The post Presenting IPAM’s 2024 Annual Newsletter first appeared on IPAM.

]]>In 2024, IPAM has explored deep connections between mathematics and physics through programs on Mathematical and Computational Challenges in Quantum Computing (Fall 2024) and Statistical Mechanics, Integrable Systems, and Geometry (Spring 2025). Both programs

were highly successful in establishing new collaborations and pushing the boundaries of scientific knowledge.

Thanks to support from our Diversity, Equity, and Inclusion endowment, IPAM was able to expand its focus on DEI in mathematics. Our new Practicum in Undergraduate Mathematics (PUMA) program, following

an idea of IPAM Trustee Edray Goins, showcases advanced mathematics to diverse groups of college students. With additional NSF funding, IPAM will be co-organizing

the Modern Math Workshop at SACNAS this Fall, and the 4th LatMath conference

in March 2025.

The post Presenting IPAM’s 2024 Annual Newsletter first appeared on IPAM.

]]>The post IPAM Seeking an Associate Director first appeared on IPAM.

]]>The Institute for Pure and Applied Mathematics (IPAM) at UCLA is seeking a second Associate Director (AD) to help lead and organize innovative programs in pure and applied mathematics. The term of appointment will be two years, beginning August 1, 2025. The appointment may be extended for a third year by mutual agreement.

The AD is expected to be an active research mathematician or scientist in a related field, with some experience in conference organization. We welcome candidates from industry and national labs as well as academia. An enthusiasm for mathematics and its applications (broadly construed) is essential. Women and minorities are especially encouraged to apply.

The AD will manage about half of the programs in coordination with the organizing committees (representing diverse scientific fields) and IPAM staff. The AD will also help set institute policy and direction, solicit and review proposals for future programs, recruit industry sponsors for the undergraduate summer program, and occasionally represent IPAM at conferences and meetings. The selected candidate will be encouraged to continue or develop his or her personal research program as well.

Applications must be submitted online at **https://recruit.apo.ucla.edu/JPF09905****. **Applications should include a CV, a short statement of the applicant’s vision for their service at IPAM, a statement on contributions to EDI, and at least three names of references. Applications will receive fullest consideration if received by **February 7, 2025. **Questions about the position/application may be sent to **ipam@ucla.edu**.

IPAM/UCLA is an affirmative action/equal opportunity employer.

IPAM is an NSF-funded national research institute in the mathematical sciences, located on the UCLA campus. Its mission is to foster interdisciplinary collaborations between mathematics and other scientific fields.

IPAM offers two long programs per year, plus several shorter programs throughout the year. Each long program has significant participation from both mathematicians and other scientists and consists of tutorials, workshops, and a culminating retreat off-campus. IPAM has funding to support both senior and junior scholars, including graduate students, as long-term scholars in-residence or as workshop participants. IPAM also sponsors a summer undergraduate industrial research program and a two-week or three-week summer school.

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]]>The post Alexander Watson Wins Journal of Mathematical Physics Young Researcher Award first appeared on IPAM.

]]>The post Alexander Watson Wins Journal of Mathematical Physics Young Researcher Award first appeared on IPAM.

]]>The post David Baker Wins Nobel Prize in Chemistry first appeared on IPAM.

]]>The post David Baker Wins Nobel Prize in Chemistry first appeared on IPAM.

]]>The post Geoffrey Hinton Wins Nobel Prize in Physics first appeared on IPAM.

]]>The post Geoffrey Hinton Wins Nobel Prize in Physics first appeared on IPAM.

]]>The post Grothendieck Shenanigans: Permutons From Pipe Dreams via Integrable Probability first appeared on IPAM.

]]>**Abstract**

If Alice and Bob each take a walk, a random one, what is the chance they will meet? If n people walk randomly in Manhattan from the Upper East Side going south or west, how often will two of them meet until they reach the Hudson? If they do not want to see each other more than once, in what relative order will they most likely arrive at the Hudson shore? If we make a rhombus like an Aztec diamond from 2 by 1 dominoes, what would it most likely look like? Ifwater molecules arrange themselves on a square grid, what angles between the hydrogen atoms will be formed near the boundaries? And if ribosomes are transcribing the mRNA, how do they hop between the sites (codons)?

The answers to these questions are all intertwined, bridging algebra with probability, integrability with stochasticity, random tilings with random permutations. The paper by Morales, Panova, Petrov, and Yeliussizov explores these connections through the study of Grothendieck multivariable polynomials and random pipe dreams, revealing deep links between algebraic structures and physical processes.

