a natural class of dynamical systems is obtained by iterating polynomial maps, which can be viewed as maps from projective space to itself. one can ask which other projective varieties admit endomorphisms of degree greater than 1. this seems to be an extremely restrictive property, with all known examples coming from toric varieties (such as projective space) or abelian varieties. we describe what is known in this direction, with the new ingredient being the "bott vanishing" property. joint work with tatsuro kawakami.

we give an introduction to the theory of zeta functions for function fields of characteristic p > 0. i will discuss the history of these zeta functions, what is known about their special values, and the question of the distribution of their zeros. i will also present some recent work with joe kramer miller which constitutes a riemman hypothesis for the zeta functions of "ordinary" function fields.

in this talk, we show the extension (in spirit of caffarelli-silvestre) of fractional power of operators defined on banach spaces. starting with the balakrishnan definition, we use semigroup method to prove the extension. this is a joint work with pablo raul stinga.
 

i will talk about $n_\infty$ operads, a generalization of $e_\infty$ operads in the equivariant world, following blumberg-hill. we show that the admissible sets of an $n_\infty$ operad capture the data of the norm maps of its algebra. this ties the notion of norms in the (incomplete) tambara functors, which are the commutative algebra objects in mackey functors.
 

many problems in finite geometry follow the following pattern: say we have a set of points in a plane, and require that some combinatorial property holds. can we say something about the algebraic structure of this set? and what if we impose some extra symmetry conditions?  by far the most famous example of such a theorem is segre’s beautiful characterisation of conics in a desarguesian projective plane of odd order $q$ (1955): every oval (which is a set $c$ of $q+1$  points such that no line contains more than $2$ points of $c$) is the set of points of a conic.  in this talk, we will explore some classical results about ovals and hyperovals and present more recent results of the same flavour about km-arcs and quasi-quadrics.

when considering topological spaces with algebraic structures, there are certain properties which are invariant under homotopy equivalence, such as homotopy-associativity, and others that are not, such as strict associativity. a natural question is: which properties, in general, are homotopy invariant? as this involves a general notion of "property", it is a question of mathematical logic, and in particular suggests that we need a system of logical notation which is somehow well-adapted to the homotopical context. such a system was introduced by voevodsky under the name homotopy type theory. i will discuss a sort of toy version of this, which is the case of "first-order homotopical logic", in which we can very thoroughly work out this question of homotopy-invariance. the proof of the resulting homotopy-invariance theorem involves some interesting ("fibrational") structures coming from categorical logic.

 among the infinitely many mathematical models, which one god chooses for our universe?
what are the common features of all those models? 

we start with two of the most well-studied random matrix ensembles. the limiting eigenvalue distribution of one of which is uniform on the unit disk, and the other of which is a semicircular distribution on the real line. these two distributions have a simple relation: 2 times the real part of the uniform measure on the disk gives you the semicircular distribution. in the first part of the talk i will speak about the "heat flow conjecture" which states that this simple relation can be accomplished in the matrix level by applying the heat operator to the characteristic polynomial of one of the random matrix. then i will move to the case where we apply the heat operator to a random polynomial which has roots distributed approximately uniform on the disk. in this random polynomial case, we can prove a version of the heat flow conjecture if we replace the characteristic polynomial by the random polynomial in the statement of the conjecture. finally, i will speak about the case of heat flow on the plane gaussian analytic function (gaf). these are joint work with brian hall and joint work with brian hall, jonas jalowy, and zakhar kabluchko.

for closed curves evolving by their curvature, the theorem of gage-hamilton and grayson establishes that an embedded curve contracts to a round point. an efficient proof was later found by huisken, with improvements by andrews-bryan, which uses multi-point maximum principle techniques. we’ll discuss the use of these techniques in other settings, particularly for the long-time behaviour of curve shortening flow with free boundary.

perfectoid signature and local étale fundamental group abstract: in this talk i'll talk about a (perfectoid) mixed characteristic version of f-signature and hilbert-kunz multiplicity by utilizing the perfectoidization functor of bhatt-scholze and faltings' normalized length. these definitions coincide with the classical theory in equal characteristic. moreover, perfectoid signature detects bcm regularity and transforms similarly to f-signature or normalized volume under quasi-étale maps. as a consequence, we can prove that bcm-regular
rings have finite local étale fundamental group and torsion part of their divisor class groups. this is joint work with seungsu lee, linquan ma, karl schwede and kevin tucker.

[pre-talk at 1:20pm]

let $\gamma < g = \operatorname{so}(n, 1)^\circ$ be a zariski dense torsion-free discrete subgroup for $n \geq 2$. then the frame bundle of the hyperbolic manifold $x = \gamma \backslash \mathbb{h}^n$ is the homogeneous space $\gamma \backslash g$ and the frame flow is given by the right translation action by a one-parameter diagonalizable subgroup of $g$. suppose $x$ is geometrically finite, i.e., it need not be compact but has at most finitely many ends consisting of cusps and funnels. endow $\gamma \backslash g$ with the unique probability measure of maximal entropy called the bowen-margulis-sullivan measure. in a joint work with jialun li and wenyu pan, we prove that the frame flow is exponentially mixing.

a kakeya set is a set of points in r^n which contains a unit line segment in every direction. the kakeya conjecture states that the dimension of any kakeya set is n. this conjecture remains wide open for all n \geq 3.

together with josh zahl, we study a special collection of the kakeya sets, namely the sticky kakeya sets, where the line segments in nearby directions stay close. we prove that sticky kakeya sets in r^3 have dimension 3.  in this talk, we will discuss background of the problem and its connection to analysis, combinatorics, and geometric measure theory. 

from a directed graph q, called a quiver, one can construct what is known as a quiver variety y_q, an algebraic variety defined as a quotient of a vector space by a group defined in terms of q.  a mutation of a quiver is an operation that produces from q and new directed graph q’ and a new associated quiver variety y_{q’}.  the mutation conjecture predicts a surprising and beautiful connection between the geometry of y_q and that of y_{q’}.  in this talk i will describe quiver varieties and mutations, and show you that you are already well acquainted with some examples of these.  then i will discuss an interesting connection to the gromov—witten theory of degeneracy loci. this is based on joint work with nathan priddis and yaoxiong wen.