The Australian Direction¹

• Christophe Breuil (virtual talk) - Some open questions on multivariable (phi, Gamma)-modules

  Let K be an unramified extension of Qp. I will describe some results and open questions on multivariable (phi, O_K*)-modules in relation to the p-adic Langlands program for GL2(K). When K=Qp all answers are known thanks to the work of Colmez, Emerton, Kisin and others. This is far from being the case when K is not Qp.
• Frank Calegari - A new proof of the transcendence of π

  A G-function, in the sense of Siegel, is a power series with rational coefficients that on the one hand has nice arithmetic properties (the LCM of the denominators of the first n coefficients grows at most exponentially) and on the other hand has nice analytic properties (it is the solution of a linear ODE with coefficients in Q(x)). These functions are expected to have rich connections with geometry.
  
  In recent years, Dimitrov, Tang, and the speaker have been exploring precisely the tension between these analytic and arithmetic properties. In certain prescribed contexts, we are able to prove a "holonomy bound" showing that power series with certain properties must be G-functions of small order. The goal of this talk is to explore the limits of our results, and to explain some new applications, including a new proof of the transcendence of π.
• Ana Caraiani - Igusa stacks and intersection cohomology

  I will discuss joint work with Linus Hamann and Mingjia Zhang, where we use the Igusa stack diagram and the work of Fargues-Scholze to study the intersection cohomology of Shimura varieties with both torsion and characteristic 0 coefficients.
• Pierre Colmez - TBA

  TBA
• Yiwen Ding - Kisin's Local-Global Compatibility on Emerton's Jacquet Modules

  Kisin's work on overconvergent modular forms initiated the study of local-global compatibility on eigenvarieties, which later developed into the global triangulation theory. Emerton developed a theory of locally analytic Jacquet modules, which in particular allows one to construct eigenvarieties from completed cohomology. In this talk, I will discuss how to deduce various local-global compatibility results in the p-adic Langlands program by combining global triangulation theory, Emerton's Jacquet-module formalism, and related representation theory.
• Hélène Esnault - The Grothendieck-Katz p-curvature conjecture for privileged local systems.

  We generalize to higher dimension the notion of physically local systems due to Katz on opens of the projective line, and prove the Grothendieck-Katz p-curvature conjecture for them. Katz proved it on opens of the projective line as a consequence of his main theorem stating that rigid local systems are of geometric origin. In higher dimension geometricity is an unsolved conjecture of Simpson. One main point in the proof is the comparison between the Galois action on the coefficients of the complex local system and the crystalline Frobenius on the crystals constructed out of the connection steming from the complex Riemann-Hilbert correspondence. Joint work with Michael Groechenig.
• Toby Gee - Prescribed lifts of 2-dimensional representations

  I will discuss the problem of producing lifts of irreducible, totally odd, 2-dimensional mod p representations of the absolute Galois group of a totally real field. This is joint work with Matt Emerton, Lue Pan, and Xinwen Zhu.
• Mathilde Gerbelli-Gauthier - Equidistribution of root numbers

  Distributions of root numbers describe the typical behavior of arithmetic objects, such as ranks of elliptic curves. In this talk, we show equidistribution of root numbers in the weight aspect, for (conjugate) self-dual symplectic automorphic representations of GL_N of sufficiently large level, with a consequence for the associated Galois representations in the conjugate self-dual case, where they are known to exist. Both the motivation and the method are endoscopic: these signs appear in Arthur's multiplicity formula and govern which members of a packet are automorphic, and the results rely on the stable trace formula. This is joint work with Rahul Dalal.
• Daniel Le - Three-dimensional p-restricted deformation rings

  In 2008, Kisin constructed potentially semistable $p$-adic Galois deformation rings and determined their dimensions. Combined with a refinement of the Taylor--Wiles patching method, this work led to major advances in the modularity of Galois representations. Although a detailed understanding of the geometry of potentially semistable deformation rings is crucial for many arithmetic applications, relatively little is known when the Hodge--Tate weights span an interval of length greater than $p$. In this talk, I will give explicit presentations of three-dimensional crystalline deformation rings over unramified extensions of $\mathbf{Q}_p$ when the gaps between consecutive Hodge--Tate weights lie between $4$ and $p-4$. The proof combines Kisin--Taylor--Wiles patching, the Emerton--Gee stack, and recent developments in integral $p$-adic Hodge theory. This is joint work in progress with Tong Liu.
• Wiesława Nizioł - Conjecture C_st for analytic varieties