**Highlight**

If Alice and Bob each take a walk, a random one, what is the chance they will meet? If n people walk randomly in Manhattan from the Upper East Side going south or west, how often will two of them meet until they reach the Hudson? If they do not want to see each other more than once, in what relative order will they most likely arrive at the Hudson shore? If we make a rhombus like an Aztec diamond from 2 × 1 dominoes, what would it most likely look like? If water molecules arrange themselves on a square grid, what angles between the hydrogen atoms will be formed near the boundaries? And if ribosomes are transcribing the mRNA, how do they hop between the sites (codons)?

The answers to the above questions are all intertwined, bridging algebra with probability, integrability with stochasticity, random tilings with random permutations, as revealed in the recent work^{1} [MPPY24]

Figure 1. **(a)** a pipe dream of the permutation w = 241653, **(b)** a permuton in the shape of an ice cream, **(c)** an example of a layered permutation, **(d)** illustration of the corresponding configuration in the 6-vertex model lying inside the frozen region.

The story started with a problem of Richard Stanley from 2017 – what is the (asymptotically) maximal value of a principal specialization of a *Schubert polynomial*. Schubert polynomials in n variables, indexed by permutations , present cohomology classes of the flag variety and are central objects of study at the intersection of Algebraic Combinatorics and Algebraic Geometry. However, the problem can be stated as a simple tiling model: let us tile the triangle in the square grid given by i + j ≤ n + 1, i, j = 1, . . . , n, with crosses or elbows (see Figure 1 **(a)**), which form a network of lines aka “pipe dreams”. Moreover, impose a special *noncrossing condition* that no two pipes can cross more than once. The configuration in Figure 1 **(a)** satisfies the *noncrossing condition*, and counting those is a statistical mechanics problem involving long-range interactions. It is notoriously difficult to study, we do not even know that the number of such configurations asymptotically grows as for some . Curiously, the bounds on the upper and lower limits we have so far give c ∈ [0.29, 0.37]. The lower bound comes from a specific construction with so-called *layered permutations*, where explicit product formulas give the count. The upper bound follows by embedding noncrossing pipe configurations into the so-called “reduced bumpless pipe dreams”, a particular subset of the Alternating Sign Matrices (known in statistical mechanics as square ice with domain-wall boundary conditions and uniform weights; it is a particular case of the Six-Vertex model).

In our quest to resolve Stanley’s problem, we explored the generalization of Schubert polynomials, the Grothendieck polynomials , which capture the -theory of the flag variety. They can also be studied as partition functions (generating functions) of the pipe dreams described above, except that each double and further intersection is “penalized” by the factors . When = 0, we recover the Schubert polynomials; when = 1, the model corresponds to pipe dreams that bounce off each other after the first intersection. For which permutations is maximal? What is the asymptotic behavior of the number of -weighted pipe dreams? When = 1, finding the asymptotically maximal value is easy by counting the number of tilings (pipe dreams) without constraints. There are such tilings (two choices of each of the tiles) distributed among permutations. Thus, the maximal value is still around . But what is the maximal permutation? What does a typical permutation look like? To answer the second question, we realized that the trajectories (pipes) are in bijection with histories of hopping particles in the *Totally Asymmetric Simple Exclusion Process (TASEP)*, initially developed in 1968 to understand RNA translation:

Analyzing the behavior of hopping particles utilizes Integrable Probability. That is, we express certain bservables of the particle system via *Schur functions* (a central object in Algebraic Combinatorics), then turn to analysis and probability to extract asymptotics of the observables via contour integrals. This produces a very interesting permuton (a continuous version of a permutation matrix) in the shape of an… ice cream cone with one scoop. See Figure 1 **(b)**.

So what about layered permutations, which give large Schubert polynomials? In the Grothendieck
model, they are also close to maximal (see Figure 1 **(c)**). In this case, there are no product
formulas, so how do we compute the values and maximize them? The Grothendieck model can
also be related to the Six-Vertex model. For = 1, it corresponds to the 2-enumeration of
Alternating Sign Matrices, which are in 2-1 correspondence with domino tilings of the Aztec
diamond. A layered permutation has its last part reversed, making it a frozen region of the
2-ASM/Aztec diamond. The existence of the frozen boundary of the Aztec diamond was one of
the early feats at the intersection of combinatorics with statistical mechanics from 1996. So when
our layered permutation has the last layer close to the tangent to the arctic circle, the number of
configurations is still asymptotically maximal, that is, . See Figure 1 **(d)**.

