A complete Riemannian manifold is called asymptotically hyperbolic if its ends are modeled on neighborhoods of infinity in hyperbolic space. There is a notion of mass for this class of manifolds defined as a coordinate invariant computed in a fixed asymptotically hyperbolic end and measuring the leading order deviation of the geometry from the background hyperbolic metric in the end. Asymptotically hyperbolic manifolds arize naturally in mathematical general relativity, in particular, as slices of asymptotically Minkowski spacetimes, in which case the mass of the slice coincides with the Bondi mass of the spacetime. Having reviewed these and related concepts, we will discuss our proof of the positive mass theorem in the asymptotically hyperbolic setting, which relies on the original ideas of Schoen and Yau and involves a blow-up analysis of the so-called Jang equation, a geometric PDE of mean curvature type

Cubic fourfolds are one of the most intensely studied objects in Algebraic Geometry, especially due to their role in the construction of hyperkähler manifolds. A big open problem surrounding hyperkähler manifolds is the construction of new examples: currently there are only 4 known deformation types. One approach is to consider finite symplectic group actions of a known Hyperkaehler manifold, and study the symplectic resolution (if it exists) of the quotient. In this talk, we will relate automorphisms of Hyperkaehler manifolds of OG10 type to automorphisms of cubic fourfolds, and use the geometry of the cubic in order to complete the classification of involutions Hodge theoretically. More specifically, we compute the algebraic sublattice of the middle cohomology of a cubic fourfold with a certain involution explicitly. If time permits, we discuss several consequences; namely a cubic fourfold with a specific involution is rational.

Can you escape? In a mansion haunted by several bored logicians, you will encounter perilous situations where one’s life depends on common knowledge and quick
thinking.

Don’t worry, your life won’t actually be in danger. You will work through the
puzzles in small groups and learn how to survive if you ever find yourself in a situation
where your life depends on figuring out the color of your hat.
All are welcome.

Come solve some riddles and win some candy!

What is the maximum number of arcs we can fit onto a surface with punctures, and how many distinct systems of arcs are there? By an arc, we mean a continuous map of the open interval (0,1) to the sphere where the endpoints limit to a puncture.

One way we can make this question mathematically rigorous is by considering arcs up to isotopy and ask that in minimal position, the arcs pairwise intersect no more than \(k\) times. We call this a \(k\)-system. In this talk, we give a sketch of the classification of 0 and 1-systems on the 4-punctured sphere. The topology we'll be using is the notion of homotopy (of paths), the Euler characteristic of a punctured surface.

Abstract: We consider a reinsurance problem for a mean-variance Stackelberg game with a random time horizon, in which an insurer and a reinsurer are the two players. The reinsurer computes its premium according to the mean-variance premium principle with parameters $$(\theta, \eta) \in \mathbb{R}^2_+$$. First, for any pair $$(\theta, \eta) \in \mathbb{R}^2_+$$, we compute the per-loss reinsurance strategy that maximizes a mean-variance functional of the insurer’s surplus at the end of the random horizon. This reinsurance strategy is excess-of-loss reinsurance with constant proportional reinsurance for losses above the deductible. Then, given the information of what the insurer will choose when offered any mean-variance premium principle, we determine the optimal pair $$(\theta^*, \eta^*) \in \mathbb{R}^2_+$$ that maximizes the reinsurer’s expected surplus at the end of the random horizon. We show that the equilibrium deductible and coinsurance are both independent of the risk aversion parameter of the insurer. We also show that if the claim severity is light-tailed, in the sense that its hazard rate function is non-decreasing, then the equilibrium reinsurance is pure excess- of-loss reinsurance with no loading for the variance of the loss. Moreover, we show that if the claim severity is heavy-tailed, in the sense that its hazard rate function is decreasing, then the equilibrium reinsurance has a non-trivial coinsurance, with a positive loading for the variance of the loss. Finally, we solve four examples to illustrate these latter two, important results.

Webex meeting link: https://uconn-cmr.webex.com/uconn-cmr/j.php?MTID=m697aa6b3c9ad62c070970df62017d89a
Meeting number: 2621 819 0875 Password: UConn

Speaker's short bio: Prof. Young is Cecil J. and Ethel M. Nesbitt Professor of Actuarial Mathematics at the University of Michigan. Most of my current research deals with financial and insurance decision making of individuals and insurance companies. She obtained her Ph.D. from the University of Virginia in 1984. Please visit her website for more information: https://lsa.umich.edu/math/people/faculty/vryoung.html

Mazur-Tate elements provide a convenient method to study the analytic Iwasawa theory of p-nonordinary modular forms, where the associated \(p\)-adic \(L\)-functions tend to have unbounded coefficients. The Iwasawa invariants of Mazur-Tate elements are well-understood in the case of weight \(2\) modular forms, where they can be related to the growth of \(p\)-Selmer groups and decompositions of the \(p\)-adic \(L\)-function. At higher weights, less is known. By constructing certain lifts to the full Iwasawa algebra, we compute the Iwasawa invariants of Mazur-Tate elements for higher weight modular forms with \(a_p=0\) in terms of the plus/minus invariants of the \(p\)-adic \(L\)-function. Combined with results of Pollack-Weston, this forces a relation between the plus/minus invariants at weights \(2\) and \(p+1\).

Link to online event will be distributed in advance

Geometry is about shapes and number theory is about integers. These might seem unrelated, but over 100 years ago Minkowski discovered many interesting links between them and he called his ideas "the Geometry of Numbers." It allowed him
to use geometry to prove theorems in number theory and number theory to prove theorems in geometry. We'll see some examples relating the geometric properties of shapes on grids of points in the plane (and in higher dimensions) to number-theoretic properties of primes, integers, and rational numbers.


Note: Free refreshments. The talk starts at 5:45.

Abstract: TBA

Join us in the Logic Colloquium for a talk by Sarah Murray (Cornell).

All welcome!