  The relationship between p-adic (pro)-etale cohomology and de Rham cohomologies is well understood now for algebraic varieties over local fields of mixed characteristic. For analytic varieties it remains largely conjectural though much progress has been made in the last decade especially in the Stein case. I will survey the results as well as the conjectures. This is based on a joint work with Pierre Colmez and Sally Gilles.
• Lue Pan - Completed cohomology of Hilbert modular varieties below middle degree

  I will discuss several vanishing results for the locally analytic completed cohomology of Hilbert modular varieties below middle degree. This talk will be based on joint work with Kai-Wen Lan and joint work with Yuanyang Jiang.
• George Pappas - Characteristic classes of automorphic p-adic local systems

  Let L be an 'etale Q_p-local system on an algebraic variety X over a number field. Using Lazard's results on the continuous cohomology of p-adic groups, one can define characteristic classes l_n(L), n>0, in the continuous absolute 'etale cohomology H^{2n-1}(X, Q_p). I will describe some results about these characteristic classes when X is a Shimura variety and L is an automorphic local system over X. This is joint work in progress with A. Petrov.
• Vytautas Pašku̅nas - TBA

  TBA
• Michael Rapoport - Integral Hecke correspondences

  Shimura varieties admit vector bundles with liftings of the Hecke correspondences (automorphic vector bundles). These vector bundles have a natural extension to p-integral models of good reduction of Shimura varieties. I will address the problem of determining the optimal power of p by which one has to multiply the Hecke correspondences in the generic fiber to obtain integral cohomological correspondences. Joint work with U. Görtz and X. He.
• Ananth Shankar - TBA

  TBA
• Sug Woo Shin - TBA

  TBA
• Naomi Sweeting - Some new bounds on adjoint Selmer groups for Hilbert modular forms

  Let f be a cuspidal Hilbert modular eigenform of parallel weight 2, without CM. The Selmer groups for the adjoint p-adic Galois representations attached to f have a rich history, including (to name just a few) works of Flach, Wiles, Taylor-Wiles, Kisin, and more recently Newton and Thorne. In particular, these Selmer groups are now all known to be finite. However, it is still not known in general that their lengths are given by appropriate L-values or congruence numbers, as predicted by the Tamagawa number conjecture. In this talk, I will report on joint work in progress with Chris Skinner, in which we bound the adjoint Selmer group for f in terms of the congruence number in new cases. Most notably, we are able to get a near-sharp bound even when the level of f is arbitrarily bad at p. The proof uses a system of auxiliary Galois cohomology classes, constructed by combining an idea of Skinner and Venkatesh with level-raising techniques inspired by bipartite Euler systems.
• Carl Wang-Erickson - Ordinary categorical Langlands correspondence for modular curves

  We discuss the categorical Langlands correspondence conjecture for modular curves, proposed by Emerton and Zhu, in generic cases that are residually reducible, odd, and ordinary. The main aim is to draw connections between Iwasawa theory and the conjecture.
• Xinwen Zhu - TBA

  TBA

• Monday, August 10th

  • 9:00-9:15 - Opening remarks
  • 9:15-10:00 - Talk 1
  • 10:00-10:45 - Break
  • 10:45-11:30 - Talk 2
  • 11:45-12:30 - Talk 3
  • 12:30-2:00 - Lunch break
  • 2:00-2:45 - Talk 4
  • 3:00-3:45 - Talk 5
  • 3:45-4:15 - Break
  • 4:15-5:30 - Lightning talks I (4x15 minutes)
• Tuesday, August 11th

  • 9:15-10:00 - Talk 6
  • 10:00-10:45 - Break
  • 10:45-11:30 - Talk 7
  • 11:45-12:30 - Talk 8
  • 12:30-2:00 - Lunch break
  • 2:00-2:45 - Talk 9
  • 3:00-3:45 - Talk 10
  • 4:00-5:30 - Poster session (with light refreshments) [LCB loft]
• Wednesday, August 12th

  • 9:15-10:00 - Christophe Breuil (virtual)
  • 10:15-11:00 - Talk 12
  • 11:15-12:00 - Talk 13
  • 12:00-6:30 - Free afternoon
  • 6:30-9:30 - Conference banquet
• Thursday, August 13th

