In recent years, there have been some  breakthroughs in extending Schmidt's subspace theorem to non-linear hypersurfaces, made by Corvaja-Zannier, and Evertse-Feretti, etc. In this talk, I will discuss my recent joint work with P. Vojta based on their works, as well as its applications.

A description of the structure of knowledge in the brains of animals, with an outline of the phylogeny of human intelligence. The theory supports some old ideas in the philosophy of mind and it may have applications in artificial intelligence. I will define finite algebraic structures, that I call mental models. They are intended to represent in a realistic way the descriptive component of knowledge (as opposed to its evaluating component). I will define the notions of  mental object and of concepts in a mental model. Concepts can be real, like these representing objects and processes of everyday life or studied in science, or they may be imaginary, like these in tales and pure mathematics. Our definitions yields a precise form of Ockham's principle of economy of concepts. The theory suggests some ideas in ontology and pedagogy. Theories of the mind that ignore biology or brain science, or do not propose sufficiently clear notions of meaning, will be criticized.

This talk will outline exactly how and why the Atiyah-Singer index theorem can be viewed as a corollary of Bott periodicity. The precise argument proceeds by introducing an abelian group K_0(.) obtained by considering pairs (M, E) where M is a closed Spin-c manifold and E is a a vector bundle on M. "Spin-c manifold" will be carefully defined. Most of the oriented manifolds which occur in practice are Spin-c. The above is joint work with Erik van Erp.

Let ${\displaystyle \Gamma }$ be a compact Hausdorff second countable topological group.  ``second countable ``the topology of ${\displaystyle \Gamma }$ has a countable base" Irr(${\displaystyle \Gamma }$) denotes the set of equivalence classes of irreducible representations of ${\displaystyle \Gamma }$. The vector space for each irreducible representation of ${\displaystyle \Gamma }$ is a finite dimensional vector space  over the complex numbers.  Irr(${\displaystyle \Gamma }$) is a countable set  ---  and we give it the discrete topology.

Let ${\displaystyle X}$ be a locally compact Hausdorff topological space with a given (continuous) action of ${\displaystyle \Gamma }$ on ${\displaystyle X}$.

For each ${\displaystyle x}$ in ${\displaystyle X}$, ${\displaystyle {\Gamma }_x}$ denotes the isotropy group of ${\displaystyle x}$. If ${\displaystyle x}$ and ${\displaystyle y}$ are in the same orbit for the given action of ${\displaystyle \Gamma }$ on ${\displaystyle X}$ , then Irr(${\displaystyle {\Gamma }_x}$) and Irr(${\displaystyle {\Gamma }_y}$) are canonically in bijection.  The usual quotient, denoted ${\displaystyle X/Gamma}$, is obtained by collapsing each orbit (for the given action of ${\displaystyle \Gamma }$ on ${\displaystyle X}$) to a point.  The extended quotient,  denoted ${\displaystyle X//\Gamma }$, is constructed by replacing the orbit of ${\displaystyle x}$ (for the given action of ${\displaystyle \Gamma }$ on ${\displaystyle X}$) by Irr(${\displaystyle {\Gamma }_x}$).
This talk will describe how extended quotient enters into the representation theory of Lie groups and reductive p-adic groups. For Lie groups this is due to Mackey-Higson-Afgoustidis. For reductive p-adic groups,  this is due to Aubert-Baum-Plymen-Soll

TBA

Would you like to play a game? It's the day before Halloween, so let's get a little spooky. Open this door with the key of imagination, and step into a world beyond your usual understanding. A world where the axiom of choice fails, but not horrendously so. A world where every set of real numbers is Lebesgue measurable. A world where we can't turn a single sphere into two with the same volume or well-order every set. But it will all start with a simple game

The goal of this talk is to provide an introduction to an alternative axiom of set theory, one which contradicts the axiom of choice but shares some of that axiom’s implications, as well as its own unique, interesting, and even paradoxical consequences. We will compare this axiom to choice, see how they contradict, and if time permits we may even discuss implications of our considerations for foundations of mathematics.

This talk will take the outline provided in Part 1 and fill in details so as to provide a complete proof.