The post Grothendieck Shenanigans: Permutons From Pipe Dreams via Integrable Probability first appeared on IPAM.

]]>The post White Paper: “Geometry, Statistical Mechanics, and Integrability” first appeared on IPAM.

]]>In the last 20-30 years, probability theory and statistical mechanics have been revitalized with the introduction of various tools, notably conformal geometry and discrete analyticity, as well as algebraic geometry and integrable systems.

Many familiar statistical mechanics models, such as the Ising model, dimer model, spanning tree model, and their cousins, have an intrinsic underlying geometric structure. For example, discrete analytic geometry was used by Kenyon, Lawler, Schramm, Werner, Smirnov, and Chelkak in their proofs of conformal invariance of scaling limits. Other work on the dimer model has led to connections with hyperbolic geometry, Lorentzian geometry, and symplectic geometry.

Recent connections between classical and discrete geometric structures on surfaces and combinatorial models such as the dimer and six-vertex models have revealed a significant connection to integrable systems and discrete geometry.

There are well-known connections between statistical mechanics models, algebraic combinatorics and representation theory: they have a common interest in Young tableaux, Gelfand–Tsetlin patterns, Knutson–Tao puzzles, and Littlewood–Richardson coefficients and their generalizations, for example. The Bethe Ansatz and Yang–Baxter equations were developed for the six-vertex model but are now fundamental tools in combinatorial representation theory, also giving explicit connections with integrability.

The application of conformal field theory (CFT) to statistical mechanics has been another pivotal area of research. CFT has proven to be a powerful tool for describing the scaling limits of critical lattice models. Schramm Loewner Evolution (SLE) and its variants were instrumental in understanding conformal invariance and scaling limits. Extending these methods to higher-rank models remains a complex and open question.

The program brought together many researchers from this somewhat disparate realm of ideas, united by the underlying themes of geometry and statistical mechanics. Activities included introductory tutorials, four workshops, and eight working groups. The groups studied recent literature and open problems in areas such as adapted geometric embeddings, vertex models, and asymptotic algebraic combinatorics.

Multiple directions for future research emerged from the program. Higher rank versions of statistical mechanics models have mysterious and unexplored connections with more sophisticated CFTs including W-algebras, as well as deeper connections with representation theory. Recent progress connecting the six-vertex model with symmetric polynomials gives a new direction of exploration towards a better understanding of asymptotic behavior of structure constants and other algebraic quantities.

The progress connecting geometries to free fermionic models leads us to analogous questions for the six-vertex model and beyond: is there an appropriate discrete geometry underlying any critical statistical mechanics model? There is clearly much to be explored!

The post White Paper: “Geometry, Statistical Mechanics, and Integrability” first appeared on IPAM.

]]>The post In memoriam: Jim Simons first appeared on IPAM.

]]>From 1968 to 1978 he was chair of the math department at Stony Brook University in New York, where his mathematical breakthroughs are now instrumental to fields such as string theory, topology and condensed matter physics.

In 1978, Jim founded the hedge fund Monemetrics, which would eventually become Renaissance Technologies. The hedge fund pioneered quantitative trading and became one of the most profitable investment firms in history. Jim then focused on making a difference in the world through the Simons Foundation, Simons Foundation International, Math for America, and other philanthropic efforts. IPAM has received generous support from the Simons Foundation over the past decade, providing funds to advance the careers of many graduate students and postdocs in the mathematical sciences.

For more about Jim and his legacy, please visit the Simons Foundation website:

- Simons Foundation Co-Founder, Mathematician and Investor Jim Simons Dies at 86 (Simons Foundation)
- Remembering the Life and Careers of Jim Simons (Simons Foundation)
- Science Lives: Jim Simons on His Career in Mathematics (Simons Foundation)

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]]>The post Spotlight on Two Newly Elected NAS Members first appeared on IPAM.

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Read the full list of newly elected NAS members here.

The post Spotlight on Two Newly Elected NAS Members first appeared on IPAM.

]]>The post The Passing of Keith Julien first appeared on IPAM.

]]>More about Keith’s passing can be found here.

The post The Passing of Keith Julien first appeared on IPAM.

]]>The post UCLA Newsroom Profile on Professor Gangbo first appeared on IPAM.

]]>The full article can be found here.

The post UCLA Newsroom Profile on Professor Gangbo first appeared on IPAM.

