Calendar

**Note special time & location: 10 – 11 AM ET in Room G02**

https://cmsa.fas.harvard.edu/event/algebraic-geometry-in-string-theory/

Reproducing the integer count of black hole micro-states from the gravitational path integral is an important problem in quantum gravity. In the first part of the talk, I will show that, by using supersymmetric localization, the gravitational path integral for 1/16-BPS black holes in supergravity can reproduce the index obtained in the string theory construction of such black holes. A more refined argument then shows that not only the black hole index but also the total number of black hole microstates within an energy window above extremality that is polynomially suppressed in the charges also matches this string theory index. In the second part of the talk, I will present a second perspective on this state count and show how the BPS Hilbert space can be obtained by directly preparing states using the gravitational path integral. While such a preparation naively gives rise to a Hilbert space of BPS states whose dimension is much larger than expected, I will explain how non-perturbative corrections in the overlap of such states are again responsible for reproducing the correct dimension of the Hilbert space.

“TTbar” deformed 2d quantum field theory is a non-local theory in which Minkowski space is deformed in a state-dependent but consistent manner. For a massive theory this is equivalent to each particle acquiring a width proportional to its mass in its rest frame, giving rise to simple CDD factors dressing the $S$–matrix, but for deformed conformal field theories the spectrum becomes quite complicated, and the question of modular invariance of the torus partition function is non-trivial. I will show that this leads to a theory of TTbar deformed modular forms in general. Maass forms turn out to play an important role as eigenforms of the deformation.

The Math Picture Language seminar will be held at 9:30 a.m. Boston time.
Click the link for a Zoom Link for Tuesday Math Picture Language Seminars.
Recorded seminars can be viewed on the Mathematical Picture Language YouTube channel.

Inspired from ideas in topology, Koszul modules and the associated resonance varieties turned out to have important algebro-geometric applications for instance to (i) Green’s Conjecture on syzygies of canonical curves, (ii) stabilization of cohomology of projective varieties in arbitrary characteristics and (iii) Chen invariants of hyperplane arrangements. I will discuss new developments related to this circle of ideas obtained in joint work with Aprodu, Raicu and Suciu.

Finding, applying and learning from the appropriate frameworks to use as lenses for educational research is an important task. In this talk I will share the three theoretical frameworks that I have used to study entry-level mathematics endeavors including: widespread change at a university (that involved implementation of TA training, course coordination, and many other efforts); leaders selected for funded efforts to incorporate active learning; and viability of mathematics tutoring centers. While I will give specific examples from my own research, each of the frameworks could be used to study a variety of university entities.

The local global approach to the study of rational points over number fields begins by embedding the set of rational points on a variety X into the set of its adelic points. The Brauer-Manin pairing cuts out a subset of the adelic points, called the Brauer-Manin set, that contains the rational points. If the set of adelic points is non-empty but the Brauer–Manin set is empty then we say there’s a Brauer–Manin obstruction to the existence of rational points on X.  Computing the Brauer–Manin pairing involves evaluating elements of the Brauer group of X at local points.  If an element of the Brauer group has order coprime to p, then its evaluation at a p-adic point factors via reduction of the point modulo p. For elements of order a power of p this is no longer the case: in order to compute the evaluation map one must know the point to a higher p-adic precision. Classifying Brauer group elements according to the precision required to evaluate them at p-adic points gives a filtration which we describe using work of Kato. Applications of our work include addressing Swinnerton-Dyer’s question about which places can play a role in the Brauer–Manin obstruction. This is joint work with Martin Bright.

We will discuss enumerative and structural properties of two families of standard Young tableaux, the realizable rectangular tableaux and the realizable staircase tableaux.  The realizable rectangular tableaux come from tropical rank 1 matrices, whereas the realizability condition for staircase tableaux comes from geometric realizability for sorting networks and the Edelman-Greene bijection.  The two notions of realizability are connected via the study of coherent monotone paths on
the permutahedron, as established by Black and Sanyal in their study of flag polymatroids.  As a consequence of providing tight asymptotic bounds on the size of these families, we make progress on the related studies of random sorting networks, realizable allowable sequences, and sorting algorithms.  Based on joint work with Igor Araujo, Alexander E. Black, Yibo Gao, Robert A. Krueger, and Alex McDonough.
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For information about the Combinatorics Seminar, please visit…

http://math.mit.edu/seminars/combin/

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I will connect the following questions and answers.

1. 42 = (12602123297335631)^3 + (80435758145817515)^3 + (-80538738812075974)^3.

2. https://people.math.harvard.edu/~alpoge/fun/fruit%20for%20thought.jpeg , aka:
Are there positive integers x, y, z such that:
x / (y + z) + y / (x + z) + z / (x + y) = 4?

3. The Birch and Swinnerton-Dyer conjecture.

4. Hilbert’s tenth problem, aka:
Is there a computer program which “solves all Diophantine equations”?

5. 0% of integers are a sum of two squares (integral or rational).

6. A positive proportion of integers are a sum of two rational cubes.

For more information, please see: https://people.math.harvard.edu/~ana/ons/

This seminar will be broadcast over Zoom: https://harvard.zoom.us/j/7855806609

I will prove that certain “motivic abelian groups” are motivic infinite loop spaces.

As 7-manifolds with special holonomy, examples of compact G2-manifolds were first constructed by Joyce as resolutions of flat G2-orbifolds. Later Walpuski constructed non-trivial G2-instantons over Joyce’s manifolds via gluing techniques. In this talk, I will first explain how to define a deformation invariant of G2-orbifolds by counting flat connections, then describe the moduli space of instantons over certain non-compact G2-manifolds that appeared in Joyce’s construction, with the aim to give a complete description of moduli spaces over some examples in Joyce’s list.