Calendar

We consider two related questions about the extremal statistics of Wigner matrices (random symmetric matrices with independent entries). First, how much can their eigenvalues fluctuate? It is known that the eigenvalues of such matrices display repulsive interactions, which confine them near deterministic locations. We provide optimal estimates for this “rigidity” phenomenon. Second, what is the behavior of the maximum of the characteristic polynomial? This is motivated by a conjecture of Fyodorov-Hiary-Keating on the maxima of logarithmically correlated fields, and we will present the first results on this question for Wigner matrices. This talk is based on joint work with Paul Bourgade and Ofer Zeitouni.

Pflueger showed that these loci are non-empty when the expected codimension is at most g-3. By studying linear series on chains of elliptic curves, we give a new proof of a slightly refined version of this result. We can also look at the behavior of the generic curve in the locus.

An interesting conjecture of Auel and Haburcak states that these loci are distinct and not contained in each other, unless they come from adding or removing fixed points. Their proof made use of curves contained in K3 surfaces and was sufficient to prove the result in small genus. Using chains of elliptic curves, we can obtain additional information.

For more information, please see https://researchseminars.org/seminar/harvard-mit-ag-seminar

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Fibered ribbon knots vs. major 4D conjectures

Location: Harvard University Science Center Hall A & via Zoom webinar

Dates: Feb 20 & 22, 2024

Time: 4:00-5:30 pm

Directions and Recommended Lodging

Registration is required.

• In-person registration: Harvard Science Center Hall A
• Zoom Webinar registration

Maggie Miller is an assistant professor in the mathematics department at the University of Texas at Austin and a Clay Research Fellow.

This will be the fourth annual Math Science Lecture Series held in Honor of Raoul Bott.

Fibered ribbon knots vs. major 4D conjectures

Feb. 20, 2024

Title: Fibered ribbon knots and the Poincaré conjecture

Abstract: A knot is “fibered” if its complement in S^3 is the total space of a bundle over the circle, and ribbon if it bounds a smooth disk into B^4 with no local maxima with respect to radial height. A theorem of Casson-Gordon from 1983 implies that if a fibered ribbon knot does not bound any fibered disk in B^4, then the smooth 4D Poincaré conjecture is false. I’ll show that unfortunately (?) many ribbon disks bounded by fibered knots are fibered, giving some criteria for extending fibrations and discuss how one might search for non-fibered examples.

Feb. 22, 2024

Title: Fibered knots and the slice-ribbon conjecture

Abstract: The slice-ribbon conjecture (Fox, 1962) posits that if a knot bounds any smooth disk into B^4, it also bounds a ribbon disk. The previously discussed work of Casson-Gordon yields an obstruction to many fibered knots being ribbon, yielding many interesting potential counterexamples to this conjecture — if any happy to bound a non-ribbon disk. In 2022, Dai-Kong-Mallick-Park-Stoffregen showed that unfortunately (?) many of these knots don’t bound a smooth disk into B^4 and thus can’t disprove the conjecture. I’ll show a simple alternate proof that a certain interesting knot (the (2,1)-cable of the figure eight) isn’t slice and discuss remaining open questions. This talk is joint with Paolo Aceto, Nickolas Castro, JungHwan Park, and Andras Stipsicz.

Talk Chair: Cliff Taubes (Harvard Mathematics)

Moderator: Freid Tong (Harvard CMSA)

Raoul Bott (9/24/1923 – 12/20/2005) is known for the Bott periodicity theorem, the Morse–Bott functions, and the Borel–Bott–Weil theorem. For more info, please see the article “Remembering Raoul Bott”  from the American Mathematical Society.

https://harvard.zoom.us/j/95706757940?pwd=dHhMeXBtd1BhN0RuTWNQR0xEVzJkdz09
Password: cmsa

Given an elliptic curve E over a global field K the abelian group E(K) is finitely generated, and so much effort has been put into trying to understand the behavior of E(K), as E varies. Of note, it is a folklore conjecture that, when all elliptic curves E/K are ordered by a suitably defined height, the average value of their ranks is exactly 1/2. One fruitful avenue for understanding the distribution of E(K) has been to first understand the distribution of the sizes of Selmer groups of elliptic curves. In this direction, various authors (including Bhargava-Shankar, Poonen-Rains, and Bhargava-Kane-Lenstra-Poonen-Rains) have made conjectures which predict, for example, that the average size of the n-Selmer group of E/K is equal to the sum of the divisors of n. In this talk, I will report on some recent work verifying this average size prediction, “up to small error term,” whenever n=2 and K is any global *function* field. Results along these lines were previously known whenever K was a number field or function field of characteristic  >5, so the novelty of my work is that it applies even in “bad” characteristic.

For more info, see https://ashvin-swaminathan.github.io/home/NTSeminar.html

Please note the special time.

I will discuss some regular subdivisions of the permutahedron in R^n, one for each Coxeter element in the symmetric group S_n. These subdivisions are “Bruhat interval” subdivisions, meaning that each face is the convex hull of the permutations in a Bruhat interval (regarded as vectors in R^n). Bruhat interval subdivisions in general correspond to cones in the positive tropical flag variety by a combination of results of Joswig-Loho-Luber-Olarte and Boretsky; the subdivisions indexed by Coxeter elements are finest subdivisions and so correspond to a subset of the maximal cones. For a particular choice of Coxeter element, we recover a cubical subdivision of the permutahedron due to Harada-Horiguchi-Masuda-Park. Applications of these subdivisions include new formulas for the class of the permutahedral variety as a sum of Richardson classes in the cohomology ring of the flag variety. This is joint work-in-progress with Mario Sanchez.

