condition on sectional curvature, which is an analogue of various matrix harnack inequalities obtained by hamilton and li-cao in time-dependent settings.

short-firm video (sfv) has been exploding on digital platforms recently. the vast amount of videos and fast-evolving trends on digital platforms pose technical challenges in making personalized recommendations. in this work, we introduce a new pure exploration problem on sfv platforms. we propose an adaptive data acquisition method, called adaptive acquisition tree (aat), to jointly account for heterogeneity in user preferences and high-dimensional product characteristics. we adaptively divide users based on preference similarity and then learn a personalized transductive bandit policy that can be used on partially or even unobserved arms to accommodate the fast-evolving and emerging trends on sfv platforms. we analytically characterize the prediction error, which is determined by both the sample size and the impurity of parameters within a group. we further derive the sample complexity for identifying an optimal set for a single user and for all users. we evaluate the algorithm via numerical experiments on data collected from the netease platform. our result demonstrates that the proposed policy, compared with several state-of-the-art benchmarks, performs significantly better in four transductive scenarios for both spotlight recommendation (i.e., best-arm identification) and top-k recommendations. with the potential to improve the expected view time by 20-25\%, our method pertains to both academic and practical values, given the increasing popularity of short-form videos and, more broadly, online user-content generation platforms.

l\'{e}vy matrices are symmetric matrices whose entries are random variables with infinite variance; they are governed by a parameter $\alpha \in (0, 2)$ dictating the power law decay of their entries. for $\alpha <1$, they are believed to serve as one of the few examples of a matrix model exhibiting a mobility edge, also called an anderson transition, that separates chaotic (goe) eigenvalue spacing statistics from ordered (poisson) ones. in this talk we describe results concerning the statistics for the eigenvalue spacings and eigenvector entries of l\'{e}vy matrices. in particular, for $\alpha \in (1, 2)$ their eigenvalue statistics asymptotically follow those of the goe throughout the spectrum, and for $\alpha < 1$ the same statement holds around small eigenvalues.
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these describe joint works with patrick lopatto, jake marcinek, and horng-tzer yau.

dynamical sampling is a term describing an emerging set of problems related to recovering signals and evolution operators from space-time samples. for example, one problem is to recover a function f from space-time samples $\{(a_{t_i}f)(x_i)\}$ of its time evolution $f_t = (a_t f)(x)$, where $\{a_t\}_{t \in t}$ is a known evolution operator and $(x_i,t_i) \in r^d \times r^+$.
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another example is a system identification problem when both $f$ and the evolution family $\{a_t\}_{t\in t}$ are unknown. applications of dynamical sampling include inverse problems in sensor networks, and source term recovery from physically driven evolutionary systems. dynamical sampling problems are tightly connected to frame theory as well as more classical areas of mathematics such as approximation theory, and functional analysis. in this talk, i will describe a few problems in dynamical sampling, some results and open problems.

coclasses in a galois cohomology group $h^1(k, t)$ parametrize
extensions of a number field with certain galois group. it is natural to
want to count these coclasses with general local conditions and with
respect to a discriminant-like invariant. in joint work with brandon
alberts, i present a novel tool for studying this: harmonic analysis on
adelic cohomology, modeled after the celebrated use of harmonic analysis
on the adeles in tate's thesis. this leads to a more illuminating
explanation of a fact previously noticed by alberts, namely that the
dirichlet series counting such coclasses is a finite sum of euler
products; and we nail down the asymptotic count of coclasses in
satisfying generality.

the goal will be to talk about dasgupta and kakde's recent results on understanding the maximal abelian extensions for totally real number fields.

i will give a brief overview of sofic groups, some of their equivalent characterizations, and their relation to the surjunctivity conjecture. time permitting, i will discuss other similar groups, such as hyperlinear or linearly sofic groups.

this talk will discuss moduli spaces of algebraic varieties and will introduce k-stability as a tool to construct moduli spaces of fano varieties.

let g be a connected reductive group with maximal torus t, and let v
and e be two representations of g. then e defines a vector bundle on
the orbifold v//g; let x//g be the zero locus of a regular section.
the quasimap i-function of x//g encodes the geometry of maps from $p^1$
to x//g and is related to gromov-witten invariants of x//g. by
directly analyzing these maps from $p^1$, we explain how to relate the
i-function of x//g to that of v//t. our formulas prove a mirror
symmetry conjecture of oneto-petracci that relates the quantum period
of x//g to a certain laurent polynomial defined by a fano polytope.
finally, we describe a large class of examples to which our formulas
apply, examples that are the orbifold analog of quiver flag varieties.
question for the audience: what else can one investigate with these
examples?

we discuss periods in the classical sense, with motivation from transcendental number theory, and relate them to the study of differential equations and values of l-functions.