The Bartnik quasi-local mass is defined to measure the mass of a bounded manifold with boundary, where a collection of geometric boundary data — the so-called Bartnik boundary data— plays a key role. Bartnik proposed the open problem whether, on a given manifold with boundary, there exists a stationary vacuum metric so that the Bartnik boundary conditions are realized. In the effort to answer this question, it is important to prove the ellipticity of Bartnik boundary conditions for stationary vacuum metrics.
In this talk, I will start with an introduction to the Bartnik quasi-local mass and the moduli space of stationary vacuum metrics. Then I will explain the ellipticity result for the Bartnik boundary conditions and, as an application, derive a local result to the existence question.

Abstract: tSNE has become the standard method of dimensionality
reduction in the medical sciences (clustering cell types, gene
expressions, etc.); amusingly (or maybe not amusingly), the
mathematical theory did not exist until recently. Even a basic
understanding of the mathematics immediately implies a series
of improvements -- I will survey the existing arguments and discuss
a number of interesting problems; given the prevalence of tSNE
in medicine, even small improvements can affect a lot of research
within a very short period of time. Joint work with George Linderman.

Pythagorean Triples are integer solutions to $$x^2+y^2=z^2$$, like 3, 4, and 5. The study of this equation goes back to ancient Greece. Number theorists are interested in similar quadratic equations, allowing other coefficients (like $$3x^2+2y^2=5z^2$$) or more variables (like $$x^2+y^2-z^2=5w^2$$). Before the 20th century, people like Lagrange, Legendre, Gauss, and Minkowski had developed a theory that predicted when such equations have nonzero rational solutions.

In his 11th Problem, Hilbert proposed totally resolving the theory of such quadratic equations that allow any number of variables and coefficients that are not just rational numbers but algebraic numbers. His problem was ultimately solved by Hasse using the idea of a "local-global principle.''

In this talk I'll give examples of variants of the Pythagorean Triples problem, illustrate how to approach it, and give a sense of what a "local-global principle'' is.

Note: Free pizza and drinks!

There are 1-dimensional models inspired by equations of fluid mechanic
which exhibit quite interesting behavior. Lesson learned from some of
the 1d models can be important for the original equations. We will
discuss a few examples, focusing mostly on the De Gregorio
modification of the Constantin-Lax-Majda mod

Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let $$X_1,\ldots,X_q$$ be either real or complex $$C^1$$ vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., $$C^m$$, or $$C^\infty$$, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".

UConn History of Analytical Philosophy Workshop

Join us for a talk by Philip A. Ebert (Stirling) on Frege!

Details t.b.a.

https://rossberg.philosophy.uconn.edu/histanalytic/

TBA

Abstract: The Eisenbud-Green-Harris (EGH) conjecture states that a homogeneous ideal in a polynomial ring K[x1, . . . , xn] over a field K that contains a regular sequence with given degrees a1, . . . , an has the same Hilbert function as a lex-plus-powers ideal containing the powers of the variables xi with the degrees ai. We discuss a case of the EGH conjecture for homogeneous ideals generated by quadrics containing a regular sequence of full length and talk about our result that gives an affirmative answer to EGH when n = 5. This is a joint work with Mel Hochster.

We will introduce d-bar Neumann operator and subelliptic estimates on pseudoconvex domains in $$\mathbb{C}^n$$ and we will try to see geometric applications that are very closely related to complex geometry, sub-Riemannian geometry, and CR-geometry and algebraic geometry.

Join us in the Logic Colloquium for a talk:

Peter Pagin (Stockholm)

Indexicals, Time, and Compositionality

Abstract:

Kaplan’s official argument in “Demonstratives” for Temporalism, the view that some English sentences express propositions that can vary in truth value across time, is the so-called Operator Argument: temporal operators, such as “sometimes”, would be vacuous without such propositions.

Equally important is the argument from compositionality. Without temporal propositions, the sentences

(1) It is raining where John is
(2) It is raining where John is now

would express the same proposition. But they embed differently:

(3) Sometimes, it is raining where John is
(4) Sometimes it is raining where John is now

(3) and (4) express distinct propositions, so if they both are of the form “Sometimes, p”, and if (1) and (2) express the same proposition, we have a violation of compositionality.

In this talk it is shown that with Switcher Semantics, which allows for a generalized form of compositionality, we can have the result that (1) and (2) agree in content when unembedded (assertoric content) but differ in content when embedded under temporal operators (ingredient sense).

We can also show that Switcher Semantics, over Kaplan’s models, preserves the validities in the Logic of Demonstratives. All in all, the arguments for Temporalism are substantially undermined.

https://logic.uconn.edu/calendar/

All welcome!