  • 9:15-10:00 - Talk 14
  • 10:00-10:45 - Break
  • 10:45-11:30 - Talk 15
  • 11:45-12:30 - Talk 16
  • 12:30-2:00 - Lunch break
  • 2:00-2:45 - Talk 17
  • 2:45-3:15 - Break
  • 3:15-4:30 - Lightning talks II (4x15 minutes)
• Friday, August 14th

  • 9:15-10:00 - Talk 18
  • 10:15-11:00 - Talk 19
  • 11:15-12:00 - Closing remarks

• Register here. Registration is free, but only registered participants may attend the conference.
• The deadline to apply for funding has passed! If you filled out the funding application that was previously found here, you have been contacted about your funding status. The funding question on the registration form is now meaningless too (we used it before to alert people about the funding application).

• Location: The conference will take place in the JFB and LCB buildings on Presidents Circle.
  
• Walking/Shuttle: From the University Guest House, it is a 1.2 mile downhill walk through campus. The campus shuttle Red Line also stops in front of the guest house every 15 minutes and will take you for free to Presidents Circle (Kingsbury stop) very close to JFB --- see https://uofubus.com/routes for a live schedule (it is also possible but slightly slower to take the Blue line from a nearby stop).
• Airport: Salt Lake City International Airport is a major international airport and a Delta hub; in addition to a large selection of domestic flights, there are direct flights to/from Amsterdam, London, Paris, and Seoul.
• Lodging: The closest hotel to the event is the University Guest House; it is about a 20 minute walk through campus to the conference venue and there is also a shuttle (see above). There are not many dining options near the guest house; however, there is a TRAX (lightrail) station very close by that makes it easy to get downtown for more choices. The TRAX station also provides an easy way to get to/from the airport with a single transfer downtown.
  Another good hotel option is the Little America Hotel, which is closer to downtown and well-connected to campus via public transport (TRAX or bus).
  There are also many options closer to the conference venue through AirBnB/VRBO.
• Surroundings: For participants looking to extend their trip, there are 8 major national parks³ located within five hours driving distance of Salt Lake City as well as many closer natural attractions. In particular, the University of Utah is nestled against the Wasatch mountains, and the Wasatch front affords abundant recreation opportunities at cooler temperatures a short drive from campus by going up Millcreek Canyon, Little and Big Cottonwood Canyons, and American Fork Canyon. Park City is a 25 minute drive away on the other side of the Wasatch.

¹In the study of the cohomology of algebraic varieties, there are two different "directions" in play: the "German direction," where one considers the coefficients of the equations defining an algebraic variety as varying in a family, and the "Australian direction," where one considers the coefficients of the cohomology theory itself as varying in a family. This terminology was introduced by Laurent Fargues [F] (in his talk at the 2015 summer algebraic geometry institute in Salt Lake City!) in the context of moduli of p-divisible groups to compare and contrast the moduli problems studied by Kisin with those studied by Rapoport and Zink. A broader comparison of these two directions has been emphasized, among other places, in Scholze's 2018 ICM talk [S], and has since grown into a useful dichotomy for understanding the many recent advances in the field, including the fundamental contributions of Emerton and Kisin. In this conference, we interpret the term "Australian direction" more expansively as including all areas of mathematics where the mathematicians Emerton and Kisin have made significant contributions.
[F] Laurent Fargues. From local class field to the curve and vice versa. In Algebraic geometry: Salt Lake City 2015, volume 97.2 of Proc. Sympos. Pure Math., pages 181-198. Amer. Math. Soc., Providence, RI, 2018.
[S] Peter Scholze. p-adic geometry. Proceedings of the ICM 2018.

²Matt and Mark, still spring chickens, are both turning 55 in 2026. But, together that makes 110! So, we're either a bit early or very late. And did you know that Benson Farb, perennial foil to Mark and Matt, even had a conference when he turned 50?

³Four of the national parks in Utah (Arches, Canyonlands, Capitol Reef, and Zion) can get quite hot in August, but Bryce Canyon (Utah) and Great Basin (Nevada) are at a higher altitude and typically still nice. Grand Teton and Yellowstone in Idaho/Montana/Wyoming should be beautiful, though crowded. There are also many other nearby state parks and natural destinations that will be less crowded than national parks (we recommend, e.g., the Sawtooth mountains in Idaho).