]]>The post White Paper: “Mathematical Challenges and Opportunities for Autonomous Vehicles” first appeared on IPAM.

]]>Autonomous vehicle (AV) research and development has achieved a similar status in terms of resources invested, societal excitement, and media coverage as space travel and exploration. At the same time, AV research is not rocket science; it is more complicated: while in itself, an AV is no more complex than a spacecraft, it must reliably interact and communicate with many other agents, particularly humans both inside and outside of the vehicle, much of it in a decentralized fashion. Hence, AVs, and their impact on us humans and our transportation systems, incur some of the most complicated science and engineering challenges that society will face in the near future. At the same time, there is some disconnect across the various research communities: professional product development is highly opaque, and public expectations and media communications have frequently been inaccurate or exaggerated.

**Operation and Performance of Automated Vehicles**

Cars in the future (even those designed to be autonomous) will likely build upon existing industry-wide vehicle architectures for data and control. Therefore, research in the area of vehicle automation must build on these industry standards and be compatible with accepted vehicle platforms. However, many research questions are still unanswered and there is room for improvement and innovation in developing reliable, safe, and efficient automated driving systems as outlined below.

- Modeling for autonomous driving
- Deep learning in autonomous driving
- Co-design of AVs individually and in systems
- Safety

**Human Aspects of Automated Driving**

For the foreseeable future, humans will continue to be involved in the operation of automated vehicles. Therefore, there is a need to understand how humans will interact with, and use, automated driving systems. Some aspects that will need to be considered include how human drivers will drive in the proximity of AVs, as well as how the human operator will (1) interact with the vehicle, and (2) share control of the driving tasks with the AV. These aspects of human interactions with automated driving systems will need to be understood to enable safe and efficient deployment in an automated future. The following subtopics were discussed in the program.

- Human interaction with automation
- AVs and land use/urban
- How to test AV safety with human agents

**Technical Systems Interactions for Automated Driving**

The adoption of autonomous vehicles will not simply mean that humans that drive their cars manually will immediately vanish from the road. Rather, increasing vehicle automation and connectivity will fundamentally change the traffic patterns on our roads and also affect how safe our roads are. Thus, one major concern is ensuring that Connected and Automated Vehicles (CAV) will improve safety and enhance the overall traffic flow.

**Understanding and predicting how connectivity and automation will influence traffic flow**Mixed traffic streams remain a significant area of interest in terms of understanding and predicting how connectivity and automation will influence traffic flow. The major research areas can be divided into (1) modeling and simulation of vehicle agents and (2) changes in design and operation of the transportation network.

**Leveraging autonomy and connectivity to improve traffic operations**

**Societal Impacts of Automated Driving**

While it is widely believed that roads with autonomous vehicles are likely to be the norm in a “reasonably near” future, the way we get to this future and how it evolves involves a number of tradeoffs and decisions. This discussion focuses on the need for the research, development and policy-making community to explicitly consider the societal impacts of a future with AVs. In particular, a key question is how we can quantitatively frame relevant discussions with a focus on the societal impact.

**Lessons from ride-hailing****Impact of AI**

In conclusion, autonomous vehicles are on their way to revolutionize how we think about transportation systems. However, before they are ubiquitous, a wide range of technical challenges must be addressed and resolved as more and higher automated vehicles are being deployed on our roadways. These problems range from control of individual vehicles to how they interact with surrounding vehicles and pedestrians to how they meet the needs of larger geographic regions. Good mathematical models serving as “what-if machines”, simulations, measurements, and guarantees of performance are also needed in order for society and decision makers to think clearly about often-competing goals and objectives.

Automation of transportation, combined with its surrounding research needs, is a fantastic area for interdisciplinary research that spans the whole pipeline from mathematical foundations, over academic areas like engineering, computer science, and also social sciences, to industry, public stakeholders, etc. The IPAM long program, with participants from many of these fields, has shown that a productive interplay of all these areas is possible, and existing collaboration of participants have demonstrated that great outcomes can result from these interactions. The authors of this white paper would like to stress these significant opportunities for cross-disciplinary research with high broader and societal impact, and also to the opportunities for cross-disciplinary programs and initiatives, to funding agencies, public stakeholders, and education and research institutions. We hope that we can all help shape a successful transition to a better, safer, more efficient, more fair, and more enjoyable, future of transportation.

The post White Paper: “Mathematical Challenges and Opportunities for Autonomous Vehicles” first appeared on IPAM.

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