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For more info, see https://math.mit.edu/combin/

Given a Fano manifold, Iritani proposed that the asymptotic behavior of solutions to the quantum differential equation of the Fano should be given by the so-called ‘Gamma class’ in its cohomology ring. Later, Abouzaid-Ganatra-Iritani-Sheridan reformulated the ‘Gamma conjecture’ for Calabi-Yau manifolds via the tropical SYZ mirror symmetry and proposed that values of the Riemann Zeta function at positive integers have geometric origins in the tropical periods and singularities of the SYZ geometry. In this talk, we will first review the content of the Gamma conjecture. Then, we will discuss the first step of generalizing AGIS’ approach to Gamma conjecture for the Gross-Siebert mirror families of a Fano manifold in dimension 2 cases, based on joint work with Bohan Fang and Junxiao Wang.

Zoom: https://harvard.zoom.us/j/93338480366?pwd=NEROWElhWStQVjVLRVZFSm1tV1ZCdz09

Maxime Ramzi speaks on Topological cyclic homology

View from the CMSA Events Page

Fibered ribbon knots vs. major 4D conjectures

Location: Harvard University Science Center Hall A & via Zoom webinar

Dates: Feb 20 & 22, 2024

Time: 4:00-5:30 pm

Directions and Recommended Lodging

Registration is required.

• In-person registration: Harvard Science Center Hall A
• Zoom Webinar registration

Maggie Miller is an assistant professor in the mathematics department at the University of Texas at Austin and a Clay Research Fellow.

This will be the fourth annual Math Science Lecture Series held in Honor of Raoul Bott.

Fibered ribbon knots vs. major 4D conjectures

Feb. 20, 2024

Title: Fibered ribbon knots and the Poincaré conjecture

Abstract: A knot is “fibered” if its complement in S^3 is the total space of a bundle over the circle, and ribbon if it bounds a smooth disk into B^4 with no local maxima with respect to radial height. A theorem of Casson-Gordon from 1983 implies that if a fibered ribbon knot does not bound any fibered disk in B^4, then the smooth 4D Poincaré conjecture is false. I’ll show that unfortunately (?) many ribbon disks bounded by fibered knots are fibered, giving some criteria for extending fibrations and discuss how one might search for non-fibered examples.

Feb. 22, 2024

Title: Fibered knots and the slice-ribbon conjecture

Abstract: The slice-ribbon conjecture (Fox, 1962) posits that if a knot bounds any smooth disk into B^4, it also bounds a ribbon disk. The previously discussed work of Casson-Gordon yields an obstruction to many fibered knots being ribbon, yielding many interesting potential counterexamples to this conjecture — if any happy to bound a non-ribbon disk. In 2022, Dai-Kong-Mallick-Park-Stoffregen showed that unfortunately (?) many of these knots don’t bound a smooth disk into B^4 and thus can’t disprove the conjecture. I’ll show a simple alternate proof that a certain interesting knot (the (2,1)-cable of the figure eight) isn’t slice and discuss remaining open questions. This talk is joint with Paolo Aceto, Nickolas Castro, JungHwan Park, and Andras Stipsicz.

Talk Chair: Cliff Taubes (Harvard Mathematics)

Moderator: Freid Tong (Harvard CMSA)

Raoul Bott (9/24/1923 – 12/20/2005) is known for the Bott periodicity theorem, the Morse–Bott functions, and the Borel–Bott–Weil theorem. For more info, please see the article “Remembering Raoul Bott”  from the American Mathematical Society.

The last decade has produced a number of remarkable discoveries, such as the first direct observation of gravitational waves by the LIGO/Virgo collaboration and the first black hole image taken by the Event Horizon Telescope. These discoveries mark the beginning of a new precision era in black hole physics, which is expected to develop further by future experiments such as LISA, the Einstein Telescope and Cosmic Explorer.

Friday, Feb. 23rd at 12pm, with lunch, lounge at CMSA (20 Garden Street). Also by Zoom: https://harvard.zoom.us/j/92410768363

A common theme throughout mathematics is the search for “constructive” solutions to problems as opposed to mere existence results. For problems over R and other well-behaved spaces, this idea is nicely captured by the concept of a Borel construction. In particular, one can investigate Borel solutions to classical combinatorial problems such as graph colorings, perfect matchings, etc. The area studying these questions is called descriptive combinatorics. As I will explain in the talk, many facts in graph theory that we know and love—for example, Brooks’ theorem—turn out to be inherently “non-constructive” in this sense. The main result of this talk is that Borel versions of various classical combinatorial theorems nevertheless hold on graphs that can, in some sense, be easily decomposed into subgraphs with finite components. No prior familiarity with Borel combinatorics or descriptive set theory will be assumed. Based on joint work with Felix Weilacher.

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For more info, see https://math.mit.edu/combin/

The Vafa-Witten equations (with or without a mass term) constitute a non-linear, first order system of differential equations on a given oriented, compact, Riemannian 4-manifold. Because these are the variational equations of a functional, the linearized equations at any given solution can be used to define an elliptic, first order, self-adjoint differential operator. This talk will describe bounds (upper and lower) for the spectral flow between respective versions of this operator that are defined by the elements in diverging sequences of reducible solutions. (The spectral flow is formally the difference between the respective Morse indices of the solutions when they are viewed as critical points of the functional.) In some cases, the absolute value of the spectral flow is bounded along the sequence, whereas in others it diverges. This is a curious state of affairs.